DRIFT AND YIELD MECHANISM BASED SEISMIC DESIGN AND UPGRADING OF STEEL MOMENT FRAMES. Sutat Leelataviwat

Transcription

1 DRIFT AND YIELD MECHANISM BASED SEISMIC DESIGN AND UPGRADING OF STEEL MOMENT FRAMES by Sutat Leelataviwat A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Civil Engineering) in The University of Michigan 1998 Doctoral Committee: Professor Subhash C. Goel, Co-chair Assistant Professor Bozidar Stojadinovic, Co-chair Professor William J. Anderson Professor Antoine E. Naaman

2 Dedicated to my parents and my brothers; Santi, Surang, Sutee, and Surat Leelataviwat. ii

3 ACKNOWLEDGMENTS The author wishes to express his profound gratitude to Professor Subhash C. Goel, co-chairman of the doctoral committee, for providing guidance and care both personally and professionally throughout the course of this study at the University of Michigan. The author is deeply appreciated for countless hours that he spent mentoring the author, without which this dissertation could not have been completed. Appreciation is also extended to Professor Bozidar Stojadinovic, co-chairman of the doctoral committee, for his invaluable guidance throughout the course of this study. The author also wishes to express his sincere thanks to his doctoral committee members, Professor Antoine E. Naaman and Professor William J. Anderson for their helpful suggestions. The author is most indebted to his parents and his brothers for their love and encouragement throughout his study, or in fact, throughout his life. The author can not find any proper words to describe his appreciation. The author also acknowledges the Rackham predoctoral fellowship from the School of Graduate Studies at the University of Michigan for their financial support. This study was greatly facilitated by the generous help from many of the author s colleagues in the Department of Civil and Environmental Engineering, who over the years have become the author s close friends. The author would like to thank those friends, notably Dr. Kyoung-Hyeog Lee, Dr. Madhusudan Khuntia, and Arnon Wongkaew. The help from the technicians at the Structures Laboratory, Robert Spence and Robert Fischer, is also greatly appreciated. Last, but not least, the author would like to express special appreciation to Amornratana Charuratna, Chonawee Supatgiate, and Supana Saivongnual, for their sincere and wonderful friendship that makes his experience in Ann Arbor a memorable one. iii

4 TABLE OF CONTENTS DEDICATION...ii ACKNOWLEDGMENTS...iii LIST OF TABLES...vii LIST OF FIGURES...ix LIST OF APPENDICES... xv NOTATION...xvi CHAPTER 1. INTRODUCTION Background and Motivation Objectives and Organization of the Dissertation A REVIEW OF SEISMIC DESIGN OF STEEL MOMENT FRAMES Introduction Equivalent Lateral Static Force Procedure (UBC-1994) Design Base Shear Distribution of Lateral Forces Drift Requirements Beam and Column Strength Requirements for Controlling the Collapse Mode Equivalent Lateral Static Force Procedure (UBC-1997) Review of Related Research Experimantal Studies Analytical Studies The Study Building Nonlinear Analyses of the Study Building Methods of Analysis Analytical Modeling of the Study Building Nonlinear Static Pushover Analysis Nonlinear Dynamic Analysis Summary and Concluding Remarks iv

5 3. DRIFT AND YIELD MECHANISM BASED DESIGN OF MOMENT FRAMES Introduction Principle of Energy Conservation Input Energy in Multi-Degree of Freedom Systems Energy-Based Design Base Shear Design Energy Level Design Base Shear for Ultimate Response Design Base Shear for Serviceability Plastic Design of Moment Frames Design of Beams Design of Columns Parametric Study of the Proposed Design Procedure Variation in Number of Stories Variation in Design Target Drift Comparison between the Current and the Proposed Design Procedures Comparison of Seismic Response Comparison of Design Forces Performance-Based Plastic Design Summary and Concluding Remarks SEISMIC UPGRADING OF MOMENT FRAMES USING DUCTILE WEB OPENINGS Introduction Concept of Moment Frames with Web Openings Testing of Steel Beams with Openings Test Set-Up Instrumentation and Test Procedure Material Properties Specimen Specimen Specimen Specimen Specimen Analysis of Test Data Overstrength of the Diagonal Members Overstrength of the Chord Members Ultimate Shear Strength of the Openings Modeling of the Openings under Cyclic Loading Summary and Concluding Remarks v

6 5. SEISMIC DESIGN AND BEHAVIOR OF MOMENT FRAMES WITH DUCTILE WEB OPENINGS Introduction Proposed Design Approach Design of Chord Members Design of Diagonal Members Design of Vertical Member Design of Welds Required Strength of the Opening under Gravity Loads Detailing of the Opening The Study Building Nonlinear Analyses of the Study Building Inelastic Static Pushover Analysis Inelastic Time-History Dynamic Analysis Experimental Program Test Set-Up Design of the Girder and the Web Opening Instrumentation and Test Procedure Material Properties Test Results Evaluation of the Proposed Design Procedure and the Analytical Modeling Summary and Concluding Remarks SUMMARY AND CONCLUSIONS Summary Introduction Conventional Moment Frame Behavior Drift and Yield Mechanism Based Design Seismic Upgrading with Beam Web Openings Seismic Behavior of Upgraded Frames Concluding Remarks and Suggested Future Studies Drift and Yield Mechanism Based Design Moment Frames with Ductile Web Openings APPENDICES BIBLIOGRAPHY vi

7 LIST OF TABLES Table 2.1. Floor Masses of the Study Building UBC Design Lateral Forces for the Original frame Design Story Shears and Story Drifts Characteristics of Earthquake Records Design Parameters (2% Drift Limit) Design Lateral Forces (in kips) Design Parameters Design Lateral Forces (in kips) Performance Criteria Earthquake Design Levels Average Yield Stress of Key Members Shear Force Contributed by Chord Members Shear Force Contributed by Diagonal Members Comparison between Expected and Experimental Ultimate Shear Strength Design of Web Openings Member Sizes of the Modified Frame with Web Openings Comparison Between Design and Attained Overstrength Values Average Yield Stress of Key Members A1. Distribution of Beam Strength B1. Weights of the Equivalent One-Bay Frame B2. Design Lateral Forces B3. Calculation of Beam Proportioning Factors vii

8 B4. Minimum Weight Beam Sections B5. Lateral Forces at Ultimate Drift Level B6. Axial Forces in an Exterior Column (kips) viii

9 ix

10 LIST OF FIGURES Figure 1.1. Organization of the Dissertation Plan View of the Study Building A Typical Three-Bay Moment Frame in the N-S Direction Scaled Pseudo-Velocity Spectra of the Earthquakes Used in This Study (5% Damping) Four Selected Earthquakes Used in this Study The Original Frame and the Equivalent One-Bay Idealized Model Base Shear - Roof Drift Response from Pushover Analysis Sequence of Inelastic Activity from Pushover Analysis Distribution of Beam Moment in Columns at the Second Floor Joint Maximum Floor Displacements due to the Four Selected Earthquakes Maximum Story Drifts due to the Four Selected Earthquakes Location of Inelastic Activity and Rotational Ductility Demands due to the Four Selected Earthquakes Roof Displacement Time Histories under the Four Selected Earthquakes Distribution of Column Strength along the Height Maximum Column Moments Due to the Four Selected Earthquakes Typical Response of Structures Design Pseudo-Acceleration and Pseudo-Velocity Spectra (UBC-94) Equivalent One-Bay Frame at Mechanism State Drift and Yield Mechanism Based Design Base Shear Coefficients Expected Response of a Structure Designed to Satisfy Serviceability Frame with Global Mechanism x

11 3.7. Frame with Soft-Story Mechanism Free Body Diagram of the Column in the Equivalent One-Bay Frame Typical Story of the Study Frames Member Sizes of the 2-, 6-, and 10-Story Frame with 2% Target Drift Base Shear versus Roof Drift Response of the Study Frames Location of Inelastic Activity in the Three Frames at 3% Roof Drift Maximum Story Drifts of the 2-, 6-, and 10-Story Frames Distribution of Maximum Story Shears from Dynamic Analyses Three Six-Story Frames with 1.5%, 2.5%, and 3% Target Drifts Base Shear versus Roof Drift Response of the Study Frames Location of Inelastic Activity in the Three Study Frames at 3% Roof Drift Maximum Story Drifts under the Four Selected Earthquakes Comparison between Design Target and Attained Maximum Drifts Distribution of Story Shears from Dynamic Analyses Member Sizes of the Original Frame and the Redesigned Frame Base Shear versus Roof Drift of the Original and the Redesigned Frames Sequences of Inelastic Activity under Increasing Lateral Forces Maximum Story Drifts of the Original and the Redesigned Frames Location of Inelastic Activity under the Four Selected Earthquakes Comparison of Design Base Shear Coefficients Recommended Performance Objectives, Adapted from [SEAOC 1995] A Possible Quantification of the Performance-Based Design Space Design Base Shear for Different Performance Objectives xi

12 4.1. Yield Mechanism of Special Truss Moment Frame and Moment Frame with Girder Web Opening Schematic Diagram of a Typical Test Set-Up Typical Test Set-Up Test Specimen Loading History 1 of Specimen Loading History 2 of Specimen Specimen 1 before Removal of Diagonal Members Specimen 1 after Removal of Diagonal Members Hysteretic Loops of Specimen 1 with Diagonal Members Hysteretic Loops of Specimen 1 without Diagonal Members Yielding and Buckling in Specimen 1 with Diagonal Members Yielding in Specimen 1 without Diagonal Members Cracking of the Chord Member Test Specimen Loading History for Specimen Hysteretic Loops of Specimen Yielding and Buckling in Specimen Cracking in the Chord Member of Specimen Test Specimen The Opening in Specimen Loading History for Specimen Hysteretic Loops of Specimen Deformation of the Test Specimen 3 (Positive Direction) Deformation of the Test Specimen 3 (Negative Direction) Local Buckling of Chord Members xii

13 4.26. Test Specimen A Close-Up View of Specimen Loading History of Specimen Hysteretic Loops of Specimen Local Buckling and Fracture of Specimen Test Specimen Close-Up View of Specimen Loading History of Specimen Hysteretic Loops of Specimen Deformation of the Test Specimen (Negative Direction) Deformation of the Test Specimen (Positive Direction) Comparison of Strain Hardening Values Comparison of Yield Stresses Equilibrium of Internal Forces in the Opening Axial Hysteretic Model for Diagonal Members [ Jain et al. 1978] Analytical Modeling of Specimen Equilibrium of Forces at the Middle Joint The Modified Frame with Beam Web Openings The Modified Frame and its Analytical Model Base Shear Roof Drift Response of the Original and the Modified Frames(Based on Expected Yield Strength) Sequences of Inelastic Activity of the Modified Frames Maximum Floor Displacements of the Modified and the Original Frames Maximum Interstory Drifts of the Modified and the Original Frames xiii

14 5.8. Location of Inelastic Activity in the Modified Frame under the Four Selected Records Maximum Overstrength Values Under the Four Selected Records Overall View of the Test Set-Up Close-Up View of the Test Specimen Lateral Bracing of the Test Specimen Beam-to-Column Connection of the Test Specimen Dimensions of the Test Specimen Dimensions of the Web Opening in the Test Specimen Close-Up View of the Special Opening Diagonal-to-Chord Junction Vertical-to-Chord Junction First Loading History Second Loading History Hysteretic Loops from the First Loading History Hysteretic Loops from the Second Loading History Deformation of the Test Frame (Positive Displacement) Deformation of the Test Frame (Negative Displacement) Inelastic Activity in the Opening Yielding of the Chord and the Diagonal Members Fracture in the Chord Member Analytical Model of the Test Specimen Analytical Simulation of the Experiment with the First Loading History Analytical Simulation of the Experiment with the Second Loading History xiv

15 A1. Typical Story of the Six-Story Frame Used to Calibrate β i A2. Four Six-Story Frames Used to Calibrate β i A3. Distribution of Maximum Story Shears under the Four Selected Records A4. Variation of Error Function X A5. Comparison between β i = ( Vi / Vn ) and Relative Shear Distributions from Dynamic Analyses B1. Drift and Yield Mechanism Based Design Procedure Flowchart B2. Internal Forces in the Roof Beam B3. Distribution of Moment in an Exterior Column (Units in kips and ft.) B4. Member Sizes of the Redesigned Frame xv

16 LIST OF APPENDICES Appendix A. CALIBRATION OF BEAM PROPORTIONING FACTOR B. DESIGN EXAMPLE C. ABSTRACT xvi

17 NOTATION a Normalized design pseudo-acceleration (with g ) a e a o Base shear coefficient for serviceability (elastic) level Mass-proportioning damping coefficient a g (τ ) Ground acceleration at time τ a g A A e b b f Ground acceleration Design pseudo-acceleration Design pseudo-acceleration for serviceability Numerical factor for beam proportioning factor Flange width of beam B 1, B 2 Amplification Factors used to determining M ux for combined bending and axial force design c Viscous damping coefficient [C] Damping matrix C C, C a d b d c E E E e v Seismic coefficient (UBC-94) Seismic coefficients (UBC-97) Depth of beam Depth of column Input energy form earthquake Young s Modulus Elastic vibrational energy, the sum of kinetic energy and elastic strain energy E es E p E k Elastic strain energy Cumulative hysteretic energy Kinetic energy xvii

18 E d f a f s F ab F ac F cr F i F iu F t Damping energy Axial compressive stress in column Restoring force Actual yield stress of beam Actual yield stress of column Critical stress Equivalent inertia force applied at level i of the structure Equivalent inertia force at level i at ultimate response Concentrated force applied at the top floor of the structure F yb Nominal yield stress of beam F yc g G G a, h h h 1 G b h i, h j Nominal yield stress of column Acceleration due to gravity Shear Modulus Ratio of column stiffness to beam stiffness for column design Height Total height of structure Height of the first story Height of floor level i (or level j ) of the structure above the ground h s Story height H Horizontal force in the story used to calculate B 2 I I c I e k k x, k y l Importance factor (UBC-94, UBC-97) Moment of inertia of chord member Earthquake intensity Effective length factor Effective length factor for buckling about x-axis (or y-axis) Unbraced length of column xviii

19 l x L L 0 m Unbraced length of diagonal member Span Length Length of special segment, Length of opening Mass of single degree of freedom system [M ] Mass matrix M Total mass of the system M c (h) Moment in the column at a height h above the ground M ch Plastic moment of chord member M lt M n Required flexural strength in member due to lateral translation Nominal Flexural Strength M nt Required flexural strength in member assuming no lateral translation M p Plastic moment M pb i, M pb j Plastic moment of beam at level i (or level j ) M pb r Reference plastic moment of beams M pc Plastic moment of columns at the base of the equivalent one-bay frame M pz Beam moment when panel zone shear strength reaches the value specified in the UBC M y Yield moment of beam M ux Required flexural strength for x-axis bending n N a N v p Number of stories Near source acceleration factor (UBC-97) Near source velocity factor (UBC-97) Fraction of cumulative plastic energy dissipated at peak response P c (h) Total axial force in column at a height h above the ground xix

20 P cg (h) Axial force in column due to gravity loads at a height h above the ground P n P u P v P xy P xc r x r y R R b R w S S v t t f Nominal compressive strength Required axial strength Axial force in the vertical member Tensile yield force of the diagonal member Buckling force of the diagonal member Radius of gyration about x-axis Radius of gyration about y-axis Structural system coefficient (UBC-97) Reaction force from cross beam Response modification factor (UBC-94) Site coefficient (UBC-94) Pseudo-velocity Time Flange thickness of beam t wc Web thickness of column t w T V V c V ei Web thickness of beam Fundamental period of the structure Design base shear Ultimate shear provided by the chord members Maximum earthquake-induced story shear in story level i V eij Maximum earthquake-induced story shear in story level i in case j V en Maximum earthquake-induced story shear in the top story (level n ) xx

21 V enj Maximum earthquake-induced story shears in the top story (level n ) in case j V i V n Static story shear at level i due to the equivalent inertia forces Static story shear at the top story (level n ) due to the equivalent inertia forces V o V p V u V x Ultimate shear strength of opening Shear in the panel zone Base shear at ultimate Ultimate shear provided by the diagonal members w, w Weight of the structure at level i (or level j ) i j W x x& & x& X Z Z c Z b α β i δ δ e δ i Total weight of the structure Displacement in the x direction Velocity in the x direction Acceleration in the x direction Error function use to calibrate beam proportioning factor Seismic zone factor (UBC-94, UBC-97) Plastic modulus of column Plastic modulus of beam Design base shear parameter Beam proportioning factor at level i Story displacement Serviceability drift level Step function for calculation of column moment and axial force δ pl Inelastic story drift δ p δ y m Story displacement due to panel zone deformation Yield story drift Expected maximum inelastic drift (UBC-97) xxi

22 s φ φ c η Elastic drift due to design level forces (UBC-97) Resistance factor for bending Resistance factor for axial compression Strain-hardening factor µ Rotational ductility θ p,max Maximum plastic rotation θ p θ x θ y τ ω, ξ c ξ i ξ s ξ x ζ ω n Plastic rotation, Inelastic drift Angle between the diagonal and the chord members Yield rotation Time instant Natural circular frequency Overstrength factor for the chord members Overstrength of the beam at level i Overstrength due to strain hardening Overstrength factor for the diagonal members Damping as a fraction of the critical value xxii

23 CHAPTER 1 INTRODUCTION 1.1 BACKGROUND AND MOTIVATION Moment-resisting steel frames have long been regarded as one of the best structural systems to resist seismic forces. The load-carrying mechanism of these frames depends on the capability of their moment-resisting joints to transfer the applied forces between members. Therefore, the strength and ductility of these joints play a crucial role in the seismic response of these frames. Unfortunately, an unprecedented number of beam-to-column connections and other failures were reported in the aftermath of the 1994 Northridge and the 1995 Kobe earthquakes [SAC 1995c, Nakashima et al. 1998]. These incidents clearly show that our knowledge about seismic behavior of momentresisting frames at present is not adequate. It creates a profound impact that can be felt by everyone involved in the design and construction of moment-resisting frames. After three years of intensive research, the engineering community remains shrouded in doubts. It is not apparent how safe the existing moment-resisting frames are, how existing momentresisting frames should be retrofitted, or how new moment-resisting frames should be designed. At a glance, the design of moment-resisting frames involves only fundamentals of structural analysis and simplified structural dynamics. After a closer look, however, the design of moment-resisting frames requires a clear understanding of earthquake-structure interaction and inelastic distribution of stresses, both at the member and at the system levels. 1

24 2 At the member level, recent studies at the University of Michigan [Goel et al. 1997, Lee et al. 1998] have shown that the stress distribution at beam-to-column connections in moment-resisting frames defies the classical beam theory. The design of these connections requires a clear understanding of the stress paths and the boundary effects. At the system level, studies [Goel and Leelataviwat 1998] have shown that moment resisting frames designed by the elastic method using equivalent static forces may undergo inelastic deformations in a rather uncontrolled manner, resulting in uneven and widespread formation of plastic hinges. Thus, combined lack of ductility of the connections and the use of unrealistic design approaches could hold a major key in explaining the recently observed poor performance of steel moment frames. The research work presented herein focuses on answering two imminent questions: how a new moment frame should be designed and how an existing moment frame could be retrofitted. The behavior of a moment-resisting frame designed by the conventional method was studied using extensive nonlinear static and nonlinear dynamic analyses. Guided by the performance of this conventionally designed frame, a new design concept was proposed based on the principle of energy conservation and theory of plasticity. This study was then extended to include seismic upgrading of existing steel moment frames for future earthquakes. 1.2 OBJECTIVES AND ORGANIZATION OF THE DISSERTATION The objectives of this study were: 1) To investigate the behavior of momentresisting frames designed by conventional methods; 2) To propose a new design procedure that addresses explicitly the ultimate drift and the yield mechanism of moment frames; 3) To propose a new upgrading scheme for existing moment frames. The organization of the dissertation can be best summarized by the chart presented in Figure 1.1. The results of this study are presented in the following five chapters and two appendices:

25 3 DRIFT AND YIELD MECHANISM BASED SEISMIC DESIGN OF STEEL MOMENT FRAMES CHAPTER 1:INTRODUCTION CHAPTER 2: A REVIEW OF SEISMIC DESIGN OF STEEL MOMENT FRAMES NEW MOMENT FRAMES EXISTING MOMENT FRAMES CHAPTER 3: DRIFT AND YIELD MECHANISM BASED DESIGN OF MOMENT FRAMES CHAPTER 4: SEISMIC UPGRADING OF MOMENT FRAMES USING DUCTILE WEB OPENINGS CHAPTER 5: SEISMIC DESIGN AND BEHAVIOR OF MOMENT FRAMES WITH DUCTILE WEB OPENINGS CHAPTER 6: SUMMARY AND CONCLUDING REMARKS Figure 1.1. Organization of the Dissertation.

26 4 Chapter 2 focuses on the behavior of conventionally designed moment frames. This chapter presents a review of the underlying concepts behind current design procedures for steel moment frames based on the Uniform Building Code [UBC 1994, UBC 1997]. The implications of the current design philosophy for steel moment frames are discussed. An existing moment frame structure designed by the conventional design method was taken as a study case. Nonlinear static and nonlinear dynamic analyses were used to identify potential problems. The results of these analyses are presented and discussed. The findings in Chapter 2 led to the development of a new design procedure presented in Chapter 3. Chapter 3 presents a new drift and yield mechanism based (DYMB) seismic design procedure for steel moment frames. In this procedure, the structure is designed at the ultimate level. The ultimate design base shear for plastic analysis is derived by using the input energy from the design pseudo-velocity spectrum, a pre-selected yield mechanism, and a target drift. The procedure also includes a step to determine the design forces in order to meet specified target drifts in the elastic stage under moderate ground motions. The results of nonlinear static and nonlinear dynamic analyses of an example steel moment frame designed by the proposed method are presented and discussed. The implications of the new design procedure for future generation of seismic design codes are also discussed. In Chapter 4, a possible scheme to modify seismic behavior of existing moment resisting frames to have a ductile yield mechanism is proposed. This upgrading scheme consists of creating ductile rectangular openings reinforced with diagonal members in the beam web near the middle of the span. These openings are designed such that, under a severe ground motion, inelastic activity will be confined only to the yielding and buckling of the diagonal members and the plastic hinging of the chord members of the opening, while other members in the frame will remain elastic. This chapter presents the experimental and analytical development of the ductile web opening system. Results

27 5 of reduced-scale experiments are presented. Based on the results of these experiments, behavior of key members is discussed. Guided by the experimental results in Chapter 4, a design procedure for seismic upgrading of steel moment frames is proposed in Chapter 5. The moment frame structure in Chapter 2 was used again as an example structure. It was modified using the proposed upgrading procedure. The response of the upgraded frame under severe ground motions is presented and discussed. Finally, results from a full-scale test of a one-story subassemblage are shown. These results were used to verify the proposed modification procedure and to verify the results from computer analyses. Chapter 6, the final chapter, presents the summary and the concluding remarks of this study. Suggestions for future studies are also presented. Appendix A describes the calibration of beam proportioning factor, which is an important factor used in the drift and yield mechanism based design presented in Chapter 3. Appendix B presents a design flowchart that summarizes the drift and yield mechanism based design procedure. This appendix also provides a detailed design example of a five-story moment frame using the proposed design procedure. Appendix C contains the abstract of this dissertation.

28 CHAPTER 2 A REVIEW OF SEISMIC DESIGN OF STEEL MOMENT FRAMES 2.1 INTRODUCTION For the last three decades, extensive experimental research and post-earthquake investigations have been carried out to better understand the response of multistory buildings subjected to earthquake excitations. Many analytical and numerical procedures as well as nonlinear finite element analysis codes have been developed to more accurately estimate the response of structures. Despite all the advances in the field of earthquake engineering, building codes and design provisions for earthquakes in the United States and many other countries remain relatively unchanged. For example, the Uniform Building Code [UBC 1994, UBC 1997], although it has gone through many revisions, is still based on the 1959 recommendations of the Structural Engineers Association of California [Seismology Committee 1959]. Similarly, the National Earthquake Hazards Reduction Program or NEHRP provisions [NEHRP 1991] are based on the 1978 ATC 3-06 [ATC 1978] provisions. The primary design procedure for regular structures specified in most building codes is still based on the Equivalent Lateral Static Force concept. Equivalent design lateral forces are derived from expected maximum seismic forces assuming elastic behavior, modified by suitable response reduction factors that depend mainly on the ductility of the structural systems. The design work strives for providing adequate strength and limiting lateral drifts to permissible values at the design (reduced or working) level. The underlying philosophy is that the strength and drift criteria at the 6

29 7 design level assure that structures remain elastic and serviceable during small and frequent earthquakes, and that, structural safety during a severe earthquake depends on the capability of structures to dissipate the input energy in the inelastic range. This chapter presents a review of underlying concepts behind current design procedures for steel moment frames based on the Uniform Building Code. The implications of current design philosophy for steel moment frames are discussed. An existing moment frame structure designed by the conventional design method was taken as a study case to investigate potential problems. This frame was subjected to an in-depth study including nonlinear static and nonlinear dynamic analyses. The response of the study building due to static forces as well as selected earthquakes is presented and discussed. 2.2 EQUIVALENT LATERAL STATIC FORCE PROCEDURE (UBC-1994) Design Base Shear The minimum design base shear, V, for allowable stress design is given by (UBC-94 Equation 28-1): ZICW V = (2.1) R w where W is the seismic weight, Z is the seismic zone factor, I is the importance factor, C is the elastic seismic coefficient, and R w is the response modification factor. The seismic weight is the weight of the building mass that induces inertia forces which, according to the UBC, includes the total dead weight. For some structures, the seismic weight must include 25% of the live load and snow load if it is greater than 30 lb./sqft. The factor Z is the seismic zone factor representing the peak ground acceleration (PGA) of the design level earthquake at the building site. The peak ground acceleration depends on the seismic zonation, originally adopted in ATC 3-06 [ATC 1978]. In high seismic

30 8 region, the zone factor, Z, has a value of 0.4. The factor I represents the relative importance of the facility. I has a value of 1.0 for standard occupancy structures. For essential or hazardous facilities, I is equal to In essence, the UBC attempts to increase the level of safety by increasing the magnitude of design forces, thereby increasing strength and limiting the deformation of structure during earthquakes. The factor C is the elastic seismic coefficient defined as: 1.25S C = (2.2) 2 / 3 T where S is the site coefficient and T is the fundamental period of vibration of the structure. Factor C need not exceed 2.75 but the ratio of C / R must be greater than The site factor, S, accounts for the ground motion amplification due to local soil conditions. The value of S ranges between 1.0 and 2.0 depending on the soil profile. The fundamental period of the structure, T, can be estimated using an empirical formula. For steel moment frames, the fundamental period in seconds is approximated by: w 3 / 4 T = 0.035h (2.3) where h is the total height of the structure in feet. The response modification factor, R w, accounts for the ductility and energy dissipation capacity of the structural system. The underlying basis of the response modification factor is that ductile structures can dissipate a significant amount of energy by means of inelastic material behavior. Hence, they can be designed to have a strength smaller than required to remain elastic and to dissipate part of the input energy by using inelastic material behavior. Ductile systems such as steel moment frames are assigned larger values of Rw than non-ductile system. In UBC, R w is taken as 12 for special steel moment resisting frames and 6 for ordinary steel moment frames. The values of R w are based on experience and performance of moment frames in past earthquakes. It should be

31 9 noted that many studies have questioned the suggested lower values of R w [Bertero 1986, Riddell et al. 1989]. R w values specified in the code and have Distribution of Lateral Forces where The distribution of lateral forces over the height of the structure is given by: n V = F t + F i (2.4) i= 1 F i is the equivalent lateral force applied at level i, F t is an additional concentrated force applied at the top floor of the structure, and n is the number of stories. The force F t increases story shears in the upper stories to account for the contributions from higher modes of vibration. F t is calculated as: F t = 0.07TV if T > 0. 7 sec. (2.5) F t = 0 if T 0. 7 sec. (2.6) The force applied at each level, F i, is given by: where wihi F i = (V Ft ) n (2.7) w h j= 1 j j w i is the weight of the structure at level i and h i is the height of level i. For a structure with equal story mass and story height, lateral forces increase linearly from the base to the top floor, corresponding to an assumed linear shape of the first mode of vibration. The effect of torsion must also be included in the design. The torsional design moment at a given story can be found from the moment resulting form eccentricities between applied lateral forces at levels above that story and the load-resisting elements in that story plus additional moment due to accidental torsion. Accidental torsional moment is calculated by assuming an additional eccentricity of 5% of the building dimension.

32 Drift Requirements The UBC requires that structures must have sufficient lateral stiffness. The UBC imposes drift limit in an attempt to keep the story drifts within an acceptable limit under both small and frequent earthquakes as well as severe ones. Under design-level forces (Equation 2.4), for a structure with a fundamental period less than 0.70 second, the story drift is limited to the smaller of 0.04/ R or For a structure with a fundamental w period greater than 0.7 second, the story drift is limited to the smaller of 0.03/ R or w By using this working level drift limit, the maximum inelastic story drift under a design level earthquake expected by UBC should be in the order of 2-2.5% [Roeder et al. 1993]. It should be noted that the drift limit for special steel moment frames, with R w of 12, is very stringent. In most cases, this drift limit dictates the member sizes Beam and Column Strength Requirements for Controlling the Collapse Mode A widely accepted design philosophy for moment frames is that columns should be relatively stronger than beams. In other words, the inelastic activity should be confined to beams only. This type of frame is generally known as a strong column - weak beam frame (SCWB). The UBC imposes a condition that at any beam to column joint, the following relationships be satisfied: where Z ( Fyc f a ) / ZbF > yb 1.0 (2.8) c Z ( Fyc f a ) / 1.25 M pz > c 1. 0 (2.9) Z c is the plastic modulus of column, Z b is the plastic modulus of beam, f a is the axial compressive stress in the column, M is the beam moment when the connection panel zone shear strength reaches the value specified in the code, of beam, and F yc is the yield strength of the column. pz F yb is the yield strength

33 11 It has been shown that, although these strength requirements are necessary, they are not sufficient to prevent flexural yielding in columns during a major earthquake because these rules are not derived from a global limit state but rather from a localized one [Lee 1996]. Very often, they do not prevent the occurrence of an undesirable collapse mechanism. To date, no explicit checks of column yielding at ultimate load condition are required by code. The UBC code also provides exceptions when Equations 2.8 and 2.9 do not have to be satisfied, which essentially means that a weak column-strong beam behavior is permitted. This can be done if the axial force in the column is less than 40% of the column yield force, if the shear resistance of the story is more than 50% greater than that of the story above, and if the column is not part of the lateral load resisting system. 2.3 EQUIVALENT LATERAL STATIC FORCE PROCEDURE (UBC-1997) Some significant changes have been introduced in 1997 version of the UBC. The major changes include: 1) The change from a working stress-based design to a strength-based design. 2) The introduction of new design coefficients, notably the near source factors and the reliability/redundancy factor. In UBC-97, both working stress design and strength design are allowed. The forces prescribed in UBC-97 are for strength design, and a factor of 1.4 is used to reduce the magnitude of the forces if working stress design is to be used. The redundancy factor accounts for the redundancy of the lateral load resisting system. The lower the degree of redundancy, the higher the prescribed earthquake forces. The near source factors are a result of recent findings that ground motions at sites close to a fault can be significantly amplified. The near source factor is directly related to the distance of the structure to the nearest fault. The closer to the fault, the higher the prescribed forces.

34 12 The deign base shear formula in UBC-97 is similar to that UBC-94 except for several new coefficients and can be expressed as: where CvIW 2.5CaIW V = (2.10) RT R C v is the seismic coefficient (ranges between 0.32N v and 0.96N v in seismic zone 4) as per Table 16-R of UBC-97, C a is the seismic coefficient (ranges between and 0.32N a 0.44N a in seismic zone 4) as per Table 16-Q of UBC-97, R is the structural system coefficient, N a is the near source acceleration factor (ranges between 1.0 and 1.5) as per Table 16-S of UBC-97, and N v is the near source velocity factor (ranges between 1.0 and 2.0) as per Table 16-T of UBC-97. The base shear must not be less than: V = 0.11C IW (2.11) a In addition, for seismic zone 4, the total base shear must not be less than: 0.8ZN viw V = (2.12) R The structural system coefficient, R, is similar to the response reduction factor, R w, in the UBC-94 but the value has been reduced to account for the change to strength based design. In UBC-97, the value of R for special moment resisting frames is 8.5, while for ordinary moment resisting frames, R is 4.5. The soil profile factors have also been revised. In UBC-97, there are six categories of soil profiles depending on the shear wave velocity as opposed to four categories in UBC-94. The design base shear of the UBC-97 depends considerably on the near source factors, the redundancy factor, and the soil profile factor. Generally, the design base shear from the UBC-97 is larger than that computed from UBC-94, when compared at the same strength-based design level or working stress-based design level.

35 13 The drift limits have also been changed to reflect the strength-based design. The expected maximum inelastic drift limit is computed from the design level drift by using an empirical formula: where m is the expected maximum inelastic drift and level forces. The drift limit for = 0.7R (2.13) m s s is the elastic drift due design m is given as for structures having a fundamental period less than 0.7 second and 0.02 for structures with a fundamental period greater than 0.7 second. If the drift limits for a special steel moment resisting frame are backcalculated to allowable stress design level as prescribed in the UBC-94, it becomes apparent that drift limits in UBC-94 and UBC-97 are very similar. However, the drift limits from UBC-97 are given in a more rational form and can be compared directly to results of a time history analysis. 2.4 REVIEW OF RELATED RESEARCH Many analytical and experimental studies have been carried out in the past to study the implications of various design codes on the seismic behavior of steel moment frames. Although most of the studies in the literature focus on the moment frames designed by earlier versions of building codes, they can serve to evaluate moment frames designed by newer codes since the underlying concepts of most building codes have not been substantially changed. The focus of these studies ranges from cyclic tests of building components and dynamic tests of full-scale and reduced-scale models to analytical investigations of various aspects of seismic behavior of steel moment frames. The major findings are summarized in the following sections Experimental Studies Modern codes allow the use of both strong column-weak beam (SCWB) and weak column-strong beam (WCSB) framing systems, as mentioned in Section 2.2.4, despite the

36 14 results of many cyclic tests of frame components that clearly show the superiority of SCWB system. Tests of beam-column assemblages representing WCSB frames [Schneider et al. 1993, Popov et al.1975] show that hysteretic behavior depends strongly on the magnitude of axial loads in columns. During tests, columns with high axial loads exhibited hysteretic behavior with rapid deterioration. Popov et al. [Popov et al. 1975] suggested that the WCSB frames can be adequately used if axial forces are kept below 50% of the yield force. However, this suggestion is based on an assumption that the ductility demands of SCWB and WCSB frames are similar, which is usually not the case as will be discussed further. One shaking table test of a small-scale three-story WCSB frame [Takanashi and Ohi 1984] has been reported and the frame collapsed during the test. Although, this frame was designed according to Japanese standards and might not directly reflect moment frames designed by U.S standards, the result strongly suggests that WCSB frames should be avoided Analytical Studies Many analytical studies on the difference in seismic behavior between weak column-strong beam frames (WCSB) and strong column-weak beam frames (SCWB) have been carried out in the past. Roeder et.al. [Roeder et al. 1993] studied the seismic response of 3-, 8-, and 20-story moment frames designed with these two different philosophies according to the UBC-88 standards, which are essentially identical to the UBC-94 requirements. The results of inelastic time history analyses of these frames under three earthquake records, the 1940 El Centro, the 1971 Pacoma Dam, and the 1979 Imperial Valley College, were reported. The results of these analyses indicated that the SCWB frames are superior to WCSB in terms of both the global response and the local damage capacity. The major findings were: 1) WCSB frames produced concentration of inelastic activity in a limited number of elements, especially in columns, whereas SCWB frames distributed the inelastic

37 15 activity over many more elements. The local ductility demand and element damage potential were much higher in WCSB frames. 2) The maximum interstory drifts of WCSB frames were very sensitive to the increase in earthquake intensity. An increase in story drifts as much as 200% was reported when the intensity of El Centro was increased by 50%, while only about 20% increase was observed for the story drifts in SCWB frames. 3) Some plastic hinges formed in columns even when the frame was designed according to SCWB requirements. The effect of plastic hinges in columns of SCWB on the seismic behavior was not obvious in that study. 4) Both SCWB and WCSB frames experienced inelastic story drifts larger than the 2% expected by the code, especially for frames with short periods. This suggests that the design base forces of the UBC may not be large enough. Osman et al. [Osman et al. 1995] studied the response of frames designed according to Canadian standard and reported similar results about the seismic behavior of WCSB and SCWB frames. The damage was found to be mostly concentrated in the first story for WCSB frame, with highest plastic rotation at the base of the frame. The SCWB frame had a better damage distribution over the height but the damage still mainly localized in the first floor especially at the base of the frame. It is important to note that, although some plastic hinges were observed in the columns of the SCWB frames in both studies, no particular attention was paid to investigate further. It was probably because the intensities of earthquakes used in those studies were not so strong, therefore, the consequences of yielding in columns were not obvious. More recent studies [Lee 1996, Park and Pauley 1975] have shown that the requirements for SCWB in the UBC may not be adequate in preventing formation of a soft story. The conditions set fourth by the building codes are very localized. They do not recognize the actual distribution of plastic beam moments in columns. In some cases, the elastic distribution may underestimate the demands by as much as 100%.

38 16 Lee [Lee 1996] studied the response of a six-story steel frame designed according to ATC 3-06 under increasing static forces (pushover analysis) and concluded that the ratio of sums of plastic moments implemented by the code can not prevent the occurrence of plastic hinges in columns. It was observed that the ratio as high as 1.8 could not prevent the formation of column hinges. The distribution of moments in columns changed drastically from the elastic distribution after the formation of beam hinges. The abrupt increase of moments in columns below joints and decrease of moments above joints were observed and led Lee to propose a three-quarter rule for SCWB design. Essentially, this rule means that three quarters of the sum of girder plastic moment should be taken by the lower column. Many similar findings have also been made by others [Park and Pauley 1975, Goel and Itani 1994, Bondy 1996]. Goel and Itani [Goel and Itani 1994] observed that moment frames designed by modern practice experienced unevenly distributed yielding among the members of the frames. The reason for this uneven distribution can be attributed to the difference in the distribution of internal forces at the ultimate level and at the design level. This difference is due mainly to the redistribution of internal forces after some significant yielding which typical elastic analysis can not capture. Park and Paulay [Park and Pauley 1975] showed that the distribution of moments in columns under dynamic excitations does not support a typical design assumption that the points of contraflexure are located at mid-height of column. They suggested that the sum of girder plastic moment should be resisted by only one column with an adjusting factor, which takes into account the effect of higher modes and ranges from 0.8 to 1.3. Bondy [Bondy 1996] also arrived at the same conclusion and proposed a method to design a column based on incremental displacement analysis using a pushover method. All the methods recently proposed except that of Bondy, though based on extensive analyses, are still based on localized joint behavior and do not recognize actual distribution of internal forces.

39 17 In conclusion, an SCWB system provides much better seismic response than a WCSB system. The WCSB should be avoided since it can result in serious damage including the collapse of the building. In addition, the use of localized joint rule to ensure SCWB in modern design codes is insufficient and a more rational method involving global plastic distribution of moments should be used. Such method, based on plastic analysis to determine the distribution of moments, will be discussed further in Chapter THE STUDY BUILDING An existing six-story moment frame was selected to study the seismic response of conventionally designed moment frames. This frame was a part of the lateral load resisting system of a building located near the epicenter of the 1994 Northridge earthquake. The frame suffered significant damage during the earthquake. More detailed description of the damage has been reported elsewhere [Hart et al. 1995]. The damage has raised serious questions regarding the performance of steel moment frames and has clearly shown how ineffective the current design procedure can be. The plan view of the study building is shown in Figure 2.1. The lateral stiffness in the N-S direction of the frame is provided by four perimeter special moment-resisting frames. Each of the moment frames is responsible for a quarter of the total mass. The bottom story is below grade with extensive outside and interior basement walls. The floor masses of the building are presented in Table 2.1. One of these three-bay frames, along with its member sizes, is shown in Figure 2.2. Table 2.1. Floor Masses of the Study Building. Floor Floor Mass (kip in/sec 2 ) Weight (kips) Roof

40 18 Figure 2.1. Plan View of the Study Building.

41 19 Figure 2.2. A Typical Three-Bay Moment Frame in the N-S Direction. The UBC-94 lateral forces were used to represent the design forces for each frame. The frame is a special moment resisting frame, thus R w =12. Other important constants used to calculate the design forces were Z =0.4 (seismic zone 4), I =0.1 (standard occupance), and S =1.5 (soil type S3). The estimated period of the frame from Equation 2.3 was 0.86 seconds. The total design base shear coefficient ( V / W ), including the torsion effect prescribed in the code, was The design lateral forces at each floor level are summarized in Table 2.2. The computed story drifts under the UBC forces are shown in Table 2.3. As can be seen, the frame satisfied the drift limits prescribed by UBC-94.

42 20 Floor h i (ft.) Table 2.2. UBC Design Lateral Forces for the Original frame. w i h i w i h i / w j h j F t F i F i /frame F torsion +5% (kips) (kips) (kips) Ecc. Total F i (kips) (kips) Roof UBC Design Base Shear Coefficient (V/W) = 0.09 Table 2.3. Design Story Shears and Story Drifts. Story Story Shear (kips) Story Drift (%) UBC Drift Limit = 0.03/12 = (0.25%) 2.6 NONLINEAR ANALYSES OF THE STUDY BUILDING Methods of Analysis Inelastic static as well as inelastic dynamic analyses were carried out to evaluate the study frame. A nonlinear finite element code SNAP-2DX [Rai et al. 1996] developed at the University of Michigan was used to perform the analyses. Inelastic static (pushover) analysis was carried out by applying increasing lateral forces representing the distribution of UBC design lateral forces. The purpose of the pushover analysis was to determine the lateral load capacity, the failure mechanism, the sequence of inelastic

43 21 activity leading to collapse, and the progressive change in the internal force distribution. For the inelastic dynamic analyses, the study frame was subjected to four selected earthquake records. The 1940 El Centro, the 1994 Northridge (Sylmar Station), the 1994 Northridge (Newhall Station), and one synthetic ground motion were scaled and used as base excitations. These records were chosen because of different characteristics of ground shaking. The El Centro record is a classic base excitation and it contains a broad frequency range. The 1994 Sylmar and 1994 Newhall records are recent records from the Northridge earthquake. They were selected because of their near-source characteristics, typically characterized by few large pulses concentrated over a relatively short duration. The synthetic record was used to represent an ideal design level earthquake. This record was generated in such a way that its response spectrum matches closely with that of the UBC-94 [Gasparini 1976]. The other three actual earthquake records were scaled so that their intensities are the same as the design earthquake. The definition of the design earthquake is still somewhat vague. Many procedures have been proposed for scaling earthquake records to represent a design level earthquake. In this study, the scaling procedure was based on the definition of spectrum intensity by Housner [Housner 1959]. The spectrum intensity of an earthquake is defined as the area under damped elastic pseudo-velocity spectrum curve for periods between 0.1 to 2.5 seconds. The earthquake intensity can be defined mathematically as: 2. 5 I = S dt (2.14) e 0. 1 v where S v is the pseudo-velocity of a single degree of freedom system. For a particular ground acceleration, the pseudo-velocity for a lightly damped system can be evaluated from: S v = t 0 a g ( τ )sin( ω( t τ )) e ζω ( t τ ) dτ max (2.15)

44 22 where a g ( τ ) is the ground acceleration at time τ, ω is the natural circular frequency of the system, ζ is damping as a fraction of critical damping, and t is the time at which the integral is evaluated. The symbol f (τ ) denotes the absolute value of the mathematical function. The records used in this study were scaled to have the same earthquake intensity as that computed from the UBC-94 design spectrum (with S =1.5 and I =1.0). The pseudo-velocity spectra of the four scaled records (with 5% damping) and the one corresponding to the UBC design acceleration spectrum are shown in Figure 2.3. The scaled records are shown in Figure 2.4. Table 2.4 summarizes the characteristics and the scaling factors of the four records S v (in./sec) UBC Sylmar Newhall El Centro Synthetic Period (sec.) Figure 2.3. Scaled Pseudo-Velocity Spectra of the Earthquakes Used in This Study (5% Damping).

45 El Centro Acceleration (g) Time (sec.) Sylmar 0.8 Acceleration (g) Acceleration (g) Acceleration (g) Time (sec.) Newhall Time (sec.) Synthetic Time (sec.) Figure 2.4. Four Selected Earthquakes Used in this Study.

46 24 Table 2.4. Characteristics of Earthquake Records. Earthquake Peak Acc. (g) Intensity (g sec 2 ) Scaled Peak Acc. (g) Duration Used (sec.) 1940 El Centro Newhall Sylmar Synthetic UBC Spectrum Intensity (Soil Type S3) = g sec Analytical Modeling of the Study Building An equivalent one-bay five-story frame of the original three-bay frame was used in this study. The one-bay frame approach has been shown to represent the behavior of the whole multi-bay frame well and has been used successfully in some past studies [Itani and Goel 1991, Basha and Goel 1994]. The one-bay frame is a frame with average properties of the original frame. The elastic properties (moment of inertia, area, and modulus of elasticity) and the yield moment of beams in the one-bay frame are the same as those of beams in the original frame. The elastic properties and the yield moment of columns in the one-bay frame are equal to one-sixth of the sum of those in the original frame. The frame was modeled as a five-story frame with fixed supports at the ground level because its bottom story is below grade and consists of basement walls. The original three-bay frame was assigned one quarter of the total mass of the building, resulting in one-twelfth of the total mass in the one-bay frame model. The floor masses were lumped at the beam-to-column connection nodes. The damping was taken as 2% of the critical value and was taken proportionally to the mass matrix only as: [ C ] = a 0[ M ] (2.16) where [C] and [M ] are the viscous damping and mass matrices of the system, and a 0 is the mass-proportional damping coefficient. With this damping model, the higher modes of response were given very little damping. The mass-proportional damping coefficient

47 25 was calculated using the estimated period from UBC (Equation 2.3) and can be found from: a = 0 2ζω (2.17) n where ζ is the damping as fraction of the critical damping, 0.02 in this case, and ω n is the natural circular frequency. For the equivalent one-bay model, the period was estimated as 0.86 second, resulting in a 0 of The beams and columns in the frame were modeled by using the beam-column element from the SNAP-2DX element library. This element is a concentrated plasticity element with the ability to form plastic hinges only at its ends. The plastic hinge model takes into account the interaction between the axial force and the plastic moment. Elasticplastic hysteretic behavior with 2% strain hardening was used to represent the inelastic response of beam-column hinge. The panel zone deformations of the frame were not considered in the analysis because the main purpose was to evaluate the global response. The three-bay frame and the idealized one-bay frame are shown in Figure 2.5. The effect of gravity loads was assumed to be small and was neglected in the analyses. This is justified because the frame is at the perimeter of the building, therefore, the lateral loads are much larger than the gravity loads. Original Frame One-Bay Idealized Model Figure 2.5. The Original Frame and the Equivalent One-Bay Idealized Model.

48 Nonlinear Static Pushover Analysis The plot of the base shear coefficient versus roof drift is shown in Figure 2.6. Figure 2.7 shows the sequence of inelastic activity under increasing lateral forces. As can be seen, the response of the frame was elastic up to a drift level of about 1% when the first set of plastic hinges formed at the base of the frame. The inelastic activity then quickly spread out into the beams resulting in significant reduction in lateral stiffness. The first set of plastic hinges in the beams was at the fourth floor. It was almost instantly followed by the formation of hinges in the second floor. The mechanism formed at the roof drift level of about 1.5% when two plastic hinges formed at the top of the first story columns creating a soft story type mechanism. Beyond this drift level, the resistance came primarily from the strain hardening of the material at plastic hinges. The ultimate strength of the frame was approximately 5 times the UBC design base shear. The response of this study frame is typical of a conventional, elastically designed, frame. Such response is generally characterized by early formation of plastic hinges at the base, high degree of overstrength, and a soft story type collapse mechanism. Early formation of plastic hinges at the column base can mean large ductility demands at a rather critical location. The formation of a soft story mechanism can lead to more serious consequences including collapse in some cases. The consequence of early formation of base hinges was evident in the 1995 Kobe earthquake when numerous failure of column base connections were observed. Both the early formation of base hinges and high overstrength are the direct consequences of the inconsistency between the prescribed strength and the drift limitation.

49 27 Base Shear Coefficient (V/W) Mechanism 1 First Plastification UBC DESIGN V = 0.09 W Roof Drift (%) Figure 2.6. Base Shear - Roof Drift Response from Pushover Analysis. 0.40V 0.23V 0.18V V 0.07V Figure 2.7. Sequence of Inelastic Activity from Pushover Analysis.

50 28 In most cases, the member sizes of moment frames are governed by the drift limits. Therefore, to increase the stiffness of the frame, designers generally increase the sizes of beams while the sizes of columns remain relatively the same. The degree of reserved strength in beams is, therefore, larger than that in columns. As the lateral forces increase due to an earthquake, plastic hinge will form at the point where the degree of reserved strength is lowest. For columns, the point where the reserved strength is lowest is at the base because the applied moments are generally largest there. Consequently, plastic hinges will form at the base at an early stage leading to large plastic rotational demands. For a well designed frame, the sizes of beams and columns should be proportioned to allow the plastic rotational demands to be more evenly distributed throughout the structure. The formation of a soft story type mechanism comes as a result of two major factors. The first factor is that, as mentioned earlier, the code allows the use of WCSB framing system in some cases when the axial load is not large. It should be emphasized once more that WCSB frames, although have been found by experiments to have a stable hysteretic response if the axial load is small, should not be used since the ductility demands and the story drifts will generally be larger than those in SCWB systems. Moreover, even though the joint requirements are satisfied, it does not necessarily mean that the plastic hinging in columns will be prevented as mentioned earlier. The second major factor is the drastic change in the internal distribution of forces after the beam yielding has occurred. As Lee [Lee 1996] pointed out in his study, the moment in the column below a joint may increase abruptly while the moment above the joint decreases. To illustrate this, the distribution of beam moment in columns at the joint of the second floor as the roof drift increases is shown in Figure 2.8.

51 29 Distribution of Beam Moment Below Joint Above Joint Roof Drift (%) Second Floor Joint Figure 2.8. Distribution of Beam Moment in Columns at the Second Floor Joint. As can be seen, the moment above and below the joint started out at about 50% of the beam moment. This corresponds to the assumption usually taken during elastic design that the point of contraflexure is at the mid-height of the column. As yielding starts, the distribution deviates more and more from the distribution. Furthermore, the moment above the joint decreases while the moment below the joint increases. This eventually leads to the formation of a soft story mechanism. This analysis clearly shows that the elastic analysis can not accurately represent the distribution of moments in the inelastic state Nonlinear Dynamic Analyses Several parameters were studied to investigate the performance of this conventionally designed steel moment frame. These parameters, including the maximum floor displacement, the maximum interstory drift, and the rotational ductility demand, are

52 30 presented in Figures 2.9 through The envelopes of maximum floor displacements are shown in Figure 2.9. The envelopes of maximum story drifts are shown in Figure The location of inelastic activity along with the rotational ductility demands at plastic hinges are shown in Figure The rotational ductility demand, in this study, is defined as the ratio of the maximum end rotation of a member to the end rotation at the elastic limit. The elastic limit rotation is the rotational angle developed when the member is subjected to anti-symmetric yield moments at the ends. The rotational ductility, µ, can then be calculated as: where θ p, max θ y + θ p,max θ p,max µ = = 1+ (2.18) θ θ y y is the maximum plastic rotation at plastic hinge and θ y is the yield rotation. 6 5 Floor Level 4 3 El Centro Sylmar Newhall Synthetic Floor Displacement (in.) Figure 2.9. Maximum Floor Displacements due to the Four Selected Earthquakes.

53 Story Level El Centro Sylmar Newhall Synthetic Story Drfit (%) Figure Maximum Story Drifts due to the Four Selected Earthquakes. (1.04) (1.04) (1.27) (1.27) (2.11) (2.11) (1.24) (1.24) (2.70) (2.70) (1.82) (1.82) (1.37) (1.37) (1.20) (1.95) (1.95) (1.20) (1.38) (3.26) (3.26) (1.38) (1.88) (1.88) (1.20) (1.20) (1.03) (1.03) (1.24) (1.82) (1.82) (1.24) (1.61) (1.61) (1.08) (1.08) (1.02) (1.35) (1.35) (1.02) (1.15) (1.15) (1.87) (1.87) (2.09) (2.09) (1.08) (1.08) (2.23) (2.23) (2.03) (2.03) (3.63) (3.63) (1.75) (1.75) El Centro Newhall Sylmar Synthetic Note: Ductility Demands Shown in Parentheses Figure Location of Inelastic Activity and Rotational Ductility Demands due to the Four Selected Earthquakes.

54 32 As can be seen, the maximum floor displacements of the frame under the four records were approximately in the same level. The story drifts under the four records were kept at about 2% due to the large overstrength of the frame. The distribution of story drifts over the height of the frame were similar in all cases, but more significantly, the maximum story drift of the fourth story under the Newhall record and that of the first story under the Sylmar record were almost twice of the others. This is because a story mechanism formed during the excitation, as shown in the Figure In fact, the soft story mechanism was observed in three out of these four cases. As mentioned earlier, the formation of a soft story mechanism can significantly affect the response of a frame and it can lead to serious consequences such as collapse. Another important effect of a story mechanism on the response of moment frames is the permanent displacement after the excitation. Figure 2.12 shows the roof displacement time histories of the study frame. It can be noticed that the frame had large permanent displacements after the El Centro, Newhall and Sylmar records, but almost none after the synthetic earthquake. This permanent displacement can seriously affect the function of the building and impede or prevent repair work after major earthquakes. The major reason for the formation of story mechanisms can be traced back, as discussed earlier, to the unrealistic assumptions used during the design process. Column design is usually carried out by sizing the columns based upon elastic distribution of moment or simply assuming mid-height inflection points for plastic state and checking the strength requirements at joints (Equations 2.8 and 2.9). Unfortunately, non-linear inelastic time history analyses [Park and Pauley 1975, Bondy 1996] have shown that the current design methods may underestimate the moment demands in columns especially when beams have gone into the inelastic state.

55 33 Roof Displacement (in.) Roof Displacement (in.) Roof Displacement (in.) El Centro Time (sec.) Newhall Time (sec.) Sylmar Time (sec.) Roof Displacement (in.) Synthetic Time (sec.) Figure Roof Displacement Time Histories under the Four Selected Earthquakes.

56 34 The distribution of beam plastic moments for column design as conventionally assumed in practice (assuming mid-height inflection points) and the final, as provided, column strength of the study frame are shown in Figure The design moments are compared with the maximum moments from the time history analyses in Figure The actual column strength is larger than the design moment due to several design provisions that introduce reductions due to axial forces. Nevertheless, under strong earthquake excitations, the strength eventually comes close to the design values due to the reduction from axial force-moment interaction. In most cases, the design moments expected by the designer were far lower than those computed during the time history analysis, as can be seen in Figure The columns, therefore, yielded under strong earthquakes. Figure 2.14 clearly substantiates the findings by Park and Pauley [Park and Pauley 1975] and Bondy [Bondy 1996] that the conventional design procedure significantly underestimates the moments in the columns. Although the structure was able to continue to dissipate the input energy after the soft story mechanism had formed, its ability to do so under a more severe earthquake is still questionable. Height Above Ground (ft.) Design Moment Provided Strength Moment (kip-in) Figure Distribution of Column Strength along the Height.

57 35 Height Above Ground (ft.) Design Moment El Centro Sylmar Newhall Synthetic Moment (kip-in) Figure Maximum Column Moments Due to the Four Selected Earthquakes. In addition to formation of soft-story mechanism, high ductility demands were found in columns of the frame. As expected from the static pushover analysis results, rotational ductility demands were largest at the column bases. This suggests that column bases may fail under strong earthquakes, leading to a loss of ability to resist overturning moment. In conclusion, the results of the time history analyses showed that, although the response of the frame was far from a collapsing state, the performance of the frame is not satisfactory. Many subtle flaws exist in the current design practice and can lead to more serious consequences if they are not treated properly. 2.7 SUMMARY AND CONCLUDING REMARKS The current design procedures for steel moment resisting frames was discussed in this chapter. Related experimental and analytical studies found in the literature were briefly presented. An actual moment frame building located near the epicenter of the

58 Northridge earthquake was used as a study case to further investigate the performance of conventionally designed moment frames. Nonlinear static and nonlinear dynamic time-history analyses were carried out and the results were discussed. The major findings are: (1) SCWB frames are superior to WCSB frames. WCSB frames have been found to produce concentration of inelastic activity in a limited number of elements, especially in columns. SCWB frames have been found to distribute the inelastic activity over many more elements. The ductility demands and damage potential are likely to be much higher in WCSB frames than in SCWB frames. (2) The maximum interstory drifts of WCSB frames have been found to be sensitive to the increase in earthquake intensity. This is due to the formation of undesirable mechanisms. (3) Some plastic hinges can form in columns even when the frame is design according to SCWB requirements. The use of localized joint strength requirements, although important, is not sufficient to prevent the formation of plastic hinges in the columns. The distribution of moments in the columns after some beam yielding has occurred was found to be drastically different from the elastic distribution. The consequences of this redistribution are widespread inelastic activity and uncontrolled mechanism. (4) The response of a conventionally designed moment frame is typically characterized by early formation of plastic hinges at the base, high degree of overstrength, and a soft story type mechanism. Most of these characteristics are not desirable and can lead to poor response under seismic excitation. (5) Most of the problems associated with moment frames can be attributed to two major factors. The first factor is the inconsistency between the strength and drift (stiffness) criteria imposed by building codes. Most of the moment frames are designed to

59 37 conform to the drift requirements leaving the sizes of beams relatively large compared to the sizes of columns. The inelastic activity, therefore, tends to occur in columns. The second factor is the inability of the elastic design method to capture the distribution of internal forces in the inelastic stage. Combination of these two factors leads to the formation of undesirable yield mechanisms. It is clear that new methods to design moment frames should be developed in such a way that the level of force and drift requirements are compatible and the plastic distribution of internal forces is explicitly recognized. One such method, based on plastic analysis and the principle of energy conservation, will be presented in Chapter 3.

60 CHAPTER 3 DRIFT AND YIELD MECHANISM BASED DESIGN OF MOMENT FRAMES 3.1 INTRODUCTION It was shown in Chapter 2 that building structures designed by modern code procedures may undergo large cyclic deformations in the inelastic range when subjected to a design level ground motion. Nevertheless, most seismic design work around the world at present is carried out by elastic methods using equivalent static design forces. Design codes in the United States, particularly the UBC [UBC 1994, UBC 1997], attempt to provide sufficient strength and stiffness by imposing stringent drift limits at design force level without any explicit checks pertaining to the ultimate state. By doing so, the UBC offers an advantage that only elastic analysis needs to be performed. However, this often, especially for steel moment frames, results in unpredictable and poor response during severe ground motions with inelastic activity unevenly distributed among structural members. Typical seismic response of a structure designed by modern codes can be best summarized using Figure 3.1. Point 0 in Figure 3.1 corresponds to the response of an equivalent elastic system. Since modern structures are designed to undergo inelastic deformation, the actual response will be as shown by the solid lines in the figure. Points 1 and 2 in Figure 3.1 correspond to the design points as specified in the UBC-94 and the UBC-97 respectively. Point 1 is the allowable stress design level and Point 2 is the strength design level. As was pointed out in Chapter 2, structural response at ultimate state may vary significantly depending on the reserve strength and the failure mechanism Generally, the ultimate response of a structure can be as follows: 38

61 39 0 Base Shear max Figure 3.1. Typical Response of Structures. 1) The structure can develop a high degree of overstrength even though the response may be poor due to the development of a non-ductile deformation mechanism, such as a story type mechanism. This kind of behavior is depicted by Point 3 in Figure ) The structure is capable of developing a ductile mechanism but the degree of overstrength is not sufficient. The result is excessive story drift as depicted by Point 4 in Figure ) The structure is capable of developing a ductile mechanism and has adequate strength. The result is a desirable response under both small, frequent, earthquakes as well as severe ones (Point 5). It is desirable to design structures so that they behave in a known predictable manner during design level ground motions. This essentially means allowing for the formation of a preselected desirable yield mechanism with adequate strength and ductility. This chapter presents and discusses a new seismic design procedure in which

62 40 the structure is designed at the ultimate strength level (Point 5 in Figure 3.1). The inelastic design base shear is derived corresponding to a target maximum drift using the principle of energy conservation. Then, plastic (limit) design is used to design the structure to achieve a selected mechanism without explicit checks of the drift criteria at allowable stress level. Results of dynamic analyses of structures designed by the proposed method will be shown and discussed. The implications of the new design concept will also be presented. 3.2 PRINCIPLE OF ENERGY CONSERVATION The principle of energy conservation is a well-known principle and has been applied to solve many mechanics problems. In the field of earthquake engineering, the use of energy as design criterion is not new. Most energy-based approaches are derived from a concept first proposed by Housner [Housner 1956]. Further investigations were carried out by many researchers [Akiyama 1985, Kato and Akiyama 1982, Uang 1988]. However, not many of these studies have found their way into design practice. A rare example is the Japanese Seismic Design Code, which was developed by considering the concept of energy balance [Kato 1995]. Most energy design methods are based on a premise that the energy demand can be predicted, therefore, suitable member sizes can be provided to dissipate the input energy within an acceptable limit state. For a single degree of freedom system subjected to a horizontal ground motion, the equation of motion at any given time can be written as: m& x + cx& + f s = ma g (3.1) where m is the mass of the system, c is the viscous damping coefficient, f s is the restoring force, and ag is the ground acceleration. Multiplying both side of Equaiton 3.1 by dx and integrating over the duration of the ground motion: m xdx & cxdx & + f sdx = & + ma dx (3.2) g

63 41 The first term on the left-hand side of this equation can be written as: 2 dx& mx& mxdx & = m dx = mxdx & & = dt (3.3) 2 This is the kinetic energy, E k, of the system at the moment the ground motion ceases. The second term on the left-hand side of Equation 3.2 is the damping energy, The third term on the left-hand side of Equation 3.2 is the absorbed strain energy which is composed of elastic strain energy, motion and E es, and cumulative hysteretic energy, E p : E d. f sdx = Ees + E p (3.4) The fact that the integral is evaluated over the entire duration of the ground E p is irrecoverable implies that E p is the cumulative hysteretic energy dissipated during the exicitation. The term on the right-hand side of Equation 3.2 is the work done by the equivalent static force ( mag earthquake, E : ) or the total input energy from the ma g dx = E (3.5) The principle of energy conservation can be written as: E + E + E + E E (3.6) k d es p = Housner [Housner 1956] defines the energy that contributes to damage of the structure as the sum of the elastic vibrational energy, E = E + E, and the cumulative e k es hysteretic energy only: E e + E E (3.7) p The right-hand side of this inequality is the energy demand and the left-hand side is the energy supply. If the energy demand and supply can be determined, Equation 3.7 can be used to design a structure by conservatively rewriting it as: E + E E (3.8) e p =

64 INPUT ENERGY IN MUTI-DEGREE OF FREEDOM SYSTEMS The characteristics of any earthquake can be measured by its effect on a single degree of freedom system (SDOF). The maximum response of the SDOF system under a particular earthquake is directly related to the input energy from that earthquake. The response of an elastic, lightly damped, single degree of freedom system can be characterized by a mathematical function: v t ζω( t τ ) g ( τ )sin( ω( t τ ))e d 0 max S = a τ (3.9) where S v is called the pseudo-velocity, a g ( τ ) is the ground acceleration at time τ, ω is the natural circular frequency of the system, ζ is the damping as a fraction of the critical value, and t is the time at which the integral is evaluated. The symbol f ( τ ) denotes the maximum absolute value of the mathematical function. The maximum kinetic energy attained by the elastic SDOF system during the ground motion can be found as: 1 2 E k = ms v (3.10) 2 Housner [Housner 1956] showed that the plots of pseudo-velocity versus period of the system, or the pseudo-velocity spectra, of typical earthquakes tend to remain practically constant over a wide range of periods. This is particularly true for a spectrum that is obtained from averaging several response spectra of earthquakes with similar intensities. Based on this assumption, if the pseudo-velocity spectra are almost constant over a wide range of periods, then the maximum earthquake input energy for the system, on the average, is: max

65 E = (3.11) ms v Equation 3.11 is independent or only slightly dependent on the period of the system. It can also be shown mathematically [Housner 1959] that the input energy for an elastic multi-degree of freedom system is approximately: E = (3.12) MS v where M is the total mass of the system. Equation 3.12 is independent of the size, shape and stiffness of the system. It should be noted one more time that the derivation of this input energy expression is based on a key assumption that the elastic velocity spectra for several earthquakes tend to remain practically constant over a wide range of periods. Although many actual response spectra are not strictly constant, they can be assumed to be so for practical purposes. The validity of Equation 3.12 for practical applications has been verified by Akiyama [Akiyama 1985]. It should be mentioned here, at this point, that there is still a controversy about the accuracy of Equation 3.12 in predicting the energy demand. Some studies in the United States [Uang 1988, Akbas and Shen 1997] show that Equation 3.12 may sometimes significantly underestimate the energy demand. Nevertheless, this study is based primarily on Akiyama s study [Akiyama 1985] and the accuracy of Equation 3.12 is assumed. 3.4 ENERGY-BASED DESIGN BASE SHEAR Design Energy Level For energy-based design purposes, the design input energy level, as expressed by Equations 3.11 and 3.12, can be found using the elastic design pseudo-acceleration spectra given in many building codes. In this study, the design is based on the UBC [UBC 1994] design spectrum which, for elastic systems, is specified as:

66 44 A = ZICg (3.13) where A is the design pseudo-acceleration, I is the importance factor, Z is the zone factor, g is the acceleration due to gravity, and C is the elastic seismic coefficient as defined by Equation 2.2. The design pseudo-velocity can be found as: S v = A T = ag ω 2 (3.14) π where ω is the natural circular frequency and: a = ZIC (3.15) The design pseudo-acceleration spectrum and design pseudo-velocity spectrum for Z = 0. 4 (seismic zone 4), I = 1. 0 (standard occupancy) and S = 1. 5(soil type S3) are shown in Figure 3.2. As can be seen, the design pseudo-velocity spectrum has a distinct characteristic that its value tends to be relatively constant over a wide range of periods, starting from period of approximately 0.5 second. A ( in./sec 2 ) Sv (in./sec) Period (sec.) Figure 3.2 Design Pseudo-Acceleration and Pseudo-Velocity Spectra (UBC-94).

67 45 For design purposes, an average value of the design pseudo-velocity over a period range can be used. An alternative for calculating the design value of pseudo-velocity value, since it is rather constant over a wide range of periods, is to apply Equation 3.14 using an estimated fundamental period, T, for steel moment frames provided by the UBC: 3 / 4 T = 0.035h (3.16) where h is the total height of the structure in feet. After the value of pseudo-velocity has been found, the energy demand can be calculated using Equation 3.11 or Equation Although these equations are only true for elastic systems, it is postulated that the energy input for a structure remains the same even when some parts of the structure are stressed beyond the elastic limit (Housner 1956). Therefore, the principle of energy conservation for a single degree of freedom can be written as: 1 2 E e + E p = msv (3.17) 2 where m is the mass of the system. Similarly, for multi-degree of freedom systems, the principal of energy conservation can be written, in an approximate sense, as: 1 2 E e + E p = MSv (3.18) 2 where M is the total mass of the system Design Base Shear for Ultimate Response It was shown in Chapter 2 that the deformation mechanism of a structure dictates its behavior during an earthquake. Good or poor performance of the structure depends significantly on whether it has a ductile mechanism, such as a strong column weak beam mechanism, or a non-ductile mechanism, such as a soft story mechanism. Considering the

68 46 fact that large uncertainty is involved in predicting ground motions, emphasis should be placed on controlling the failure mechanism rather than on controlling the conventional working level strength and drift values. The energy should be dissipated by means of a controlled mechanism, which is capable of developing a stable hysteretic response within an acceptable margin of drift. An equivalent n-story one-bay moment frame subjected to equivalent inertia forces in its maximum drift response state with a selected global mechanism is shown in Figure 3.3. The plastic deformation of the frame takes place after the structure reaches its yield point. After the formation of the yield mechanism, the deformation of the frame is assumed to be uniform over the height of the structure and all the energy is dissipated only in plastic hinges. The inelastic drift of the structure is related to the plastic rotation, θ p, of the frame, i.e., the inelastic story drift is approximately equal to the plastic rotation of the frame ( δ θ ). The principal idea is that, by limiting the amount of plastic pl p rotation, the global drift of the structure can be controlled. F i M pbi h i θ p θ p Figure 3.3. Equivalent One-Bay Frame at Mechanism State.

69 47 where The principle of energy conservation states that: E T 2π 2 2 e + E p = MSv = M ( ag) (3.19) E e is the elastic vibrational energy, and E p is the cumulative plastic work done by the structure. Equation 3.19 is cast in terms of a because this value can be readily obtained form the building codes as opposed to S v. Akiyama [Akiyama 1985] showed that the elastic vibrational energy can be calculated by assuming that the entire structure is reduced into a single degree of freedom system, that is: 1 T V 2 Ee = M ( g ) (3.20) 2 2π W where V is the yield base shear and W is the total seismic weight of the structure ( W = Mg ). This simplification was justified by the results of several dynamic analyses [Akiyama 1985]. Substituting Equation 3.20 into 3.19 and rearranging would gives: E p 2 WT g = 2 8π ( a 2 ( V W ) 2 ) (3.21) Equation 3.21 gives the total cumulative plastic energy during the entire excitation. During the peak response of the structure, only a portion of the total cumulative plastic energy, pe p (where p <1), is dissipated by the structure. The exact determination of the amount of energy dissipated during the peak response requires a full dynamic analysis using the exact properties of both the structure and the ground motion to which the structure will be subjected. In view of the uncertainty involved in predicting the ground motions, the value of p is taken as unity for design purposes, implying that all the plastic drift is assumed to be uni-directional. Although it is extremely unlikely that the deformation will be uni-directional, the following two factors are taken into consideration in setting p equal to unity.

70 48 First, it is well known that higher modes of vibration can play an important role in the seismic response of structures. The inter-story drift, which is a suitable damage index for frame structures, is generally larger than the global drift assumed in the design process. Qi [Qi and Moehle 1991] studied the response of reinforced concrete structures due to several earthquakes and reported that the inter-story drift can be as much as 30% larger than the roof drift, in some cases. For steel moment frames, the ratio between interstory drift and the global drift can even be larger. It may be as high as 1.4 [Collins 1995] or even 2.0 for some cases [Krawinkler 1997]. This amplification is compensated by assuming uni-directional plastic drift in the design process. Second, Qi [Qi and Moehle 1991] in the same study showed that the inelastic seismic response of a single degree of freedom system in a certain period range, can be reasonably well captured by the response due to largest earthquake acceleration impulse. This equivalent impulsive loading produces mainly uni-directional plastic deformation, thereby, implying that the assumed uni-directional plastic drift might be appropriate for design purposes. This is particularly true for near field type earthquakes. Based on many trial and error analyses in this study, it was found that the use of p = 1 along with AISC-LRFD provisions produces a satisfactory design. The results from dynamic analyses will be shown later in Section 3.6. Following the assumptions stated above, the energy dissipated by plastic hinges in the structure shown in Figure 3.3 must be equal to p n i= 1 pb pc p E p or: E = ( 2M + 2M ) θ (3.22) i where M pb is the plastic moment of the beam at the level i and M i pc is the plastic moment of the columns at the base of the structure. Using the expression for E p from Equation 3.21, Equation 3.22 can be rewritten as: ( n 2M + 2M pb i pc i= 1 π 2 WT g 2 V 2 ) θ p = ( a ( ) ) (3.23) 2 8 W

71 49 After yielding, the equivalent inertia forces must be in equilibrium with the internal forces. Equating the internal work done in plastic hinges to the external work done by the equivalent inertia forces gives: where n i= 1 n 2M + 2M = F h (3.24) i pb pc i= 1 i i F i is the equivalent inertia force at level i and h i is the height of beam level i from the ground. Assuming an inverted triangular force distribution along the height of the structure, the inertia force at level i can be related to the base shear by: F i = n j= 1 w h i w j i h j V (3.25) where w (or w ) is the weight of the structure at level i (or j ). This assumed i j distribution corresponds to the assumed linear shape of the first mode of vibration. Substituting Equations 3.24 and 3.25 into Equation 3.23 gives: V and consequently: n i= 1 n i= 1 w h i w h i 2 i i 2 WT g 2 V 2 θ p = ( a ( ) ) (3.26) 2 8π W V W n 2 wihi 8 2 i 1 θ p π = 2 = ( a 2 n 2 i= 1 w h i i T g V ( ) W ) (3.27) Solving Equation 3.27 for V / W gives the required design based shear coefficient:, the admissible solution of the quadratic equation V W = α + α a 2 (3.28)

72 50 where α is a dimensionless parameter, which depends on the stiffness of the structure, the modal properties, and the intended drift level, and is given by: α = n i= 1 n i= 1 w h i w h i 2 i i 2 θ p 8π 2 T g (3.29) Equation 3.29 gives the required design base shear corresponding to an intended drift level, θ p. After the base shear has been determined the design force at each level can be found from Equation It is important to recognize two issues as follows: First, this base shear produces the associated drift level only if the global mechanism is maintained as assumed in Figure 3.3. Therefore, plastic design must be used to ensure the formation of the intended global mechanism. Detailed plastic design procedure for steel moment frames will be discussed later in Section 3.5. Second, the drift level given in Equation 3.29 is the inelastic drift. The total drift is the sum of the elastic and inelastic drift. Hence, it is important to estimate the yield elastic drift of the structure so that the value of θ p can be prescribed according to an intended ultimate drift level. For example, for a moment frame that has an estimated yield drift of 1%, if the structure is to be designed for maximum drift of 2%, the value of can be taken as 1%(0.01). This approximately corresponds to 1% plastic drift. Combining with 1% elastic drift, this gives approximately a total of 2% drift. A better method to determine the yield drift is the pushover analysis. The yield drift can be approximated as the drift corresponding to the inflection point of a bilinear approximation of the pushover roof drift versus base shear response curve. Due to the uncertainty involved in predicting earthquakes, only an estimated value of the yield drift from past experience is sufficient for design purposes. Typically, this yield drift is rather constant for steel moment frames. This is because structural steel members, especially for θ p

73 51 wide flange sections, are manufactured in such a way that their ratios of the strength to stiffness ( Z / I where Z is the plastic modulus and I is the moment of inertia) are quite constant. Therefore, for two moment frames with similar mass and size, if the yield mechanism is the same, then the yield drifts will also be similar. This will be shown later in Section 3.6. The calculated design base shear coefficients for one-bay frames with two, four, six, eight, and ten stories, each with constant story mass and constant story height of 14 feet, are shown in Figure 3.4. For all frames, the yield drifts were assumed to be 1% (0.01). The inelastic drifts ( θ p ) were selected as 0.005, 0.010, 0.015, and corresponding to assumed total target drifts of 1.5%, 2.0%, 2.5%, and 3%, respectively V/W θ p Number of Stories Figure 3.4. Drift and Yield Mechanism Based Design Base Shear Coefficients. As can be seen, when the target drift increases, the required design base shear decreases. The underlying design philosophy for the proposed method is that lateral forces are calculated corresponding to a selected displacement ductility. Choosing a

74 52 target drift is conceptually equivalent to selecting a displacement ductility. This concept has been discussed widely in the past for single-degree of freedom systems [Veletsos and Newmark 1960, Veletsos and Newmark 1965]. However, the use of this concept for multi-degree of freedom systems has not been fully appreciated, probably due to the fact that the yield mechanism directly influences the ductility [Mahin and Bertero 1981]. Hence, the ductility demand for multi-degree of freedom systems varies significantly from system to system, since they are not designed to have the same yield mechanism. The proposed design method explicitly accounts for the effect of the failure mechanism on the ductility demand by employing plastic (limit) analysis, which will be discussed in Section Design Base Shear for Serviceability Modern seismic design philosophy comprises of two levels of performance, which are: serviceability and ultimate limit states. Structures should be capable of resisting minor or moderate earthquakes without significant damage (serviceability limit state) and resisting major earthquakes without collapse (ultimate limit state). In most of the current building codes, the underlying philosophy is that structural safety during a severe earthquake depends on the capability of structures to dissipate the input energy in the inelastic range. The design forces for a structure are derived from the ultimate level elastic design spectrum reduced by a response modification factor that accounts for ductility of the structural system. The serviceability limit state is imposed implicitly by limiting the story drifts under the above design forces. However, as was shown in Chapter 2, attempting to combine both limit states into one design may result in an inconsistency between strength and drift requirements. This leads to structures with illproportioned member sizes and eventually leads to structures with undesirable yield mechanisms.

75 53 A better way to satisfy this two-level seismic design is by considering the two limit states independently, i.e., the required strength or stiffness of the structure is calculated separately for each of the limit states. The governing case can then be selected from the two cases. The methodology presented thus far is intended to calculate the required yield strength to satisfy the ultimate limit state (Equation 3.28). However, this design yield strength does not ensure the serviceability criteria. The serviceability limit state can also be considered in the design. This can be done by comparing the required yield strength of the structure that satisfies the serviceability limit state with the one that satisfies the ultimate limit state (Equation 3.28). The design can be carried out according to the larger required yield strength of the two values computed from the two limit states. The required yield strength of the structure that satisfies the serviceability limit state can be found by the following procedure. First, the serviceability design level spectrum is selected. This spectrum corresponds to an earthquake with moderate intensity, which can be assumed to be in the order of one-sixth to one-eighth the intensity of the UBC elastic design spectrum [Uang 1993]. Next, the serviceability drift limit is established. This drift limit can be selected based on an allowable level of damage of the structure. The objective of the serviceability limit state is that under the selected serviceability design level spectrum, the structure should remain within the selected drift limit. Figure 3.5 shows the expected response of a structure designed to satisfy serviceability limit state. Under the selected serviceability design level spectrum, since the structure can be expected to behave elastically, the total force acting on the structure can be estimated as: Ae ae = (3.30) g

76 54 where a e is the base shear coefficient for serviceability (elastic) level, A e is the pseudoacceleration from serviceability level spectrum calculated using an estimated period (Equation 3.16), and g is the acceleration due to gravity. From Figure 3.5, to satisfy the serviceability design objective, the required yield strength can be related to the base shear coefficient a e by using the following relation: V W δ y = a e (3.31) δ e where δ e is the target serviceability level drift, δ y is the expected yield drift of the structure. As was discussed earlier, the value of the yield drift can be estimated from past experience. It will be shown later that the value of yield drift is rather constant regardless of the strength of the structure and therefore can be predetermined. It should be noted that the total force a e is only an estimate because the strength is not known at the design stage, consequently, the period of the structure is also unknown. However, the estimated period according to UBC (Equation 3.16) should be sufficient as a first approximation. This required yield strength can be compared with the required yield strength for ultimate limit state (Equation 3.28) to determine the governing value. Force-Displacement Response a e δ y /δ e Required Strength a e Design Objective δ e δ y Figure 3.5. Expected Response of a Structure Designed to Satisfy Serviceability

77 PLASTIC DESIGN OF MOMENT FRAMES As mentioned earlier, a desirable global deformation mechanism must be maintained during the entire excitation in order to satisfy Equation In Chapter 2, it was shown that, during an earthquake, the distribution of internal forces at ultimate load is drastically different than that predicted by elastic analysis. Therefore, the theory of plastic design must be utilized since it provides internal force distribution at ultimate level corresponding to a selected failure mechanism without considering the intervening elastic-plastic range of deformation. Theory of plasticity has long been utilized in design of framed structures [Beedle 1961]. Only recently has the concept been applied in design of structures for earthquakes. For plastic design of steel structures for seismic loading, most of the studies found in the literature focus on the design of braced frames such as eccentric or concentric braced frame, with very little on the design of moment resisting frames [Hassan and Goel 1991, Englehart and Popov 1989]. Recently, Mazzolani and Piluso [Mazzolani and Piluso 1997] proposed a plastic design method for steel moment frames based on kinematics theorem of plastic collapse which includes the second order effects ( P ). Although the method proposed is one of the most sophisticated and complete methods, it is based on a premise that beam section properties are known ahead of time by designing the beams to resist vertical loads. Only the sizes of columns are determined using the procedure. Therefore, this method warrants the failure mode of the structure, but does not assure the ultimate drift of the structure under dynamic excitations since the structure designed by this method may not have enough lateral strength. From Equation 3.23, it is clear that the sizes of beams are directly related to the amount of energy required to be dissipated in order to maintain the target drift. Hence, in the method proposed herein, both the sizes of beams and columns are treated as unknown.

78 56 Generally, moment frames are so placed in a structure that the influence of gravity loads is much smaller than that of the earthquake loads. It will be shown later that this is especially true when a structure is designed by using the proposed method which uses significantly larger design lateral forces compared to typical elastic design forces required by building codes. Therefore, in many cases, the gravity loads can be safely ignored. Moreover, by controlling the ultimate drift within an acceptable limit, the second order P- effect can also be assumed to be small. Considering these two factors, the plastic design of steel moment frames can be significantly simplified. The primary aim of the proposed plastic design procedure is to eliminate the possibility of formation of plastic hinges in columns. The well-known strong columnweak beam mechanism is generally believed to be a desirable mechanism for seismic design, and is selected for this design procedure. The equivalent n-story one-bay moment frame subjected to design forces in its mechanism state is shown in Figure 3.6. It is assumed that the design forces, F i, have already been computed from Equations 3.28 and The plastic moment capacity of the beam at level i is denoted by a product β M, i pbr which will be defined later. F i β i M pb r h i M pc M pc Figure 3.6. Frame with Global Mechanism.

79 Design of Beams level i is Applying a virtual rotation d θ at the base, the work done by the external force at F i i h dθ, the internal work done at each plastic hinge in the beams is β i M pb dθ, and the internal work done at each plastic hinge at the base is M pc dθ. By r selecting a suitable distribution of plastic moment capacity of beams and equating the external work done to the internal work done in plastic hinges, the required beam strength at each level can be determined, namely: where n F h = n i i i= 1 i= 1 2β M + 2M (3.32) i pbr pc F i is the design force at level i calculated from Equations 3.25 and 3.28, h i is the height of beam level i from the ground, β i is the beam proportioning factor for beam strength at beam level i, M pb is the reference plastic moment of beams, and M r pc is the required plastic moment of columns in the first story. In Equation 3.32, the beam proportioning factor β i represents the relative beam strength at level i with respect to M pb r. The factor β i can be predetermined as will be discussed later. The product β M is the plastic moment capacity of beam at level i. Here, i pb r pbr beams at all levels. If F i, h, β, and variable in Equation 3.32 is M pb. r For frames with fixed bases, the value of desirable value of i i M is common for M pc are all predetermined, the only unknown M pc must be appropriately chosen. A M pc should be such that the story mechanism in the first story is prevented. As a first approximation, assuming plastic hinges form at the base and the top of the first-story columns, the plastic moment of the first-story columns to prevent this mechanism should be, from Figure 3.7: 1.1Vh M = 1 pc 4 (3.33)

80 58 where V is the total base shear (from Equation 3.28), h 1 is the height of the first story, and the factor 1.1 is the overstrength factor to account for possible overloading due to strain hardening as will be explained later. 1.1V M pc M pc M pc M pc h 1 Figure 3.7. Frame with Soft-Story Mechanism. With a known value of M pc, the required nominal beam strength at each level, M, which is equal to Z F where b yb Z b is the plastic modulus and pb i yield stress of the beam, can be determined from the design inequality: F yb is the nominal φm β M (3.34) pbi i pb r where φ is the customary resistance factor and is equal to 0.9. It should be noted that the resistance factor φ can be taken as 1 but it is recommended that it should be taken as 0.9 to comply with AISC-LRFD design specifications. The beam proportioning factor, β, plays an important role in the seismic response i of a structure. It represents the variation of story lateral strength and stiffness along the height of the structure. Indirectly, it reflects the variation of story drifts along the height.

81 59 Ideally, the optimum distribution of β i values is achieved when the distribution of story drifts under earthquakes over the height of the structure is uniform. In order to achieve this, the beams should be proportioned according to the story shears. The distribution of beam strength along the height should closely follow the distribution of story shear induced by earthquakes. Based on the numerical analyses presented in Appendix A, it was found that the relative distribution of maximum earthquake-induced story shears along the height can be closely approximated by using the distribution of the static story shears calculated from the design lateral forces given by Equation It was concluded that the beam at each level should be proportioned based on the square root of the ratio of the static shear at that level to the shear at the roof level (level n ): 1 / 2 V β i i = V (3.35) n where V i and V n are the story shears at level i and at the roof level (level n ) due to the design forces calculated from Equation The value of the exponent ½ was determined based on a least square minimization of the error between the actual story shear distribution and the distribution given by static story shears. More details on this process can be found in Appendix A. The actual shear distribution in some example structures from nonlinear dynamic analyses will be shown later in Section Design of Columns In order to ensure the selected strong column-weak beam plastic mechanism at the ultimate drift level, it is important that columns are designed assuming that all plastic hinges are fully strain-hardened when the drift is at the target ultimate level. The moment generated by a fully strain-hardened beam is taken into account by multiplying its nominal plastic moment by a factor called the overstrength factor, ξ. By assuming appropriate overstrength factors, generally ranging from 1 to 1.1, the design moment for each column can be calculated.

82 60 A free body diagram of a column of the frame in Figure 3.6 is shown in Figure 3.8. Since all the beams have gone into the strain-hardening range, the applied force at each level, F, must be updated to account for the increase in internal forces. The i magnitude of the updated forces, F iu, acting on one column can be found by equating the overturning moment of the column produced by the updated forces F iu to the resisting moment produced by the beams and column base. The distribution of the updated forces can be assumed as inverted triangular along the height of the frame as used earlier. F iu ξ i M pb i M c ( h ) h i M pc Figure 3.8. Free Body Diagram of the Column in the Equivalent One-Bay Frame. With the assumed inverted triangular distribution, the updated forces can be related to the updated base shear for one column V u by: F iu = n j=1 w h i j i w h j V u (3.36) where w i and h i are as defined earlier. The equilibrium equation for one column can then be written as: n i= 1 F iu h i = M pc + n i= 1 ξ M (3.37) i pbi

83 61 where M pc is the plastic moment at the base of the frame from Equation 3.33, ξ i is the overstrength factor at level i, and M pb is the nominal plastic moment of beam at level i i Substituting Equation 3.36 into 3.37 and solving for V u gives: V u n wihi n i= 1 = ( )( M pc + n ξ im pb ) (3.38) i 2 i= 1 w h i= 1 i i Upon substitution of Equation 3.38 back into 3.36, the updated force at each level can be determined as: n wihi Fiu = ( )( M pc + ξ jm pb ) (3.39) n j 2 j= 1 w h j= 1 j j In Equation 3.39, the subscript j is used to avoid the confusion between the summation over the range and the symbol denoting level i. After F iu has been determined, the distribution of moments in the column can be found by treating the column as a cantilever, namely: M n n c ( h ) = iξ im δ i Fiu ( hi h ) pbi i= 1 i= 1 δ (3.40) where M c ( h ) is the moment in the column at a height h above the ground, and ξ i, M pb i, h i, and Fiu are as defined previously. δ i is a step function defined as: δ i = 1 if h hi (3.41) δ i = 0 if h > hi (3.42) Similarly, the axial force in the column at a height h above the ground, P c ( h ), can be expressed as: n 2ξ im pbi Pc ( h) = δ i ( ) + Pcg ( h) (3.43) L i= 1

84 62 where L is the span length of the beams, P cg (h) is the axial force due to gravity loads acting at height h, and other symbols are as defined previously. After M c (h) and (h) have been determined, the column can be designed as beam-column elements by any suitable design provisions. The purpose of introducing the overstrength strength factor, ξ i, is to account for the difference in the actual and the nominal yield strength as well as to account for the increase in strength due to strain hardening. For example, typical modern A36 beams have an average actual yield stress of 49 ksi [SAC 1995b]. This substantial difference between the nominal and the actual yield stress could have detrimental effects on columns. From Equation 3.40, it is apparent that the moment in a column could be underestimated by as much as 40% if beam moments, M pb, are calculated using the i nominal yield stress. For A572 GR.50 steel, the difference between the actual and the nominal yield stress is much smaller than that of grade 36 steel. Typical A572 GR.50 steel has an average yield stress of about 55 ksi, or approximately 10% greater than the nominal value. Conceptually, the effect of this overstrength due to the difference in yield strengths can be neglected when the columns and the beams in the frame are of the same steel grade. The effect of the difference is most pronounced when the beams in the frame are A36 while the columns are A572 GR.50. Unfortunately, this is usually the case for most structures. The second source of overstrength is strain hardening. Since it is expected that all beams will be deformed well beyond their elastic limit, strain hardening of material will occur. The exact value of overstrength factor for strain hardening of beams in bending is still not quite well known due to the fact that the overstrength for this purpose is the one with respect to rotation as opposed to a more familiar, uniaxial, case. Due to lack of sufficient information on the actual moment-rotation response of typical beams, the overstrength due to strain hardening only may be taken between 1.0 to The final overstrength factor for column design is the product of the overstrength due to the P c

85 63 difference in nominal and actual yield strength and the overstrength due to strain hardening: Fab / Fyb ξ i = ξ s (3.44) F / F ac yc where F yb and respectively, F ab and F yc are the nominal yield strengths of the beams and columns, F ac are their expected yield strengths, and ξ s is the overstrength factor due to strain-hardening. For a conservative design, the ratio 3.44 could be taken as 1.0. F / F in Equation In the proposed design method, all the beams are assumed to reach their maximum overstrength at the same time. This may appear to be too conservative. However, it should be realized that the proposed method is based on an assumed linear distribution of inertia forces. In many cases, the actual distribution of inertia forces can be quite different due to higher mode effects causing some members, especially in the upper levels of the structure, to deform beyond the expected overstrength level. Therefore, using an average overstrength factor for beams uniformly over the entire height of the structure appears reasonable for calculating the design forces for the columns. It should also be noted that the moment and axial force calculated from Equations 3.40 and 3.43 are the internal forces of the column in the equivalent frame. The results can be projected back to the original multi-bay frame by considering that the internal forces of the exterior columns in multi-bay frame are the same as those in the equivalent frame. The internal forces of the interior columns in the multi-bay frame can be found by assuming that the moment is twice as large as the moment in the column of the equivalent frame, and that the axial force due to an earthquake load is practically zero. Hence, the axial force at each level in the interior column of the multi-bay frame is the axial force due to gravity loads only. This assumption is accurate enough for practical purposes provided that the bay width of the multi-bay frame is constant or nearly constant. A ac yc

86 64 flowchart summarizes the design procedure and a frame design example are presented in Appendix B. 3.6 PARAMETRIC STUDY OF THE PROPOSED DESIGN PROCEDURE In order to validate the proposed design procedure, a parametric study was carried out to investigate the effect of the number of stories and the effect of variation of the design target drift. The effect of the variation in the number of stories was studied by using 2-, 6-, and 10-story one-bay moment frames designed with the same target drift of 2%. The drift of 2% was chosen because it was consistent with the ultimate drift limit expected by the UBC (Sections and 2.3). The effect of the variation of target drift was studied by using three 6-story moment frames designed with 1.5%, 2.5% and 3% target drifts. All one-bay moment frames in this study consist of equal story mass, story dimensions, and story gravity loads. A typical story height is 14 feet and the bay width is 25 feet. The gravity loads acting in each column at each story are assumed to be 25 kips. The story weight is 190 kips. A typical story of the study frames is shown in Figure kips 25 kips W=190 kips 14 ft. 25 ft. Figure 3.9. Typical Story of the Study Frames Variation in Number of Stories The effect of the number of stories was studied first by designing the 2-, 6-, and 10-story frames with the proposed procedure. The target drifts for all three frames were set at 2%. The design parameters were calculated as shown in Table 3.1. The design force

87 65 at each level was calculated using Equations 3.25 and 3.28 based on the UBC-94 spectrum with Z = 0. 4 and I = The design forces for the three frames are shown in Table 3.2. The internal forces were calculated from the plastic analysis as presented earlier (Equations 3.32, 3.40, and 3.43). All members in the study frames were designed by using AISC-LRFD specifications (AISC 1994) assuming A572 GR.50 steel for all members. The overstrength factor was taken as 1.05 for all beams except at the roof level where the value of 1 was used. The reason for using the overstrength factor of one at the roof level is because plastic hinges are allowed at that level without affecting the global behavior at the mechanism state. Since same steel grade was used for the beams and columns, the overstrength used in the design was the one associated with strain hardening only. The final member sizes of the three frames are shown in Figure It should be mentioned that some members were selected such that their compactness ratios were below the limits given by AISC-LRFD seismic provisions. This was done in order to keep their strength as close to the required (calculated) values as possible. This exception was made for the purpose of this parametric study only. In a real design situation, the compactness limits must be strictly applied, such as the case of a study frame presented later in Section Table 3.1. Design Parameters (2% Drift Limit). Number of Estimated a Estimated θ α Design p Stories Period (sec.) Yield Drift V / W Note: Calculations based on π = and g = 386 in/sec 2.

88 66 Table 3.2. Design Lateral Forces (in kips). Floor Level 2-Story Frame 6-Story Frame 10-Story Frame W27x94 W30x116 W14x311 W36x135 W14x311 W36x150 W14x500 W27x94 W36x160 W14x500 W30x116 W14x311 W36x160 W14x550 W33x130 W14x311 W36x170 W14x550 W33x130 W14x398 W36x170 W14x550 W24x55 W36x135 W14x398 W36x182 W14x550 W24x62 W14x90 W36x135 W14x398 W36x182 W14x550 W14x90 W14x398 W14x550 Figure Member Sizes of the 2-, 6-, and 10-Story Frames with 2% Target Drift.

89 67 It should be emphasized that the sizes of beams and columns depend mainly on the story weight and the number of bays used in the design process. The assumed onebay frames were used solely for the purpose of parametric study and they do not necessarily represent moment frames in actual building structures, which are typically multi-bay. The fundamental periods of the 2-, 6-, and 10-story frames from modal analysis were 0.86, 1.02, and 1.6 seconds, respectively. It should be noted that the estimated periods given by the UBC closely approximate the modal analysis periods of the structures except for the 2-story frame, where the UBC estimate is significantly lower than the actual value. The effect of this underestimation will be discussed later. A series of non-linear analyses was carried out to study the response of the three frames. The three study frames were subjected to inelastic static analysis and inelastic time history analysis. Inelastic static analysis was carried out to investigate the yield mechanisms of the study frames by applying increasing lateral forces representing the inverted triangular distribution given by Equation For the inelastic dynamic analysis, the three frames were subjected to four different scaled ground motion records. The records used in this study were the same as used in Chapter 2, which were the Newhall record, the Sylmar record, the 1940 El Centro record, and a synthetic earthquake whose response spectrum matched with that of the UBC. All analyses were carried out by using a computer program SNAP-2DX [Rai et al. 1996] developed at the University of Michigan. Modeling assumptions were similar to those used in Chapter 2. These assumptions included: floor masses were lumped at the nodes, frame dimensions were taken at centerlines, gravity loads were neglected, yield stress of 50 ksi with 2% strain hardening with respect to rotation was used for all members, and 2% mass proportioning damping was used with the estimated period by the UBC. Figure 3.11 shows the base shear versus roof displacement plots of the three frames obtained form pushover analyses. Figure 3.12 shows the location of inelastic

90 68 activity in the three frames at 3% roof drift. As can be seen, all frames developed strong column-weak beam mechanism as intended in the design. The yield drifts of the three frames were within 1%, which were also consistent with the values assumed in the design (Table 3.1). It should be noted that the frames could be redesigned with the refined values of yield drifts from Figure However, in this study, only one design iteration was used for each frame to reduce computation time. 0.5 Base Shear Coefficient (V/W) Story 6-Story 10-Story Roof Drift (%) Figure Base Shear versus Roof Drift Response of the Study Frames.

91 69 Figure Location of Inelastic Activity in the Three Frames at 3% Roof Drift. Some selected results from the inelastic dynamic analysis of the three frames are presented briefly in Figures 3.13 and Figure 3.13 shows the envelopes of maximum story drifts of the three frames due to the four selected ground motions. The envelopes of maximum story drifts show that almost all story drifts were within the target design limit of 2%. It can be noticed that the story drifts of the 2-story frame under El Centro and Newhall records were larger than the 2% target drift. This is because the actual calculated fundamental period of this 2-story frame, which is 0.86 second, falls in the range where the pseudo-velocity spectra of the El Centro and Newhall records are significantly larger than the design spectrum, as can be seen from Figure 2.3.

92 70 El Centro Sylmar Newhall Synthetic 10 9 Story Level Target Drift Target Drift Target Drift Story Drift (%) Figure Maximum Story Drifts of the 2-, 6-, and 10-Story Frames. It is also worth noting that the period estimated using the UBC formula was much smaller than the computed period for the 2-story frame. Hence, the design input energy was too small as can be seen from the pseudo-velocity spectra shown in Figure 2.3. This trend has been observed in most short-period structures and has been reported in the literature [Reddell 1989]. Design of short-period structures can be improved by using an average (constant) value of pseudo-velocity based on the UBC to compensate for the low value of input energy calculated from the pseudo-velocity spectrum. Figure 3.14 presents the relative distribution of maximum story shears, V / V, ei en where V ei and V en are the earthquake-induced story shear at level i and at the level n, respectively, along the height of the three frames under each earthquake. Figure 3.14 also shows the distribution of beam strength and stiffness, β i, used in the design process of

93 71 the three frames. As can be seen, in most cases, the distribution given by Equation 3.35 agreed well with the distribution of story shears from time history analyses. The difference is possibly due to the higher mode effects, and can be seen clearly in the 10- story frame case. Nevertheless, the results demonstrate that the proposed design method successfully controls the yield mechanism and maximum story drifts under design level earthquakes El Centro Sylmar Newhall Synthetic Beta Story Level V ei / V en Figure Distribution of Maximum Story Shears from Dynamic Analyses.

94 Variation in Design Target Drift The effect of the variation in design target drift was studied next by designing three 6-story frames with target drifts of 1.5%, 2.5% and 3%. The target drift of 2% was used earlier (Section 3.6.1) and was, therefore, excluded. The purpose was to study the effectiveness of the proposed procedure to control maximum drift within the target drift level. The design was carried out in similar fashion as for the frames presented in the previous section. The design parameters are summarized in Table 3.3 and the design lateral forces are summarized in Table 3.4. From Table 3.3, it can be seen that the base shear becomes smaller when the target drift becomes larger. This is consistent with the result for single-degree of freedom systems studied by Chopra [Chopra 1995]. As the target ductility increases, the required strength of the system decreases. The member sizes of the three frames are shown in Figure Table 3.3. Design Parameters. Number of Estimated a Estimated θ α Design p Stories Period (sec.) Yield Drift V / W Note: Calculations based on π = and g = 386 in/sec 2. Table 3.4. Design Lateral Forces (in kips). Floor Level 1.5% Target Drift 2.5% Target Drift 3.0% Target Drift

95 73 W33x118 W24x76 W24x62 W36x150 W14x455 W27x94 W14x257 W24x84 W14x193 W36x170 W14x455 W30x99 W14x257 W27x84 W14x193 W36x182 W14x550 W30x108 W14x283 W27x102 W14x233 W40x183 W14x550 W30x116 W14x283 W27x102 W14x233 W40x183 W14x550 W30x116 W14x283 W27x102 W14x233 W14x550 W14x283 W14x233 Figure Three Six-Story Frames with 1.5%, 2.5%, and 3% Target Drifts. The actual periods were calculated from modal analysis to be 0.91, 1.45, and 1.67 seconds for the frame with 1.5%, 2.5%, and 3% target drift, respectively. It can be noticed that the periods estimated by using the UBC formula are significantly lower than the modal analysis values when the frames are designed for a large value of target drift. In the case of six-story frame the effect of this underestimation is not pronounced because the design pseudo-velocity spectrum is quite constant at this range of periods. However, care should taken when designing a low-rise structure with a large target drift, since the error in the period may result in an unconservative design. These three 6-story frames were subjected to the same series of inelastic analyses as was done previously, using the same modeling assumptions. Figure 3.16 shows the base shear versus roof displacement plots of the three frames from the results of pushover analyses. As can be seen, the approximate yield drifts of the three frames were about the same, regardless of their strength. This is due to the fact that steel members are usually manufactured in such a way that their strength and stiffness are in proportion, as was mentioned earlier. Although the yield drifts of the three frames were smaller than what was assumed during the design, no further design iteration was carried out. All three

96 74 frames developed the same global (strong column-weak beam) mechanism as intended in the design, as can be seen in Figure 3.16 where the location of inelastic activity in the three frames at 3% roof drift is shown. 0.7 Base Shear Coefficient (V/W) Roof Drift (%) 1.5% 2.5% 3.0% Figure Base Shear versus Roof Drift Response of the Study Frames. Target Drift 1.5% Target Drift 2.5% Target Drift 3.0% Figure Location of Inelastic Activity in the Three Study Frames at 3% Roof Drift.

97 75 Some selected results from the inelastic dynamic analyses are presented briefly in Figures 3.18 through Figure 3.18 shows the envelopes of maximum story drifts along the height of the three frames due to the four selected ground motions. The overall maximum story drift under each earthquake of all six-story frames, including the one with a 2% target drift in Section 3.6.1, are presented again in Figure 3.19 where the target drifts and maximum drifts are compared. The 45-degree line in Figure 3.19 represents a 1-to-1 relationship between the design target drift and the maximum drift El Centro Sylmar Newhall Synthetic Story Level Target Drift Target Drift Target Drift Story Drift (%) Figure Maximum Story Drifts under the Four Selected Earthquakes.

98 76 Maximum Drift (%) El Centro Newhall Sylmar Synthetic Target Drift (%) Figure Comparison between Design Target and Attained Maximum Drifts. Figures 3.18 and 3.19 show that almost all story drifts remained well within the target design drift limits. The story drifts of the 6-story frame designed with a target drift of 3% were slightly less than the target value. This result suggests that when the period of the structure becomes large, the displacement tends to be bounded and depends mainly on the characteristics of the ground motion and the elastic stiffness of the frame, regardless of the strength of the system. This behavior is characteristic of structures in the displacement sensitive period range. Nevertheless, in this case, the proposed design procedure will give a conservative design. This fact is a direct result of the assumption that all plastic energy is dissipated in one pulse ( p =1 in Equation 3.21) for all structures, regardless of their periods.

99 77 El Centro Sylmar Newhall Synthetic Beta 10 9 Story Level Target Drift 1.5% Target Drift 2.5% Target Drift 3.0% V ei / V en Figure Distribution of Story Shears from Dynamic Analyses. Figure 3.20 presents the distribution of maximum story shears along the height of the three frames. Figure 3.20 also shows the distribution of beam strength and stiffness, β i, used in the design process of the three frames. As can be seen, the distribution given by Equation 3.35 agreed well with the distribution of story shears from time history analyses. The deviations under some records were probably due to the higher mode effects. Nevertheless, these results agree well with the results from previous analyses.

100 COMPARISON BETWEEN THE CURRENT AND THE PROPOSED DESIGN PROCEDURES Comparison of Seismic Response In order to compare the difference between the current and the proposed design procedures, the moment frame presented in Chapter 2 was used as a study case. This moment frame was redesigned using the proposed design procedure. The objective was to compare the seismic behavior of the original and the redesigned frame through inelastic dynamic analyses. The redesign process started by calculating the design base shear according to the proposed design procedure. The period of the structure was estimated by Equation 2.3 to be 0.86 second. By assuming 1% elastic drift, the value of θ p was taken as 1%(0.01) for a total maximum drift of 2%. The parameters a (Equation 3.15) and α (Equation 3.29) were found to be 0.83 and 1.73, respectively. With these parameters, the design base shear coefficient V / W was calculated to be Design steps and detailed calculations are presented in Appendix B. The member sizes of the redesigned frame are shown in Figure 3.21 along with the member sizes of the original frame for comparison. Redesigned Frame Original Frame Figure Member Sizes of the Original Frame and the Redesigned Frame.

101 79 The base shear coefficient computed by the proposed design procedure is 0.333, which is approximately three times the UBC-94 base shear coefficient for the same building (Chapter 2). Thus, the design lateral forces in the proposed design procedures are much larger making the relative effect of the gravity loads much smaller. Consequently, the gravity loads can be safety neglected during the design process. It is also noteworthy that, even though the proposed design method requires significantly larger design base shear, the total weight of the original frame and the total weight of the redesigned frame are almost equal (154.6 kips for the redesigned frame and kips for the original frame). This is because member sizes in most of the frames designed according to current code procedures are governed by drift requirements, making the member sizes relatively large regardless of the base shear. The member sizes of the frame designed by the proposed design method are smaller for beams but are larger for columns when compared to the member sizes of the original structure. This is a result of imposing the strong column-weak beam mechanism in the design process. In order to compare the response of the frame designed by the proposed design procedure to the response of the original frame, the series of nonlinear analyses carried out in Chapter 2 was repeated. The redesigned frame was modeled using the same assumptions as used in Chapter 2 for the original frame. A One-bay, five-story, model of the redesigned frame was prepared for inelastic static analysis and inelastic time history analysis. Inelastic static analysis was carried out by applying increasing lateral forces representing the inverted triangular distribution of the UBC design lateral forces. For the inelastic dynamic analysis, the frame model was subjected to the same ground motion records as used in Chapter 2. Figure 3.22 shows the base shear versus roof displacement pushover responses for the two frames. The sequences of inelastic activity in the two frames are shown for comparison in Figure The horizontal drift at the UBC design lateral force level of the original frame satisfied the UBC limit. The drift for the redesigned designed frame,

102 80 however, was above the UBC limit. Both frames possess significant overstrength above the UBC design force level -- six times for the original frame and four times for the redesigned frame. The sequences of inelastic activity of the two frames under increasing static lateral forces were significantly different. In the original frame, as mentioned in Chapter 2, the first set of plastic hinges to form was at the base and the yield mechanism was a soft story in the first story. Both of these are not considered as good behavior. The redesigned frame, on the other hand, behaved as expected, following selected strong column-weak beam mechanism. All plastic hinges occurred only in the beams and in the column bases with the later forming last, a major improvement over the original frame where the hinges at column bases formed first. Base Shear Coefficient (V/W) Mechanism 1 First Plastification 1 First Plastification UBC DESIGN V = 0.09 W Original Redesigned 4 Mechanism Roof Drift (%) Figure Base Shear versus Roof Drift Response of the Original and the Redesigned Frames.

103 Original Redesigned Figure Sequences of Inelastic Activity under Increasing Lateral Forces. Some selected results from the inelastic dynamic analyses of the redesigned frame are presented in Figures 3.24 and Figure 3.24 shows the maximum story drifts of the redesigned frame as well as those of the original frame from Chapter 2. Figure 3.25 shows the locations of inelastic activity of the redesigned frame under selected ground motions along with the ductility demands at plastic hinges. The maximum story drifts of the redesigned frame, shown in Figure 3.24, are generally smaller than those of the original frame. More significantly, the original structure developed a soft story mechanism, as mentioned in Chapter 2, but the redesigned frame did not. The maximum drifts of the redesigned frame agreed well with the target design limit of 2%, as expected. Moreover, the inelastic activity in the redesigned frame was much better controlled and limited to the locations as intended in the design. Another significant observation was that the rotational demands at the base of the redesigned frame were significantly smaller than those in the original structure. Hence, the chance of premature failure at the column base in the redesigned frame was much smaller.

104 Story 3 2 El Centro Sylmar Newhall Synthetic 3 2 El Centro Sylmar Newhall Synthetic Story Drfit (%) Original Frame Story Drfit (%) Redesigned Frame Figure Maximum Story Drifts of the Original and the Redesigned Frames. (1.49) (1.49) (1.03) (1.03) (2.47) (2.47) (1.83) (1.83) (1.42) (1.42) (1.51) (1.51) (1.87) (1.87) (2.46) (2.46) (2.17) (2.17) (2.17) (2.17) (2.05) (2.05) (2.48) (2.48) (2.33) (2.33) (2.98) (2.98) (1.74) (1.74) (2.17) (2.17) (3.29) (3.29) (2.07) (2.07) (2.67) (2.67) (1.04) (1.04) El Centro Newhall Sylmar Synthetic Note: Rotational Ductility Demands Shown in Parentheses Figure Location of Inelastic Activity under the Four Selected Earthquakes. The results demonstrate that, even though the story drifts under static lateral forces do not satisfy the drift criteria prescribed in the UBC, as seen in the frame designed by the proposed method, the response under dynamic loading can be significantly better. This is because the inelastic activity occurs in a control manner following a desired yield mechanism.

105 Comparison of Design Forces The calculated design base shear coefficients for one-bay frame models with two, four, six, eight, and ten stories, based on the UBC-94 spectrum and a 2% target drift, are shown in Figure Also shown in the figure are the design base shear coefficients from UBC-94 and UBC-97. The UBC-94 design base shear coefficients were calculated based on Z = 0. 4, I = 1. 0, and S = They were multiplied by a factor of 1.4 to represent strength design levels. The UBC-97 design base shear coefficients were computed assuming that both the redundancy/reliability factor and the near field factor were equal to one. Design Base Shear Coefficient Number of Stories Proposed 1.4 UBC-94 UBC-97 Figure Comparison of Design Base Shear Coefficients As can be seen, the UBC-94 and UBC-97 design base shear coefficients are approximately the same. On the other hand, the design base shear coefficients based on the proposed method are about three to four times larger. This comparison shows that the

106 84 R w factor in UBC-94 or R factor in UBC-97 are unreasonably high. Structures designed using the UBC-94 and UBC-97 design base shear coefficients would likely experience large displacements during a design level earthquake. It should be noted that the final strength of structures designed using the UBC-94 and the UBC-97 is likely to be greater than the specified base shears. This is because of the drift criteria and the larger values of redundancy/reliability and near filed factors. Nevertheless, the comparison suggests that the UBC design procedure can be improved significantly by simply using smaller values of the current R (or R w ) factor, approximately one-third, and changing from an elastic design approach to a plastic design approach. As mentioned in Chapter 2, the response modification factor R (or R ) is the most controversial factor in the current design process. To date, no simple procedure to evaluate these factors exists. However, it is clear that the use of a constant response factor for all moment frames independent of their periods is not reasonable. The proposed design method is a direct design method where the need for response modification factor R is completely eliminated from the design process. w 3.8 PERFORMANCE-BASED PLASTIC DESIGN In recent years, a new design philosophy for building codes has been discussed among the engineering community. New generation of design codes for earthquakes are moving toward performance-based design framework [Vision ]. The goal of any performance-based design procedure is to produce structures that have predictable seismic performance. Within the context of performance-based design, a structure is designed such that, under a specified level of ground motion, the performance of the structure is within prescribed bounds. These bounds depend mainly on the importance of the structures.

107 85 Based on the Vision 2000 document [Vision ], performance of structures is categorized in four categories, which are 1) Fully Operational, 2) Operational, 3) Life-Safe, and 4) Near Collapse. These performance levels are to be selected corresponding to different design earthquake levels, which depend on the frequency of occurrence. SEAOC recommends four levels of design earthquakes, which are; 1) Frequent earthquakes that have 50% of probability of exceedance in 30 years, 2) Occasional earthquakes that have 50% of probability of exceedance in 50 years, 3) Rare earthquakes that have 10% of probability of exceedance in 50 years, and 4) Very rare earthquakes that have 10% of probability of exceedance in 100 years. The performance objectives can be best summarized in Figure The drift and yield mechanism based method developed in this study offers an opportunity to design a structure within a performance-based framework. Earthquake Performance Near Collapse Basic Unacceptable Life Safe Performance Basic Essential Operational Basic Essential Safety Critical Fully Operational Basic Essential Safety Critical Frequent Occasional Rare Very Rare Earthquake Design Level Figure Recommended Performance Objectives, Adapted from [SEAOC 1995]. The qualitative definition of performance objectives mentioned earlier is quite an open question at present. Many response parameters can be used to measure performance. Possible parameters include ductility demands, damage indices, and story drifts. Even

108 86 though, SEAOC suggests possible values of transient and permanent drift levels for each performance objective, these two parameters vary depending on the structural system. The exact quantitative measurement of performance is not within the scope of this study. In this study, drift levels are used as a measure of performance, based on the underlying assumption in the UBC. Under a design level earthquake, the maximum story drift should be in the order of 2-2.5%, according to the drift limits given in the UBC-97. Based on this assumption, the drift levels corresponding to each performance criterion are proposed in Table 3.5 and are used as examples in this study. It should be noted that the suggested drift levels are not based on any theoretical basis and are used only to illustrate the performance-based application of the proposed design method. Similarly, earthquake design levels are used based on Housner s intensity (Equation 2.14) presented earlier. Table 3.6 shows a possible scenario between design earthquake levels and Housner s earthquake intensity. Based on Tables 3.5 and 3.6, the performance-based design space can be quantified as shown in Figure Table 3.5. Performance Criteria. Performance Maximum Story Drift Fully Operational 0%-1% Fully Operational-Operational 1%-2% Operational-Life Safe 2%-3% Life Safe-Near Collapse 3%-4% Table 3.6. Earthquake Design Levels. Performance Intensity Frequent 0-0.5UBC Frequent-Occasional UBC Occasional-Rare UBC Rare-Very Rare UBC

109 Basic Interstory Drift (%) Basic Essential Basic Essential Safety Critical Basic Essential Safety Critical 0.0 0UBC 0.5UBC 1.0UBC 1.5UBC 2.0UBC Housner s Intensity Note: UBC Spectrum Intensity (Soil Type S3) = g-sec 2 Figure A Possible Quantification of the Performance-Based Design Space. For a given structure, the design base shear can be computed using a combination of interstory drift and earthquake intensity from Figure To illustrate a performancebased design procedure, the same 5-story moment frame used in Section is taken as an example. The design base shears were calculated, using the procedure proposed in this study, based on the combination of the following parameters: Target Drift = {0.5, 1, 1.5, 2.0, 2.5, 3, 3.5}% Earthquake Intensity = {0.5, 1.0, 1.5, 2.0}UBC After the values of design base shear for all combinations were calculated, a contour plot of equal design base shears were obtained. The contour plot of the design base shears for this particular frame is shown in Figure The contour lines in the figure can be used to select the design base shear level required to satisfy a given performance objective. Note that the yield drift of 1% is assumed in the design process. The base shear for the story drift of 0.5% is calculated using Equation 3.31.

110 88 Basic Essential Safety Critical V/W= Interstory Drift (%) UBC 0.5UBC 1.0UBC 1.5UBC 2.0UBC Earthquake Intensity Figure Design Base Shear for Different Performance Objectives. It should be noted that Figure 3.29 is unique for a given moment frame. It readily gives the minimum design value for a particular performance objective. For example, if the structure is to be designed for the essential class of structures, the minimum design base shear would be in a range of 0.6 to 0.8. On the other hand, the optimal design for the basic performance objective corresponds to a design base shear in the range of As can be seen, the proposed design method presents a direct link between the design objective and the design base shear. This direct link provides an obvious advantage over conventional design methods. 3.9 SUMMARY AND CONCLUDING REMARKS A new design procedure for steel moment frames was presented and discussed. The new design concept is based on plastic (limit) design theory. The design forces are

111 89 derived using the principle of energy conservation. In this proposed design method, the story drift is directly specified as a design parameter and no explicit checks for drift criteria under design forces are required. Also, the response modification factor ( R or R w ) is also completely eliminated, making the design procedure consistent. Nonlinear dynamic analysis was used to verify the proposed method. The results show that the proposed method can produce structures that meet a preselected performance objective. The implications of the proposed method were also presented. The major findings in this chapter are: (1) The use of plastic design principles in combination with the proposed design forces leads to structures with better seismic response. The results of a parametric study show that the proposed method produced structures with story drifts that comply well with the target drift values. The results show that the proposed method works particularly well for medium-rise structures. The proposed method, however, tends to produce overdesigned high-rise structures and under-designed low-rise structures. Another significant observation was that the estimated period values given by the UBC were too small for low-rise frames. Care should be taken when design such structures. (2) Comparing with a structure designed using a conventional method, a structure designed using the proposed method has relatively smaller beam sizes and larger column sizes. The total weights of the structures design using both methods are similar. The seismic responses of the two structures, on the other hand, are not. The sequences of inelastic activity of the two frames under increasing static lateral forces are drastically different. In the conventionally designed frame, the first set of plastic hinges to form was at the column bases and the yield mechanism was a soft story in the first story. The redesigned frame, on the other hand, behaved as expected, with the selected strong column-weak beam mechanism. All plastic hinges were limited to only the beams and at the column bases, the later forming last.

112 90 The results from dynamic analyses also showed a similar trend. The maximum story drifts of the frame designed using the proposed method were consistently smaller than those of the original frame. More significantly, the soft story mechanism was eliminated. The maximum drifts of the redesigned frame agreed well with the target design limit. In addition, the inelastic activity in the frame designed with the proposed method was much better controlled and limited to the locations as intended in the design. The plastic rotation demands at the base of the redesigned frame were much smaller compared to those in the original structure. (3) The use of elastic drift limit without considering the response at the ultimate level is not quite meaningful for seismic design. It was shown that, even though the story drifts under static lateral forces do not satisfy the drift criteria prescribed in the UBC, the response under dynamic loading can be significantly better if the inelastic activity occurs in a control manner, following a desired yield mechanism. (4) By comparing the design base shear coefficients required by the proposed method and those required by the UBC-94 and UBC-97, it was found that the required design base shears from the UBC-94 and UBC-97 were far too small. This suggests that the values of the response factors, R in the UBC-97 and R w in the UBC-94, are unrealistically large. More appropriate values should be about 3 to 4 times smaller than those currently used. (5) The proposed method can be easily presented in a performance-based design framework. The performance objectives can be defined based on the earthquake intensities and interstory drift levels. An optimal design base shear corresponding to a selected performance objective can be readily and directly obtained using the proposed design approach. The methodology presented thus far is a purely deterministic procedure. A probabilistic approach can be incorporated into the proposed design process especially in a performance-based design framework. Calibration of some important factors, especially

113 91 the p factor mentioned in Section 3.4.2, could improve the design significantly. However, probabilistic study was not within the scope of this research.

114 CHAPTER 4 SEISMIC UPGRADING OF MOMENT FRAMES USING DUCTILE WEB OPENINGS 4.1 INTRODUCTION Moment-resisting steel frames have long been regarded as one of the best structural systems to resist seismic forces. The performance of such frames under seismic forces depends primarily on the strength and ductility of their beam-to-column connections. Unfortunately, a large number of beam-to-column connection failures were observed after the 1994 Northridge Earthquake and the 1995 Kobe Earthquake. These failures clearly show that typical beam-to-column moment connections possess far less ductility than expected. Notwithstanding the question of ductile performance of connections, it was shown in Chapter 2 that moment resisting frames designed by elastic method using equivalent static forces may undergo inelastic deformations in a rather uncontrolled manner, resulting in uneven and widespread formation of plastic hinges. Thus, combined lack of ductility of the connections and the use of elastic design method could hold a major key in explaining the recently observed poor performance of steel moment frames. The drift and yield mechanism based design approach presented in Chapter 3 can be utilized to design new moment frame structures so that they will behave in a controlled and preferred manner. However, there is an urgent need for methods to retrofit and upgrade the existing moment-resisting steel frames which were designed using the pre-1994 practice. Two approaches are being employed in current design practice and 92

115 93 research studies to upgrade such frames: (1) A strengthening strategy, where the beam-tocolumn connection is reinforced to meet the strength and ductility demand [Lee et al.1997, Engelhardt and Sabol 1998]; (2) A weakening strategy, where the beam is weakened (away from the connection) in order to create a fuse that limit the force demand on the connection [Chen et al. 1997]. The strengthening strategy requires checking the adequacy of columns and other critical regions of the frame for the increased force demands. For this reason, weakening strategy (such as the dog bone solution) is becoming increasingly popular. As part of the weakening strategy, one possible scheme to modify the behavior of moment resisting frames to have ductile yield mechanism is to create rectangular Vierendeel openings in the girder web near the middle of the span. The shear capacity of the openings can be increased, if needed, by adding diagonal and vertical members into the openings to provide additional stiffness to the frame. The openings are designed so that, under a severe ground motion, the inelastic activity will be confined only to yielding, and buckling of the diagonal members, and the plastic hinging of the chords of the opening while other members in the frame remain elastic. This chapter presents the experimental and analytical development of the ductile opening system. Results of reduced-scale experiments are presented. Based on the results of these experiments, behavior of key members of the frame is discussed. 4.2 CONCEPT OF MOMENT FRAMES WITH WEB OPENINGS The concept of using openings as ductile segments is derived from a structural system known as Special Truss Moment Resisting Frame (STMF). This structural system has been studied both analytically and experimentally by Goel et al. [Itani and Goel 1991, Basha and Goel 1994] at the University of Michigan during the past ten years and has been recently incorporated into the 1996 UBC Supplement [UBC 1996] and the AISC- LRFD seismic provision [AISC 1997]. The system consists of truss frames with special

116 94 segments designed to behave inelastically under severe ground motions while other structural members of the frame remain elastic. The special segments in STMF structures can be either Vierendeel openings or Vierendeel openings with X-diagonal members, depending on the desired level of shear strength of the special panels. When a STMF structure is subjected to lateral forces induced by an earthquake, the shear forces in the floor girders are resisted by the chord members and the X-diagonals in the openings. After yielding and buckling of the diagonal members, plastic hinges will form at the ends of the chord members. After the openings in all floor girders have yielded, complete mechanism forms when additional plastic hinges occur at the column bases of the frame. From the results of extensive analytical and experimental studies, STMFs have been shown to have excellent hysteretic behavior under severe seismic forces and perform well when compared to conventional open web and solid web framing systems. Excellent energy dissipation, smaller story drifts, and smaller base shear were observed in STMF system. Full-scale tests have also shown that STMFs possess excellent energy dissipation and can sustain large cyclic displacements. The idea of STMF structural system originated from a study of cyclic behavior of the conventional open web frames. Itani and Goel [Itani and Goel 1991] carried out fullscale tests of open web frames and observed that they exhibited hysteretic response with rapid degradation due to buckling in major load carrying members. Guided by the experimental results, they developed the STMF framing system. A design procedure that explicitly accounts for the inelastic distribution of internal forces in the special segments was also developed. In this design approach, the special segments are designed first. Other structural elements are subsequently designed to remain elastic under the forces generated by the fully yielded and strain hardened special segments. Using this design approach, two full-scale special truss moment frames were designed and tested. These test frames behaved as intended and demonstrated very ductile cyclic response. Analytical study was also carried out to examine the seismic behavior of STMF framing

117 95 system. The results showed that STMF structures respond to seismic forces in an excellent manner. Based on the results of Itani and Goel s STMF study, Basha and Goel [Basha and Goel 1994] developed STMF with ductile Vierendeel opening segments. This type of special segments eliminates the use of diagonal members, therefore, only the chord members of the special opening segments are designed to resist the applied shear force. A mathematical expression to calculate the maximum force generated by a fully yielded and strain hardened opening was developed. Two full-scale test specimens, one with simulated gravity loads and one without gravity loads, were designed and tested in that study. The results were very satisfactory. The hysteretic response of the tested STMFs was very stable and ductile. The pinching phenomenon, typically found in the hysteretic response of truss-like structures, was completely eliminated. An analytical study was carried out to verify the design procedure and the seismic response of STMF structures. These results were also very satisfactory. In moment frames with web openings, the openings serve the same function as the special segments in STMF system. Under a severe ground motion, the inelastic activity will be confined only to the openings. Mainly, the inelastic activity consists of yielding and buckling of diagonal members and plastic hinging of the chord members of the openings. In this proposed system, the chord, diagonal, and vertical members should be designed such that, under their fully yielded and strain-hardened condition, the moment at every beam-to-column connection generated by the shear force in the opening will be smaller than a chosen critical value. This critical value is selected so as to reduce the risk of premature failure of connections and confines all inelastic activity only to the openings. Figure 4.1 shows a moment frame modified with girder web opening and a STMF at the ultimate (mechanism) state.

118 96 E a) Special Truss Moment Frame. E b) Moment Frame with Girder Web Opening. Figure 4.1. Yield Mechanism of Special Truss Moment Frame and Moment Frame with Girder Web Opening.

119 TESTING OF STEEL BEAMS WITH OPENINGS In order to study the viability of the mentioned scheme, five, approximately halfscale, specimens representing girders with an opening in moment frames were prepared for cyclic tests. The main objective of the test program was to obtain the information needed for the development of the design procedure and the analytical modeling of the proposed structural systems. This information includes: 1) The behavior of the opening in both the elastic range and the inelastic ranges; 2) The yield mechanism and the overstrength of the key members; 3) The best detailing scheme to meet the ductility demand; 4) The reparability of the opening after being subjected to severe deformation. Each specimen was fabricated differently to study various aspects of the proposed upgrading system. The first and the second specimens were tested in order to obtain the information about the overall behavior of the proposed upgrading scheme and also about the stiffness of the support frame. The third test was done to verify a detailing scheme around the opening region to meet the required ductility demand. The fourth test was aimed at studying the behavior of an upgrading scheme using a Vierendeel type opening. The fifth test was aimed at studying the reparability of the special opening after a severe deformation history Test Set-Up Each specimen was made to represent a girder with a special opening. Each test specimen was approximately half-scale. Postulating the anti-symmetric behavior of girders in moment frames, all specimens were made as half-span models. The test specimens were mounted on a support frame, which consisted of 2C15x50 beams. The specimens were braced against lateral torsional buckling using two lateral supports at a location below the openings.

120 98 The shear load was applied using a hydraulic actuator with 50 kips capacity and ± 5 inches stroke length. The actuator was connected between the tip of the test specimen and a reaction wall. The force generated by the actuator represents the shear force induced by an earthquake at the mid-span of a girder. A schematic diagram of a typical test set up is shown in Figure 4.2. A photograph of a typical test set-up is shown in Figure 4.3. Figure 4.2. Schematic Diagram of a Typical Test Set-Up.

121 99 Figure 4.3. Typical Test Set-Up Instrumentation and Test Procedure The specimens were loaded following a cyclic displacement pattern, consisting of cycles of increasing displacement magnitude. For each specimen, the magnitude and the direction of the applied displacement were selected based on the size and shape of the specimen. The tests were stopped when a considerable reduction in strength and stiffness of the specimen was observed. The hysteretic response of each specimen was obtained from the load cell and the displacement transducer in the hydraulic actuator. Additional data on various key members of the specimens were obtained using electrical strain gauges. Two potentiometers were mounted to the support frame to measure the base rotation.

122 Material Properties The specimens were made from dual grade W18x40 steel beams. In some specimens A36 steel bars were used for diagonal members in the special openings. The actual material properties were obtained from tensile tests of coupons from various parts of the test specimens. For a given specimen, an average yield stress of coupons from the flanges and coupons from the web was used to represent the yield stress of that specimen. The average yield stress values of various key members are given in Table Specimen 1 Table 4.1. Average Yield Stress of Key Members. Coupon Specimen Yield Stress (ksi) W18x40 1, W18x40 3,4, x 3 / 8 PL x 1 / 4 PL. 2,3, / 8 x 1 / 4 PL L1 1 / 4 x 1 1 / 4 x 1 / 4 2,3,4, L1 1 / 2 x 1 1 / 2 x 3 / L2 1 / 2 x 2 1 / 2 x 3 / Specimen 1 was designed to gather information about the basic behavior of the proposed upgrading scheme as well as to gather the information about the test set-up support frame. The most important objective was to verify the yield mechanism of the opening. The opening was designed based on the information obtained from tests of special truss moment frames. It was created by removing the material in the web using flame cutting torch. Then, the cut surfaces were smoothed using a steel grinder. Specimen 1 is shown in Figure 4.4 along with the details in the vicinity of the opening. The size of the opening was approximately 16 inches by 13.5 inches leaving about 2.25 inches for each of the chord members. The diagonal members were made from 2, 1inch by 3/8 inch, flat bars.

123 101 The test of the first specimen was done in two phases. In the first phase, the first loading history was applied to investigate the combined yield mechanism of the diagonal and the chord members. The maximum applied drift was 1.75%. Then, the diagonal members were removed by flame cutting and the second loading history was applied in the second phase of the test. The maximum applied drift was 2.75%. This was done in order to investigate the behavior of the chord members alone without the diagonal members. The two loading histories are shown in Figures 4.5 and 4.6. In Figures 4.5 and 4.6, the drift ratio is the ratio in percent of the applied displacement to the distance from the tip of the beam to the centerline of the support frame. Figure 4.7 shows the specimen before the diagonals were removed and Figure 4.8 shows the specimen after the diagonals had been removed. Figures 4.9 and 4.10 show the hysteretic loops from the first and the second phases of the test, respectively. In Figures 4.9 and 4.10, the applied displacements have been corrected to obtain equivalent fixed-end displacements by subtracting the rigid body displacements caused by the rotation of the support frame. Figure 4.4. Test Specimen 1. Note: Dimensions in inches

124 102 Drift (%) Cycle Figure 4.5. Loading History 1 of Specimen 1. Drift (%) Cycle Figure 4.6. Loading History 2 of Specimen 1.

125 103 Figure 4.7. Specimen 1 before Removal of Diagonal Members. Figure 4.8. Specimen 1 after Removal of Diagonal Members.

126 Force (kips) Corrected Displacement (in.) Figure 4.9. Hysteretic Loops of Specimen 1 with Diagonal Members Force (kips) Corrected Displacement (in.) Figure Hysteretic Loops of Specimen 1 without Diagonal Members.

127 105 In the first phase of the test, Specimen 1 with diagonal members produced a very stable hysteretic response. Some local yielding was observed prior to buckling of the diagonal members at the corrected drift of approximately 0.6%. Yielding in the chord members and in the horizontal member followed the buckling. Large degree of strain hardening was observed, as can be seen by the steep rise of the force-displacement response after yielding. Yielding and buckling in the first phase of the test are shown in Figure Some important observations from the first phase of the test were: The vertical member (horizontal member in the test set-up) suffered significant yielding. Although it can be shown that the contribution of this member to the overall shear resistance can be neglected, the result suggests that the compactness ratio of this member should be large enough to prevent its premature fracture. The buckling load of the diagonal members was much larger than expected. This suggested that the value of effective length factor was much lower than initially assumed. From an analysis of the test data, it was found that the appropriate effective length factor was in the order of 0.8 of the clear length of the braces. The results of the first phase were very encouraging. The expected yield mechanism formed, with inelastic activity confined in a few key locations only. In the second phase of the test, the diagonal members were removed, as mentioned earlier. The response was very stable at first, with yielding at the ends of the chord members only. Yielding was mainly concentrated above the brace-to-chord junctions. Plastic hinges were clearly seen before cracking occurred in a chord member right above the location where the diagonal member used to be. Cracking resulted in a reduction in the load carrying capacity, as can be seen in Figure This early fracture was probably due to high stress concentration at the corners of the opening. Yielding of the chord members in the second phase of the test is shown in Figure The cracking in the chord member is shown in Figure 4.13.

128 106 Figure Yielding and Buckling in Specimen 1 with Diagonal Members.

129 107 Figure Yielding in Specimen 1 without Diagonal Members. Figure 4.13.Cracking of the Chord Member.

130 Specimen 2 Specimen 2 was designed in a similar fashion as Specimen 1. Only a few changes were introduced in this specimen. Major modifications included the use of a more compact vertical member and the use of smaller diagonal members. The vertical member was more compact as to prevent premature fracture that occurred during the previous test. The size of the diagonal members was reduced because the results from the previous test also suggested that the capacity of the actuator might not be sufficient to impose a drift of 3% if the member sizes remained the same as in Specimen 1. Specimen 2 is shown in Figure The loading history for Specimen 2 is shown in Figure The response of Specimen 2 is shown in Figure Note: Dimensions in inches Figure Test Specimen 2.

131 109 Drift (%) Cycle Figure Loading History for Specimen Force (kips) Corrected displacement (in.) Figure Hysteretic Loops of Specimen 2.

132 110 The response of Specimen 2 was stable, similar to that of the Specimen 1. The first inelastic activity occurred in the diagonal members when one of them buckled at the corrected drift of approximately 0.3%. Following the buckling of both diagonal members, yielding in the vertical member was observed. Finally, yielding of the chord members completed the yield mechanism. As the test progressed, severe deformation of the diagonal and vertical members was clearly seen, as shown in Figure Nevertheless, the load carrying capacity continued to rise. At the drift of 1.6%, small cracks developed in both the chord and the vertical members. As the test continued, these cracks propagated until they reached the flanges of the chords, as can be seen in Figure Although the crack in the vertical member continued to grow, it did not compromise its ability to transfer the applied shear force. Overall, the performance of Specimens 1 and 2 was not satisfactory. Although both specimens could maintain their load carrying capacity, it was clear that the detailing scheme used could not sustain the high ductility demands at the critical locations. Figure Yielding and Buckling in Specimen 2.

133 Specimen 3 Figure Cracking in the Chord Member of Specimen 2. The goal of Specimen 3 was to verify if an alternative detailing scheme could sustain large ductility demands at the critical sections. The size of the opening was larger than those in Specimens 1 and 2 to reduce the ductility demand. It has been shown by Basha and Goel [Basha and Goel 1994] that the ductility demand is inversely proportional to the square of the size of the opening. Results form tests of special truss moment frames and tests of Specimens 1 and 2 suggest that the size of the opening in the order of 20% of the span length is most desirable. Hence, the size of the opening in Specimen 3 was set at about 20% of the span length, if it is taken as the distance from the tip of the specimen to the center of the support frame. The detailing scheme near the corners had also been improved. Schematic drawing of Specimen 3 is shown in Figure The opening of Specimen 3 is shown in Figure The loading history is shown in Figure 4.21 and the response under this loading history is shown in Figure 4.22.

134 112 Note: Dimensions in inches Figure Test Specimen 3. Figure The Opening in Specimen 3.

135 113 Drift (%) Cycle Figure Loading History for Specimen Force (kips) Corrected Displacement (in.) Figure Hysteretic Loops of Specimen 3.

136 114 The response of Specimen 3 was excellent. The hysteretic loops were very stable, as can be seen from Figure Similar to previous specimens, the inelastic activity started with buckling of the diagonal members. Yielding of the chord and the vertical members was observed in subsequent cycles. At the corrected drift of 2%, small cracks developed at the ends of the vertical member, however, the load carrying capacity remained unaffected. In later cycles, local buckling occurred at the end of the chord members. This local buckling resulted in a small reduction in the load carrying capacity. No fracture was observed in the chord members at the end of the loading history. Fracture eventually developed after additional small displacements cycles, which were applied to the specimen after the loading history shown in Figure 4.21 was complete. Additional loading was applied in order to observe the failure mode only. No significant data were recorded during these additional cycles. The deformation of the test specimen is shown in Figures 4.23 and Local buckling and the deformed shape of the diagonal members are shown in Figure Figure Deformation of the Test Specimen 3 (Positive Direction).

137 115 Figure Deformation of the Test Specimen 3 (Negative Direction). Figure Local Buckling of Chord Members.

138 116 Specimen 3 clearly shows that it is possible to detail the opening to meet the high ductility demand. Subsequent analysis shows that the ductile behavior of the opening depends on both the geometry and the local detailing of the opening. An important factor that influences local ductility is the geometry of the welds around the corners. It is important to provide some free distance from the ends of welds to the edges of the chord members, as can be seen in Figure 4.20, to allow plastic deformation of the material near the edges. Such gaps should be maintained at all weld locations Specimen 4 Specimen 4 was designed to study if a Vierendeel opening can be used as an alternative to the X-braced openings. This type of openings may be useful as it provides space for ducts. The disadvantage in this case is that the chord members must provide all the shear resistance. For this reason, additional plates were needed to reinforce the chord members of Specimen 4. The additional plates were fully welded along the length of the chord members. The details of Specimen 4 are shown in Figure 4.26 and The loading history for Specimen 4 is shown in Figure The hysteretic response is shown in Figure Specimen 4 did not perform as well as the other specimens. The specimen failed prematurely at one of the corners of the opening. The additional plate suffered severe local buckling at a very early stage of the test. The strength of the opening reduced suddenly after cracking since this type of opening has very little redundancy in its load carrying mechanism. This response is different from the ductile behavior observed in special truss moment frames with Vierendeel opening. This is probably due to stress concentration at the corners of the opening. Local buckling and fracture of Specimen 4 are shown in Figure 4.30.

139 117 Note: Dimensions in inches Figure Test Specimen 4. Figure A Close-Up View of Specimen 4.

140 118 Drift (%) Cycle Figure Loading History of Specimen 4. Force (kips) Corrected Displacement (in.) Figure Hysteretic Loops of Specimen 4.

141 119 Figure Local Buckling and Fracture of Specimen Specimen 5 Specimen 5 was the last specimen in this series. The objective of Specimen 5 was to investigate the reparability of the opening after severe deformation. Previously tested Specimen 4 was moved from the test fixture and the chord members were removed by flame cutting. The chord members of the opening were then replaced by 4 angles, two for each chord member. New diagonal members were also added. This repair scheme offers an advantage that the chord members are continuous from the opening into the web of the girder making them behave similarly to the chord members in special truss moment frames. The detailed drawing of the fifth specimen and a close-up photograph are shown in Figures 4.31 and 4.32, respectively. The loading history for Specimen 5 is shown in Figure Its hysteretic response is shown in Figure The deformation of the specimen is shown in Figures 4.35 and 4.36.

142 120 Figure Test Specimen 5.

143 121 Figure Close-Up View of Specimen 5. Drift (%) cycle Note: Unsymmetrical Loading Due to Out-of-Plane Twisting Figure Loading History of Specimen 5.

144 Force (kips) Corrected Displacement (in.) Figure Hysteretic Loops of Specimen 5. Figure Deformation of the Specimen 5 (Negative Displacement).

145 123 Figure Deformation of Specimen 5 (Positive Displacement). Specimen 5 produced a very stable response under cyclic loading, similar to the response of Specimen 3. After buckling and yielding of the diagonal members, plastic hinges started to form in the chord members. The chord members of this specimen were very ductile, with plastic hinges well distributed along their length. In fact, the chord members of Specimen 5 performed better than the chord members of Specimen 3 because they were continuous far into the beam. The repaired specimen performed satisfactorily up to the drift about 2.5% when out-of-plane twisting of the opening started. Twisting intensified as the test continued. During the last few cycles, it was impossible to displace the specimen to the intended level. Therefore, only one-sided loading was applied, as can be seen in Figure It was decided to stop the test to prevent damage to the actuator. Specimen 5 shows that it is possible to replace the opening after severe deformation and damage, provided that out-of-plane buckling of the chord members is prevented. This can be done by providing lateral supports or keeping the size of the

146 124 opening sufficiently small. It can be noticed that the opening of Specimen 5 was considerably larger than those in previous specimens. The length of this opening was roughly 30% longer than that of the opening of Specimen 4. The results suggest that the length of the opening should be maintained at about 20% of the span length. Doing so provides sufficient stiffness and prevents excessive out-of-plane twisting. 4.4 ANALYSIS OF TEST DATA The results form this series of tests provide invaluable information on the behavior of the proposed upgrading scheme. As mentioned earlier, the goals of the proposed system are to confine all the inelastic activity to the openings as well as to reduce the strength demands at the beam-to-column connections. Therefore, the overstrength of the opening plays a crucial role in the design of the upgrading scheme. Any overstrength in the opening will result in the increase in strength demands in other members. It is necessary to accurately estimate the overstrength developed in the opening so that the ultimate strength of the opening can be predicted. Consequently, the demands at critical members can be controlled. The ultimate shear strength of an opening is the sum of the ultimate strengths contributed by the chord members, the diagonal members, and the vertical member. During the tests, it was observed that the reductions in the load-carrying capacity of the specimens after the cracking of the vertical members were considerably small. Therefore, by neglecting the contribution from the vertical member, the ultimate strength can be expressed as: where V 0 = ξ cvc + ξ xvx (4.1) V o is the ultimate shear strength of the opening, V c and Vx are the nominal shear strengths provided by the chord and diagonal members, respectively, and ξ c and ξ x are the overstrength factors for the chord and diagonal members, respectively.

147 125 As mentioned in Chapter 3, the overstrength occurs primarily from two sources: 1) The difference in the actual and the nominal strength; 2) Strain hardening of materials. In this proposed opening system, overstrength can be found in the diagonal members, the chord members, and the vertical member. However, only the diagonal members and chord members provide considerable resistance to the applied forces. Therefore, only overstrength of the chord and the diagonal members is of concern in this study Overstrength of the Diagonal Members In order to accurately estimate the shear strength of an opening, it is important to account for both the maximum probable tensile strength and the probable post-buckling strength of the diagonal members. During the tests, it was observed that the diagonal members exhibited uniform yielding along their length. Therefore, the overstrength due to strain hardening would not be significant. Strain hardening is typically more important when the yielding is concentrated, such as in plastic hinge regions. For the diagonal members, the difference in the actual yield stress and the nominal yield stress dominates the overall overstrength of the members. Maximum tensile strength of the diagonals can, therefore, be estimated as the product of the overstrength factor due to the difference between the actual and nominal yield stresses and the nominal tensile strength. It has been shown [Itani and Goel 1991] that flat bars exhibit large ductility capacity. Therefore, it is recommended that this kind of structural members be used as the diagonal members. Typically, flat bars are made of A36 steel. The expected yield stress for A36 is on the order of 49 ksi. Hence, the overstrength factor ξ x can be taken approximately as: ξ 49 / (4.2) x The nominal post-buckling strength of the diagonal members is 0.3 of the nominal buckling load as suggested in the AISC seismic provisions for STMFs [AISC

148 ]. As mentioned earlier, test results show that the approximate buckling load can be found by assuming the effective length factor k of With these effective length and the post-buckling strength factors, the shear contribution of diagonal members in the opening can be estimated as: V x = ξ ( P + 0.3P ) sinθ (4.3) x xy xc x where ξ x is the overstrength factor (1.40), P xy is the nominal yield force of the diagonal member, P xc is the nominal buckling force of the diagonal member, and θ x is the angle between the diagonal and the chord members Overstrength of the Chord Members An expression for the overstrength factor for the chord members was first proposed in the study of special truss moment frames by Basha and Goel [Basha and Goel 1994]. This overstrength factor is a function of the length of the opening, the section properties of the chord members, and the material properties. The overstrength factor can be expressed as: ξ c L L δη( 6EI ) + (1 η) M 2 o c ch L o = (4.4) M ch where δ is the story drift, η is the strain hardening factor, E is the young modulus, I c is the moment of inertia of the chord member, L is the span length, L o is the length of the special segment, and M ch is the plastic moment of the chord member. In Equation 4.4, all variables can be readily calculated except the strain hardening factor η. Basha recommended that the value of the strain hardening factor, η, may be taken as 0.05, if the plastic moment is calculated by using the actual yield strength, or it may be taken as 0.10, if the plastic moment is calculated by using the nominal strength.

149 127 However, these values were based on built up chord angle members and might not be applicable to wide-flange members. Numerical simulations were carried out to calibrate the strain hardening factor for wide-flange beam sections. Specimen 1, after the diagonal members had been removed, was used as the model. Static pushover analyses were carried out to find the envelopes of the force-displacement response corresponding to different values of the strain hardening factor. The objective was to compare these envelopes with the test results. The specimens were modeled in SNAP-2DX [Rai et al. 1994] using beam-column elements. The simplified model consisted of four beam-column elements representing the W18x40 beam, the chord members and the vertical member. The chord members were connected to the element representing the W18x40 beam by means of rigid links, which had the same dimension as the opening. All elements were modeled using the actual yield strength obtained from coupon specimens. Two values of strain hardening, 5% and 10%, were used. The hysteretic loops of Specimen 1 in the second phase of the test, before cracking occurred, are compared with the results from computer analyses in Figure Force (kips) Experiment 10% 5% Corrected Displacement (in.) Figure Comparison of Strain Hardening Values.

150 128 As can be seen, the strain hardening value of 10% gives a better approximation of the ultimate shear strength. It can also be noticed that the overstrength due to this strain hardening alone is considerably large. Contrary to the diagonal members, the overstrength due to the difference between the actual yield stress and the nominal yield stresses in the chord members is not significant. The results from pushover analyses using the actual yield stress and two assumed values of nominal yield stress are compared with the response from the experiment in Figure It can be seen that the difference in the ultimate shear strength from the experiment and the ones computed using the nominal values are insignificant. The result suggests that the nominal yield stress value can be used in calculating the ultimate shear strength of the opening provided that a proper value of strain hardening is used. Considering these two factors, it is recommended that ultimate shear strength can be computed using the nominal yield stress and the strain hardening value of 10% Experiment Fy = 51.5 Fyn = 50 Fyn = 36 Force (kips) Corrected Displacement (in.) Figure Comparison of Yield Stresses.

151 129 It was also found form post-experiment investigations that the plastic hinges in the chord members of most specimens were located at a small distance away from the edge of the opening. Therefore, in calculating the ultimate shear strength of an opening, it is recommended that the length of the opening be taken as 0.95 of the nominal length. Using all this information, the overstrength of an opening can be computed as: L 0.95L δ ( 0.10)(6EI ) + (1 0.1) M 2 o c ch (0.95Lo ) ξ c = (4.5) M ch or L 0.95L o 0.665δEI c M 2 ch L o ξ c = (4.6) M ch Ultimate Shear Strength of the Openings Figure 4.39 shows the equilibrium of the internal forces at the ultimate state of a frame modified with a web opening. The shear strength of the opening consists of the shear resistance contributed by the chords, the diagonals, and the vertical member. By assuming that the points of inflection of the chord members are at the middle of the opening and by neglecting the shear contribution from the vertical member, the vertical resultant force V o in the opening can be expressed as: V o 4( ξ cm ch ) = + ξ x ( Pxy + 0.3Pxc )sin( θ x ) (4.7) 0.95L o where ξ c and ξ x are the overstrength factors for the chord and the diagonal members, respectively, M ch is the plastic moment of T-section chord members, L o is the length of the opening, P xy is the yield force of the diagonal members, P xc is the buckling force of the diagonal members, and θ x is the angle between the diagonal and the chord members.

152 130 2ξ cm ch/0.95l o ξ cm ch θ x ξ xp xy 0.3ξ xp xc Resultant Force V o Using Equations 4.2, 4.6, and 4.7, the expected ultimate shear strength of Specimens 1 to 5 can be calculated. The shear forces contributed by the chord members (the first term on the right-hand side of Equation 4.7) in Specimens 1 to 5 are calculated and shown in Table 4.2. The shear forces contributed by the diagonal members (the second term on the right-hand side of Equation 4.7) in Specimens 1 to 5 are calculated and shown in Table 4.3. The calculated values of the ultimate shear strength, the sum of the shear contributed by the chord and the diagonal members, of Specimens 1 to 5 are compared with the attended loads from experiments in Table 4.4. In all calculations, it was assumed that the story drift for each specimen was approximately equal to the fixedξ cm ch 2ξ cm ch/0.95l o b) Equilibrium of Internal Forces BMD SFD a) Moment and Shear Force Diagrams Figure Equilibrium of Internal Forces in the Opening.

153 131 end drift from the experiment. The fixed-end drift was the ratio of the applied displacement to the distance from the tip of the beam to the center of the support frame. The nominal yield stresses were taken as 36 ksi for the diagonal members and as 50 ksi for the chord elements. The modulus of elasticity was assumed to be ksi. Lengths L and L o were taken as twice the specimen length and twice the opening length, respectively. This is because the test specimens were all half-span models. Table 4.2. Shear Force Contributed by Chord Members. Specimen δ I c L L o M ch ξ c V c (%) (in 4 ) (in.) (in.) (kip in) (kips) * * Note: ( * ) Drifts at first fracture. Table 4.3. Shear Force Contributed by Diagonal Members. Specimen L x A x θ x P xy P xc ξ x V x (in.) (in 2 ) (rad.) (kips) (kips) (kips) Table 4.4. Comparison between Expected and Experimental Ultimate Shear Strengths. Specimen V x V c Expected V Experiment * o Exp./ V o (kips) (kips) (kips) (kips) Note: ( * ) Average maximum positive and negative loads.

154 132 Table 4.4 shows that the expected values compare well with the results from experiments. For Specimens 1 and 2, the expected values are somewhat smaller than the actual values. This is probably because, in these specimens, the diagonal members were connected to the chord members inside the opening. The consequence was that the plastic hinges were pushed further into the openings making the length of the opening smaller and the strain-hardening rate larger. This finally resulted in additional shear forces. Although Equation 4.6 recognizes the rigid zones at the ends of the chord members by introducing a factor of 0.95, this value was calibrated primarily from Specimen 3 test results. Thus, the detailing scheme used in Specimen 3 is recommended for future use Modeling the Openings under Cyclic Loading Cyclic behavior of a beam with an opening can be captured by using a finiteelement based computer code that has the capability to model beam-column elements as well as axial compression elements. One of the issues involved in the modeling of any truss-like structure under cyclic loading is the modeling of the axial compression elements. Axial compression elements exhibit a complex post-buckling strength degradation pattern, which affects the overall hysteretic response of the structure. Pinching observed in the hysteretic loops of the test specimens is due primarily to the strength degradation after buckling and the increase in the member length after significant yielding [Jain et al. 1980]. In this study, test specimens were modeled using SNAP-2DX [Rai et al. 1997]. The axial compression elements in SNAP-2DX uses the Jain s hysteretic model [Jain et al. 1978], which is capable of modeling the post-buckling strength and stiffness as well as the elongation in the element. The model uses several straight-line segments depending on several control parameters in compression, but it is bilinear in tension. In compression, the compressive strength reaches the buckling strength in the first cycle. In subsequent cycles, the compressive strength reduces to the post buckling strength specified by a

155 133 strength reduction factor. The strength reduction factor of 0.3 has been found to correlate well with the results from experiments. In tension, the element yields at the yield strength of the element. After the yield point, the load carrying capacity remains at the same level without strain hardening. Jain s hysteretic model is shown in Figure An analytical model using the same modeling assumptions mentioned in Section was created to represent Specimen 3. Axial compression elements were used to represent the diagonal members and beam-column elements were used to represent the chord members and the beam. The model was subjected to the corrected displacement from the test. The results from the analysis are shown in Figures The figure shows that the behavior of a beam with an opening can be modeled very well by using the modeling assumptions presented earlier. Such modeling is essential for the seismic evaluation of the proposed upgrading scheme, which will be presented in Chapter 5. Tension P xy Displacement αp xc 5 y P xc y = Yield Displacement in Tension α = Strength Reduction Factor Compression Figure Axial Hysteretic Model for Diagonal Members [Jain et al. 1978].

156 Experimental Analytical Force (kips) Corrected Displacement (in.) Figure Analytical Modeling of Specimen SUMMARY AND CONCLUDING REMARKS An upgrading scheme for steel moment resisting frames was proposed in this chapter. This upgrading scheme consists of creating ductile rectangular openings in the middle of the beam web to control the yield mechanism of the frame. Five half-scale halfspan specimens were tested to study the behavior of each key member of the opening. The major findings in this chapter are: (1) In moment frames with beam web openings, the openings serve the same function as the special segments in STMF structures. Under a severe ground motion, the inelastic activity will be confined only in the opening region. It consists mainly of yielding and buckling of diagonal members and the plastic hinging of the chord members of the openings. In this proposed system, the chord, diagonal, and vertical members should be designed such that, under their fully yielded and strain-hardened condition, the moment at the every beam-column connection generated by the shear force in the

157 135 opening will be smaller than a critical value to reduce the risk of premature failure of connections. (2) Results of tests of beams with openings show that the proposed upgrading scheme is feasible. Specimens with proper detailing provided a stable hysteretic response. The inelastic activity was confined only in the designated locations as intended in the design. (3) Out of five specimens, Specimen 3 was the best. Its stable response was due to the proper detailing of critical locations. Special detailing should be provided to minimize stress concentration and also to move the locations of plastic hinges away from the critical corners. The detailing scheme used in Specimen 3 provides the needed ductility and it is also practical. The opening can be flame-cut but the surface of the cut close to the corners should be smoothed out by grinding. It is desirable to have a radius in all corners, although test results suggest that it may not be necessary. The vertical members that reinforce the ends of the opening should be placed at an offset of about 0.5 in. from the ends of the opening. This is done so that the plastic hinges are pushed away from the critical areas. The diagonal bars also help in reinforcing the critical areas. In addition, all welds should have at least 0.25 in. clearance from the edges to allow for plastic flow, thus increasing local ductility. (4) Shear in the openings is primarily resisted by the chord members and the diagonal members. The ultimate shear strength of openings can be predicted by multiplying the nominal shear strength of the chords and diagonals by the corresponding overstrength factors. The overstrength factor of the diagonal members can be primarily attributed to the difference in the nominal and the actual yield strengths. The overstrength factor due to strain hardening is not significant for diagonals. A value of 1.4 has been found to be a reasonably accurate value of the overstrength factor for the chord members.

158 136 Unlike the diagonal members, the overstrength of the chord members is dominated by strain hardening. The overstrength factor for the chord members is a function of the length of the opening, the section properties of the chord members, and the material properties. A strain hardening of about 10% has been found to correlate well with the experimental data. It is recommended that ultimate shear strength be computed using the nominal yield stress value with 10% strain hardening. (5) Openings can be easily replaced after severe deformation as shown by Specimen 5. The chord members of the opening can be replaced by angles, provided that the length of the opening is not too large. Lateral instability may occur if the chord members are not properly braced against lateral movement. This is also true for the original wide-flange beam openings before retrofitted. (6) Cyclic behavior of beams with openings can be modeled using a finiteelement based software, provided that proper hysteretic behavior of the axial compression elements is used. Axial compression elements exhibit a complex post-buckling strength degradation pattern, which affects the overall hysteretic response of the structures. An analytical model in Section was shown to capture the hysteretic behavior a beam with an opening very well.

159 CHAPTER 5 SEISMIC DESIGN AND BEHAVIOR OF MOMENT FRAMES WITH DUCTILE WEB OPENINGS 5.1 INTRODUCTION The results of small-scale experiments described in Chapter 4 show that the proposed upgrading scheme can be utilized to control the strength and deformation of existing moment frames. To verify this, it is necessary to conduct an analytical investigation on the seismic response of full-scale structures as well as to experimentally verify the analytical models used to predict their response. Guided by the experimental results presented in Chapter 4, a design procedure for seismic upgrading of steel moment frames is presented in this chapter. The moment frame structure discussed in Chapter 2 was used as an example structure. It was modified using the proposed upgrading procedure. The response of the upgraded frame under severe ground motions was studied using the same series of nonlinear analyses as in Chapter 2. Finally, a full scale testing of a one-story subassemblage was carried out to verify the proposed modification procedure and to verify the results of computer analyses. 5.2 PROPOSED DESIGN APPROACH As mentioned in Chapter 4, conventional beam-to-column connections may possess far less ductility than expected [Englehardt and Husain 1993, SAC 1996]. In the system proposed herein, beam openings can be conservatively designed so that, under a severe ground motion, the connection moments generated by the shear force in the 137

160 138 opening will be smaller than a chosen critical value during the entire earthquake excitation. This critical value can be selected based on the type of connections used in the frame. For conventional beam-to-column connections (welded-flange-bolted-web connections), one possible design approach is that the openings, under their fully yielded and strain-hardened condition at about 3% of story drift, would generate moments at the beam-to-column connections smaller than their flexural yield capacity. This reduces the risk of having premature failure of those connections. The story drift of 3% is selected based on an observation that strong column moment frames would generally experience story drift less than 3% for design level earthquakes. The maximum shear strength of an opening can be estimated by using the equations given in Chapter 4. The design of an opening begins by calculating the maximum allowable shear force in the opening from the design requirement that the connection moment created by the opening shear force is approximately equal or smaller than the yield moment of the connection. Generally, moment frames are exterior frames, therefore, the effect of gravity loads is small when compared to that of the lateral loads. Therefore, it is neglected in the following design procedure. Figure 4.39 shows the internal forces in a frame with an opening due to lateral loads only. From the simplified moment and shear force diagrams shown in Figure 4.39a, assuming that the opening is placed at the mid span and neglecting the moment due to the vertical member and the axial forces in the chord members, the opening shear force, V o, should be (using center line dimensions): V o 2φM y (5.1) L where M y is the yield moment of the connection,φ is the resistance factor, 0.90, and L is the span length.

161 Design of Chord Members As shown in Chapter 4, the overstrength factor for the chord members is a function of both the length of the opening and the section properties of the chord members. The overstrength factor for the chords of a wide-flange beam with an opening, ξ c, can be expressed as: L 0.95L 0.665δEI M 2 o c ch L o ξ c = (5.3) M ch where δ is the story drift, E is the young modulus, I c is the moment of inertia of the chord member, L is the span length, L o is the length of the special opening, and M ch is the nominal plastic moment of the chord members. By substituting δ of 0.03 (3% drift), the overstrength factor for the chord members can be evaluated as: L 0.95L 0.02EI M 2 o c ch L o ξ c = (5.4) M ch The overstrength factor is directly related to the ductility demand at the plastic hinges in the chord members the larger the ductility demand, the larger the overstrength factor. Therefore, in order to prevent severe damage in the chords during a major earthquake, the overstrength factor should be maintained in the range of Overstrength values in this range have been found by experiments to be practical. From Equation 5.4, the overstrength factor of the chord members is a function of both the length of the opening and the section properties of the chord members. The design process for a chord member is based primarily on a trial and error approach to converge on a reasonable value of the overstrength factor. The design begins by determining the length of the opening. From the tests of STMF frames and the tests of small-scale specimens, the length of the opening on the

162 140 order of about 0.20 to 0.25 of the span length has been found to perform well and provide a good combination of frame stiffness and strength. After the length of the opening has been selected, since I c and M ch can be expressed in terms of the depth of the chord, the chord of the opening can be designed by varying the depth of the chord until the overstrength factor calculated by Equation 5.4 converges to the target range of Design of Diagonal Members With a known depth of the chord members, the shear contribution of the chords can be determined, and consequently, the size of the diagonal bars can be computed. From Equation 4.7, it follows that: ξ ( P x 0.3P )sinθ V 4( ξ M ) c ch xy + xc x = o (5.5) 0.95Lo where V o is the required shear force in the opening calculated from Equation 5.1. Forces P xy and P xc can be calculated by using the formulas given in the AISC- LRFD specifications [AISC 1994] by using the clear length of the diagonal members and the effective length factor, k, of The design process for the diagonal members is also based on trial and error approach to satisfy Equation Design of the Vertical Member member, With the designed chord members and diagonal bars, the force in the vertical P v, can be found from equilibrium. From Figure 5.1, where equilibrium of forces at the vertical-to-chord junction is shown, this axial force can be calculated as: P v = 1.4( P 0.3P ) sinθ (5.6) xy xc x Conservatively, the compression force in vertical member can be taken as: P = 1.4P sinθ (5.7) v xy x

163 141 Using this force, the vertical member can be conservatively designed by using the AISC-LRFD specifications, assuming a clear length and the effective length factor, k, of ξ cm ch/0.95l o ξ cm ch ξ xp xy Resultant Force V o θ x 0.3ξ xp xc ξ cm ch 2ξ cm ch/0.95l o P v 0.3ξ x P xc ξ x P xy 2ξ c M ch /0.95L o 2ξ c M ch /0.95L o Design of the Welds Figure 5.1. Equilibrium of Forces at the Middle Joint. The welds for the diagonal bars should be designed to take the fully strain hardened forces created by the diagonal members, i.e., 1.4Pxy. The welds for the vertical member should be designed such that the full plastic moment of the vertical member can be developed Required Strength of the Opening under Gravity Loads The previously mentioned design procedure for the girder web opening is a limit state design procedure, which considers the force distribution at the ultimate lateral load condition. It is based on a premise that the dead load is small. However, the opening should also be checked against the gravity load combination, 1.4DL+1.6LL, even though its effect may be small. Under this condition, all the members in the opening should

164 142 remain in the elastic range. The design philosophy used herein is based on permitting inelastic activity in the openings only in the event of extreme earthquake lateral loads Detailing of the Openings Stress concentration is a major cause of damage and cracking in steel structures. In case of the proposed upgrading scheme, stress concentration is highest at the corners of the opening. Therefore special detailing should be provided to minimize the stress concentration and to move the location of plastic hinges away from the critical corners. The detailing scheme used in Specimen 3, presented in Chapter 4, provides the needed ductility and is practical too. Therefore, it is recommended that this kind of detailing be used. The opening can be flame-cut, but the surface of the cut in the vicinity of the corners should be smoothed out by grinding. It is desirable to have a radius in all corners, although test results suggest that it might not be necessary. The vertical members at the ends of the opening should be placed at an offset of about 0.5 in. from the ends of the opening. This is done so that the plastic hinges are pushed away from the critical areas. The diagonal bars, as used in Specimen 3 described in chapter 4, also help in reinforcing the corners. All welds should have at least 0.25 in. clearance from the edges to allow for plastic flow and increase local ductility. 5.3 THE STUDY BUILDING The building selected for this study is the six-story moment frame structure used earlier in Chapters 2 and 3. In order to study the response of the proposed structural system, the moment frame structure of the example building was modified according to the proposed procedure. Taking the fourth floor girder (W36x150) as an example, and assuming grade 50 steel, the maximum allowable shear at mid span is:

165 143 2φM y L = 2(0.9)(504)(50) 300 = kips (5.8) Choosing the length of the opening as 0.20 times the span length and a target overstrength factor of 2.0 for the chord members, the appropriate depth of the chord members was found by trial and error to be 4.25 inches, with an exact overstrength value of 2.09 and a plastic moment, chords is: 4ξ M c 0.95L ch o = M ch 4(2.09)(347.6) 0.95(0.2)(300), of k-in. Therefore, the shear provided by the = kips (5.9) By taking ξ = 1. 40, the shear contribution from the diagonal members should be, from Equation 5.5: x 1.4( Pxy + 0.3Pxc )sinθ x = = kips (5.10) Using 1 7 / 8 x 1 1 / 8 bars interconnected at the mid-length, the yield force and the buckling force (with k = 0.80 and l x = 23 in.) were found to be 76.0 kips and 64.3 kips, respectively. Taking θ x to be approximately 49 degrees (Figure 5.2), the shear contribution of the diagonal members is: 1.4( Pxy + 0.3Pxc )sinθ x = kips (5.11) The total shear force provided by the opening is: V = = kips (5.12) o which is adequate. With the selected bar size, the compression force in the vertical member is: P = 1.4P sinθ 79.9 kips (5.13) v xy x = Therefore, double angles 2L2 1 / 2 x 2 1 / 2 x 3 / 8 with a calculated critical load of 95 kips were used for the vertical member. The modified frame is shown in Figure 5.2. Calculations for the other floor girders are summarized in Tables 5.1 and 5.2.

166 Figure 5.2. The Modified Frame with Beam Web Openings. 144

167 145 Beam Size ØMy (kip-in) Table 5.1. Design of Web Openings. V-allowable Chord Depth ξ c (kips) (in) V c (kips) Diagonal Members V x (kips) W27x / 2 x 3 / W36x x1 7 / W36x / 8 x1 1 / W36x x1 3 / Note: Calculations based on Fy = 50 ksi for chord members and Fy=36 ksi for diagonal members. Floor Table 5.2. Member Sizes of the Modified Frame with Web Openings. Beam Chord Depth Size (in) Opening Length (in) Diagonal Members (inxin) Vertical Members Roof W27x / 2 x 3 / 4 2L 2 x 2 x 3 / 16 5 W36x x 1 7 / 8 2L 2 x 2 x 3 / 8 4 W36x / 8 x 1 1 / 8 2L 2 1 / 2 x 2 1 / 2 x 5 / 6 3 W36x x 1 3 / 8 2L2 1 / 2 x 2 1 / 2 x 1 / 2 2 W36x x 1 3 / 8 2L2 1 / 2 x 2 1 / 2 x 1 / NONLINEAR ANALYSES OF THE STUDY BUILDING In order to study the behavior of the structure modified by the proposed scheme, nonlinear analyses were performed to compare its behavior before and after the modification. One-bay, five story, models of the original three-bay moment frame and the modified frame with web openings were prepared for inelastic static ("pushover") and inelastic dynamic analysis. SNAP-2DX [Rai et al. 1994] computer program was used for the inelastic analyses. Modeling assumptions were similar to the ones used in previous studies. These assumptions include: 1) Gravity loads were neglected; 2) Floor masses of the frame were lumped at the beam-to-column connection nodes; 3) 2% mass proportional damping was used in dynamic analyses with the estimated frame period calculated from the 1994 UBC. In this study, girders and columns were modeled using

168 146 beam-column elements with 2% strain hardening in the end moment-rotation models. The chord members were also modeled using the beam-column elements, but with 10% strain hardening. Diagonal members of the openings were modeled using axial buckling elements with Jain s hysteretic model as explained in Chapter 4. In order to accurately simulate the response of the modified frame, it is important to correctly model the overstrengths of the structural members. This is because controlling the overstrength is one of the main objectives of this upgrading system. Therefore, the yield stress for each member was taken as the expected yield stress. The yield stresses for the girders and columns were taken as 55 ksi (expected for A572 GR.50 steel). For diagonal members, the yield stress was taken as 49 ksi (expected for A36 steel). For comparison purposes, the original frame was also modeled using an expected yield stress of 55 ksi for all members. The panel zone deformations of the original and the modified frames were not considered since the main purpose was to compare the overall behavior of the modified frame to that of the original frame. The analytical models of the modified frame with openings and the original frame are shown in Figures 5.3 and 2.5, respectively. Figure 5.3. The Modified Frame and its Analytical Model.

169 147 Similar to the analyses performed in Chapter 2, the static pushover analysis was carried out by applying lateral forces representing the UBC distribution of design lateral forces. For the inelastic dynamic analysis, these two models were subjected to the four scaled earthquake records used in previous chapters. These records were: the 1940 El Centro record, the 1994 Northridge (Sylmar Station) record, the 1994 Northridge (Newhall Station) record, and one synthetic record. The results from the analyses are presented and discussed in the following sections Inelastic Static Pushover Analyses The results from static pushover analyses are summarized in Figures 5.4 and 5.5. Figure 5.4 shows the base shear versus roof displacement plots for the original and the modified frames. In the modified frame, the first inelastic activity was the buckling of the diagonal members in the fourth floor girder when the roof drift was approximately 0.8%. After all the diagonal members in other floor openings had buckled, yielding of the diagonal members started. Then, it was followed by yielding in the chord members. At the roof drift of about 2.3%, last set of plastic hinges formed in the chord members of the roof girder opening. The sequence of inelastic activity up to the onset of mechanism is shown in Figure 5.5. The strength corresponding to the first significant non-linearity in the forcedisplacement plot of the modified frame was approximately half that of the original frame. Both frames showed significant overstrength above the UBC-94 design force level, approximately 6 times for the original frame and 5 times for the modified frame. The stiffness of the modified frame was somewhat smaller than that of the original frame because of the openings. However, in the original frame, the first set of plastic hinges to form was at the column bases and the yield mechanism was a soft story in the first story, as shown in previous chapters. The modified frame, on the other hand, behaved in a truly strong-column weak-beam fashion with inelastic activity essentially limited to the

170 148 openings in the girders and minor flexural yielding at the column base forming almost last in the plastic hinging sequence. The behavior of the modified frame was very similar to the plastic designed frame presented in Chapter 3. It is also noteworthy that no yielding was observed in the beamto-column connections. Therefore, in this case, the chance of having premature failure at the connections was significantly smaller than in the original case. Another objective of the pushover analysis of the modified frame was to determine the maximum overstrength values in the chord members at 3% roof drift. As mentioned earlier, one of the objectives of the proposed design procedure is to control the value of the overstrength factor within an acceptable range,1.8 to 2.0. Table 5.3 shows the overstrength values in the chord members at 3% roof drift and the design values. Maximum overstrength ratio at a plastic hinge was calculated by dividing the magnitude of the plastic moment occurred at 3% roof drift by the yield moment of the corresponding chord member. 0.8 Base Shear Coefficient (V/W) Mechanism 1 First Plastification 1 First Buckling UBC DESIGN V = 0.09 W 4 Mechanism Modified Original Roof Drift (%) Figure 5.4. Base Shear Roof Drift Response of the Original and the Modified Frames (Based on Expected Yield Strength).

171 V V V V V Note: Inelastic Activity in Openings Includes Yielding and Buckling of Diagonal Members and Plastic hinging of Chord Members. Figure 5.5. Sequence of Inelastic Activity in the Modified Frames. Table 5.3. Comparison Between Design and Attained Overstrength Values. Floor Level Beam ξ c (Analysis) Design Value Roof W27x W36x W36x W36x W36x All of the overstrength values were smaller the design values. The maximum overstrength occurred in the fourth floor girder (2.06) where the inelastic activity first started. This value agreed well with the design value (2.09) Inelastic Dynamic Analyses Selected results from the inelastic dynamic analyses of the two frames are presented in Figures 5.6 through 5.9. The envelopes of maximum story drifts and maximum floor displacements of the two frames are compared in Figures 5.6 and 5.7,

172 150 respectively. Note that the response of the original frame was slightly different than the ones shown in Chapters 2 and 3. This is because the modeling assumptions were slightly different. The floor displacements and the maximum story drifts of the modified frame with web openings were similar to those of the original frame. However, the patterns of inelastic activity in the two frames were different. A story mechanism in the first story formed in the original frame, as discussed in Chapters 2 and 3. It can be noticed from Figure 5.8, where the ductility demands and location of inelastic activity are shown, that the inelastic activity in the modified frame was limited only to the openings, as was intended in the design. No story mechanism was observed in the modified frame. In addition, no yielding in the beam-to-column connections was observed, thereby, reducing the risk of having premature failure. As emphasized earlier, design for controlled inelastic activity results in better response. Damage inspection and repair work after an earthquake would also be relatively easier and less costly Floor Level 4 3 El Centro Sylmar Newhall Synthetic Floor Level 4 3 El Centro Sylmar Newhall Synthetic Floor Displacement (in.) Modified Frame Floor Displacement (in.) Original Frame Figure 5.6. Maximum Floor Displacements of the Modified and the Original Frames.

173 Story Level 3 2 El Centro Sylmar Newhall Synthetic Story Level 3 2 El Centro Sylmar Newhall Synthetic Story Drfit (%) Story Drfit (%) Modified Frame Original Frame Figure 5.7. Maximum Interstory Drifts of the Modified and the Original Frames. (4.35) (3.42) (5.38) (5.10) (2.91) (2.77) (3.76) (6.24) (2.50) (4.93) (1.30) (3.63) (1.10) (1.44) (2.04) (2.04) (2.59) (2.47) (2.47) El Centro Newhall Sylmar Synthetic Note: Rotational Ductility Demands of the Chord Members Shown in Parentheses. Figure 5.8. Location of Inelastic Activity in the Modified Frame under the Four Selected Records.

174 152 Figure 5.8 shows that the ductility demands were quite large in the chord members of the openings. These large ductility demands were expected. They are characteristic of structural systems that have fuse elements, such as eccentrically braced frames and special truss moment frames. These high ductility demand values imply that the chord members must have large ductility capacity. Experiments have shown that the fuse elements in this upgrading system can be very ductile. It was found that the ductility demand corresponding to the overstrength on the order of 2.0 is acceptable. Figure 5.9 shows the maximum overstrength values in the chord members due to the four selected records. As can be seen, the maximum overstrength values in the chord members were well within the acceptable limit. The overstrength values were lower than the design values (Table 5.1) because the attained drifts were smaller than the value assumed during the design (3%). (1.41) (1.30) (1.54) (1.51) (1.24) (1.21) (1.34) (1.64) (1.18) (1.48) (1.04) (1.32) (1.01) (1.02) (1.12) (1.12) (1.19) El Centro Newhall Sylmar Synthetic Figure 5.9. Maximum Overstrength Values Under the Four Selected Records.

175 153 The results of the analyses show that it is possible to improve the performance of a moment frame using the proposed upgrading scheme. One critical issue is the ductility capacity of the chord members. Although results form the test program presented in Chapter 4 suggest that the ductility demand corresponding to an overstrength factor of about 2 can be satisfied with the ductility capacity created by the detailing scheme similar to the one used in Specimen 3, these results were based on small-scale tests with simplified boundary conditions. A full-scale experiment is necessary to fully verify these assumptions. The results of a full-scale test conducted as part of this study will be presented and discussed in the following sections. 5.5 EXPERIMENTAL PROGRAM In order to study the behavior of the proposed upgrading scheme experimentally, a full-scale specimen representing a one-story sub-assemblage of a moment frame with openings was designed and fabricated for a cyclic test. The objective of this test program was to verify the design and the analytical modeling procedures, which were developed based on tests of small-scale specimens. In addition, the full-scale test provided an opportunity to verify the detailing scheme used in the critical locations. The following sections describe the test procedure and the test results Test Set-Up A one-story subassemblage, consisting of a full-scale 28 feet long W24x62 beam with a web opening and two 13 feet long W14x82 columns at the ends of the beam, was designed and fabricated for this experimental study. The columns were half-story high above and below the beam with pinned ends at both the top and bottom of the columns. The specimen represented a story in a one bay frame assuming that inflection points were at the mid-height of the story. Even though this assumption is not entirely accurate as shown in Chapter 2, it provides a convenient way to simulate the cyclic behavior of the

176 154 proposed structural system in the laboratory. The load was applied at the top of one of the columns via a 100-kip actuator with a maximum stroke length of ± 5 inches. In order to simulate earthquake loading, the actuator force must be transmitted through both columns. This was accomplished by using one W12x50 link beam, which was pinconnected between the column ends. The applied load represented the story shear induced by an earthquake. Lateral braces were provided at one-third of the span to simulate the presence of cross beams in the real structure. An overview of the test set-up is shown in Figures 5.10 to The dimensions of the test frame are presented in Figure Figure Overall View of the Test Set-Up.

177 155 Figure Close-Up View of the Test Specimen. Figure Lateral Bracing of the Test Specimen.

178 156 Figure Beam-to-Column Connection of the Test Specimen.

179 157 Note: Dimensions in inches Figure Dimensions of the Test Specimen.

180 Design of the Girder and the Web Opening. In this test, the size of the girder was selected based on the capacity of the columns. These columns were not specifically designed for this experiment. Instead, they were previously designed and used for other purposes [Basha and Goel 1994, Itani and Goel 1991]. It was decided that, in order to prevent any significant yielding in the columns, the maximum actuator force should not exceed 65 kips. The size of the girder was selected such that the moment generated by this actuator force would create moments at the beam-to-column connections of about 85% of the yield moment of the beam. With the selected girder size, the opening was then designed according to the previously described procedure such that the maximum applied force at 3% story drift was at the target value (65 kips). The chords of the opening in the specimen were designed to have an expected overstrength factor of about 1.80 at 3% story drift. The dimensions of the opening are shown in Figures 5.15 and the close-up photographs of the opening are shown in Figures 5.16 to Figure Dimensions of the Web Opening in the Test Specimen.

181 159 Figure Close-Up View of the Special Opening. Figure Diagonal-to-Chord Junction.

182 160 Figure Vertical-to-Chord Junction Instrumentation and Test Procedure Specimen displacement was applied at the top of the test frame in a quasi-static manner using a predetermined cyclic displacement pattern. Two loading histories were used in this experiment. Initially, only one loading history was intended to be used. However, during the test, one of the bolts that connected the actuator to the reaction wall unexpectedly became loose, causing the actuator to twist. It was necessary to stop the experiment to replace the bolt. The test was resumed with the second loading history. The first loading history consisted of cycles of increasing displacements up to about 0.9% story drift where buckling and yielding first initiated. The second loading history consisted of cycles of large displacement amplitudes up to 3% story drift. The 3% story drift limit was used because the value assumed in the design corresponding to 3% story drift, and also because the maximum stroke of the actuator was on the order of

183 161 3% story drift of the frame. The first and the second loading histories are shown in Figures 5.19 and 5.20, respectively Story Drift (%) Cycle Figure First Loading History Story Drift (%) Cycle Figure Second Loading History.

184 162 The hysteretic response of the test frame was obtained from the load cell and the displacement transducer in the hydraulic actuator. Additional data on various key members were obtained using electrical strain gauges at selected points. Photographic records were also made throughout the experiment Material Properties The girder was made of a W24x62 dual grade steel. All diagonal web members were made of A36 steel flat bars. The material properties were obtained by means of tensile tests of coupons from various parts of the specimen. An average yield stress from coupons was used to represent the yield stress of the girder. The average yield stresses of various key members are given in Table 4.1. Table 5.4. Average Yield Stresses of Key Members. Coupon Yield Stress (ksi) W24x L2 1 / 2 x 2 1 / 2 x 3 / / 8 x 1 1 / 8 PL Test Results The hysteretic loops from the first phase and the second phase of the test are shown in Figures 5.21 and 5.22, respectively. In the first phase of the test, the response started to deviate from elastic behavior at a drift of about 0.75%. Two of the diagonal members buckled at this displacement. At the story drift of about 0.9%, some yielding in the diagonal members was observed. An actuator bolt became loose at this point. The test was resumed using the second loading history after the bolt was replaced. In the second phase of the test, after the diagonals had completely buckled and yielded, the chords of the opening started to plastify. Plastic hinges clearly formed at the end of the 1.8% story drift displacement cycles. Progressing further into larger

185 163 displacement cycles showed a mechanism pattern as intended by the design, i.e., yielding and buckling of the diagonal members followed by plastic hinging at the ends of the chord members. The deformation and the complete mechanism of the test frame are shown in Figures 5.23 to The test specimen was able to sustain many cycles of large displacements without fracture. Only local buckling in the chords and local necking in the diagonal bars due to very high local strain were observed. These local instabilities resulted in a small reduction in the load carrying capacity of the frame during the 3% story drift cycles. The chord members fractured much later after the frame was subjected to additional decreasing displacement cycles, which are not shown here. These additional displacement cycles were used to observe the failure mode only, thus no significant data were recorded. The chord member cracks stopped propagating when they reached the flange of the chord members. This fracture can be seen in Figure Story Drift (%) Load (kips) Displacement (in.) Figure Hysteretic Loops from the First Loading History.

186 Story Drift (%) Load (kips) Displacement (in.) Figure Hysteretic Loops from the Second Loading History. Figure Deformation of the Test Frame (Positive Displacement).

187 165 Figure Deformation of the Test Frame (Negative Displacement). Figure Inelastic Activity in the Opening.

188 166 Figure Yielding of the Chord and the Diagonal Members. Figure Crack in the Chord Member.