Investigation of Passive Control of Irregular Building Structures Using Bidirectional Tuned Mass Damper

INVESTIGATION OF PASSIVE CONTROL OF IRREGULAR BUILDING STRUCTURES USING BIDIRECTIONAL TUNED MASS DAMPER

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in

the Graduate School of The Ohio State University

Mariantonieta Gutierrez Soto

Graduate Program in Civil Engineering

The Ohio State University

2012

Master's Examination Committee:

Dr. Hojjat Adeli, Advisor

Dr. Ethan Kubatko

Dr. Vadim Utkin

Mariantonieta Gutierrez Soto

2012

ABSTRACT

Eight different sets of equations proposed for tuning the parameters of tuned mass dampers (TMDs) are compared using a 5-story building with plan and elevation irregularity, and a 15-story and a 20-story building with plan irregularity subjected to seismic loading. Next, the performance of bidirectional tuned mass damper (BTMD) is compared with that of the pendulum tuned mass damper (PTMD) using three different structures with plan and vertical irregularities ranging in height from 5 to 20 stories and dominant fundamental periods ranging from 0.55 sec to 4.25 sec subjected to Loma

Prieta earthquake. It is concluded that BTMD performs consistently better than PTMD for reduction of maximum displacement, acceleration and base shear. BTMD is advantageous over PTMD because it can be tuned for two modes of vibrations and therefore can be used as an alternative to using two TMDs. Then, the effectiveness of

BTMD is investigated using a 20-story building structure with plan irregularity subjected to six seismic accelerograms. Finally, the optimal placement of the BTMD is investigated using five different multi-story building structures with plan and elevation irregularities ranging in height from 5 to 20 stories and fundamental periods ranging from 0.55 sec to

4.25 sec subjected to Loma Prieta earthquake.

DEDICATION

This thesis is dedicated to all people who want to know or build safer buildings.

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ACKNOWLEDGMENTS

Many people see the development of a thesis difficult to achieve, others are filled with egocentrism unavoidable, however there are people without forgetting their merit who have acknowledged that without the support of professionals and institutions in one way or another they facilitate, contribute and expand their valuable assistance in the preparation and completion of this research. It is therefore my pleasure to use this space to be fair and consistent with them, expressing my acknowledgements.

First and foremost, my utmost gratitude is to my advisor Dr. Hojjat Adeli, The

Abba G. Lichtenstein Professor of Civil Engineering for accepting me to do this thesis under his guidance. His support and confidence in my work and his ability to guide my thinking has been an invaluable contribution, not only in the development of this thesis, but also in my development as a researcher. He has been a source for my inspiration and motivation. Own ideas, always framed in his orientation and rigor, which has been the key to the good work we have done together, which cannot be conceived without his always actively participation. I am also thankful for always providing me the means to carry out all the activities proposed for the development of this thesis. Also for his advice in courses, seminars and conferences related to this field. His leadership, patience and steadfast encouragement to complete this study; allowed me to take my first steps as a

iv professional which simultaneously became a solid foundation for my work habits with which help me to face the future. Thank you very much Dr. Adeli.

I would also like to acknowledge my sincere gratitude to the rest of my thesis committee members: Dr. Ethan Kubatko, Professor at Civil Environmental and Geodetic

Engineer Department for which I give my special thanks. The few times that we held discussions on finite element topics, generally, I was clearly outweighed and highly enriched by the quality of our conversations. Dr. Kubatko, thank you very much and I hope in the near future we have other opportunities to share our views. And last but not least, I consider it an honor to have Dr. Vadim Utkin, Professor at the Electrical Engineer

Department, as my committee member. Thank you for your encouragement and insightful comments.

Dr. Amelia Mieses, Technical Support Engineer at Computers and Structures, for her valuable assistance in the software SAP2000 and ETABS license and technical support that made this research possible.

Dr. Jose Luis Almazan, Professor of the Department of Structural and

Geotechnical Engineering at Pontificia Universidad Católica de Chile (Santiago, Chile), for his patience, availability and generosity in sharing his experience and extensive knowledge on this research, especially in bidirectional homogenous tuned mass dampers.

His collaboration was a great help. Thank you also for thoughtful and quick responses to the different concerns that emerged during the development of this work, which has also been reflected in the good results achieved.

Dr. Diego Lopez Garcia, Professor of the Department of Structural and

Geotechnical Engineering at the Pontificia Universidad Católica de Chile (Santiago,

Chile), for his expertise and assistance with damping systems subjected to earthquake loading, giving wise advice, helping with various applications, recommendation books, and so on. Despite the distance, he has painstakingly e-mailed the information I needed.

Engineering Faculty at the Ohio State University (Columbus, OH), for their untiring effort in encouraging the teaching staff to pursue professional growth, such as,

Dr. Carolyn Merry, Chair and Professor at The Civil Environmental and Geodetic

Engineer Department, whose sincerity and encouragement I will never forget, and for helping the departments to run smoothly. And Mrs. Cynthia Crawford, Department Fiscal

Officer for giving me all her feedback and advice from the first time I step at OSU.

The French Fellowship received from College of Civil, Environmental and

Geodetic Engineering supervised by Dr. William Wolfe, Chair of the Graduate

Committee in Civil Engineering. Thank you for funding much of my masters studies forgivable bestowing a scholarship from the call of the year 2010.

Dr. John Merrill, for the financial support received from the First-Year

Engineering program and the great opportunity to interact with students which have been a source of motivation and an opportunity to grow as a professional.

The Computer Systems Department from The Civil Environmental and Geodetic

Engineer Department at the Ohio State University (Columbus, OH), especially to Mr.

Daniel Vehr, Computer Systems Manager, who was always generous and willing, who shared knowledge and experience of how to improve my computer skills, especially his

vi assistance when the system broke down multiple times during the development of this research.

The Science and Engineering Library (SEL) at The Ohio State University

(Columbus, OH), for being open 24 hours 7 days a week, for having all the subject areas in relation to Engineering majors, and for assisting me in so many different ways.

My teammates, colleagues and doctors, I have only words of gratitude, especially for those times when one could be below expectations: it has been long hard roads which sometimes setting goals makes one forget the importance of human contact. However, as in all activities of life, always at the end there are some criteria that allow you to prioritize and this is why I emphasize my thanks to Adithya Jayakumar, Shawn Hall,

Rachael Meyer, Justin Shen and Nengmou Wang for their help and feedback in my moments of doubt and uncertainty.

And of course, to the most deep and heartfelt gratitude goes to my family.

Without their support, collaboration and inspiration would have been impossible to carry out this tough research. To my father Argenis, for his example of patience and generosity, my mother Nélida, for her continuous support and honesty, and my dear sister

Mariaryeni, for her tenacity, focus and a great role model in pursuing dreams!

Finally, the one above all of us, the omnipresent God, for answering my prayers for giving me the strength to plod on despite my constitution wanting to give up and throw in the towel, thank you so much Dear Lord.

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VITA

February 12, 1988 ...... Born - Caracas, Venezuela

July 2004 ...... U. E. “Luis E. Egui Arocha” High School

May 2010 ...... B.S. Civil Engineering, Lamar University

2010 ...... French Fellowship, Department of Civil,

Environmental and Geodetic Engineering,

The Ohio State University

2011 to present ...... Graduate Teaching Associate, First-Year

Engineering Program, The Ohio State

University

FIELDS OF STUDY

Major Field: Civil Engineering

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TABLE OF CONTENTS

Page ABSTRACT ...... II DEDICATION ...... III ACKNOWLEDGMENTS ...... IV VITA ...... VIII FIELDS OF STUDY ...... VIII TABLE OF CONTENTS ...... IX LIST OF TABLES ...... XII LIST OF FIGURES ...... XIV

CHAPTER 1...... 1

INTRODUCTION ...... 1

1.1 Literature Review ...... 1

1.1.1. Passive Control ...... 1

1.1.1 Semi-Active Control ...... 5

1.1.1 Active Control ...... 6

1.1.1 Hybrid Control ...... 7

1.2 Objectives and Plan of Research ...... 8

CHAPTER 2...... 12 STANDARD, PENDULUM, AND BIDIRECTIONAL TUNED MASS DAMPERS ...... 12

2.1 Introduction ...... 12

2.2 Selection of Optimum Parameters Values for a TMD ...... 28

2.3 Pendulum Tuned Mass Damper ...... 33

2.4 Bidirectional Tuned Mass Damper ...... 35

CHAPTER 3...... 39 COMPARISON OF EQUATIONS FOR OPTIMUM TUNING PARAMETERS ...... 39

3.1 Introduction ...... 39

3.2 Conclusion Remarks...... 63

CHAPTER 4...... 65 COMPARISON OF PTMD AND BTMD ...... 65

4.1 Introduction ...... 65

4.2 Summary of Comparison Results Between PTMD and BTMD ...... 68

CHAPTER 5...... 73 BTMD PLACEMENT ...... 73

5.1 Introduction ...... 73

CHAPTER 6...... 96 CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH ..... 96

6.1 Summary of Conclusions ...... 96

6.2 Recommendations for Future Research ...... 100

BIBLIOGRAPHY ...... 102

APPENDIX A ...... 117 BTMD AND PTMD DETAILED PARAMETERS FOR THE IRREGULAR STRUCTURES ...... 117

APPENDIX B: ...... 123 MATLAB CODE ROUTINE FOR OBTAINING OPTIMAL TMD TUNING PARAMETERS ...... 123

LIST OF TABLES

Table 1: Actuator Placement Review ...... 9

Table 2: Buildings and towers with TMD applications in chronological orders ...... 18

Table 3: Equations for finding the optimum tuning parameters of a TMD ...... 30

Table 4: Beam and column designs for the 5-story irregular building ...... 40

Table 5: Free vibration modal response of the 5-story irregular building ...... 42

Table 6: Maximum absolute displacement for 28 different earthquake incident angles for the 5-story irregular building ...... 46

Table 7: Optimum tuning parameters and maximum absolute displacement (cm) and acceleration (g) and base shear (kN) for the 5-story building...... 47

Table 8: Maximum absolute floor acceleration and interstory displacement (drift) for 5- story building using different equations for optimum TMD tuning parameters ...... 50

Table 9: Free vibration modal response of the 15-story irregular building ...... 54

Table 10: Maximum absolute displacement (cm) for different earthquake incident angles of 15-story building ...... 56

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Table 11: Optimum tuning parameters and maximum absolute displacement (cm), acceleration (g) and base shear (kN) response for the 15-story building ...... 57

Table 12: Free vibration modal response of the 20-story irregular building ...... 61

Table 13: Maximum absolute displacement per earthquake incident angles of 20-story building ...... 62

Table 14: Optimum tuning parameters and maximum absolute displacement (cm), acceleration (g) and Base Shear (kN) response for the 20-story building ...... 63

Table 15: Maximum displacement and acceleration response and the base shear for the 5- story building for different PTMD placement ...... 75

Table 16: Maximum displacement and acceleration response and the base shear for the

15-story building for different BTMD placement ...... 76

Table 17: Maximum absolute displacement, absolute acceleration and base shear responses for the 20-story building for different BTMD placement ...... 78

Table 18: Earthquake ground motions ...... 78

Table 19: Comparison results of 20-story building equipped with BTMD at BC-12 roof location using 6 seismic records...... 81

Table 20: Free vibration modal response of the 12-story FEMA irregular building ...... 86

Table 21: Maximum absolute displacement (m) per earthquake incident angles of 12- story with vertical and plan irregularities irregular building ...... 87

Table 22: Maximum displacement and acceleration response and the base shear for the

12-story building with vertical and plan irregularities for different BTMD placement .... 89

Table 23: Free vibration modal response of the 12-story building with vertical setback.. 93

Table 24: Maximum absolute displacement (cm) per earthquake incident angles of 12- story building with vertical setback ...... 94

Table 25: Maximum displacement and acceleration response and the base shear for the

12-story building for different BTMD placement ...... 95

Table 26: Design parameter information for the 5th story building...... 117

Table 27: Design parameter information for the 15th story building ...... 117

Table 28: Design parameter information for the 20th story building ...... 118

Table 29: Design parameter information for the 12th story FEMA building ...... 118

Table 30: Design parameter information for the 12th story podium building ...... 119

Table 31: PTMD/BTMD tuning parameters for 5th story irregular building ...... 120

Table 32: PTMD/BTMD tuning parameters for 15th story irregular building ...... 121

Table 33: PTMD/BTMD tuning parameters for 20th story irregular building ...... 122

LIST OF FIGURES

Figure 1: Taipei 101 high-rise building (http://goo.gl/TBuz8) ...... 13

Figure 2: The Aspire Tower (Courtesy of Sonya&Travis) ...... 14

Figure 3: Shanghai World Financial Center (http://goo.gl/leB2I) ...... 14

Figure 4: SDOF system with a Tuned Mass Damper (TMD) ...... 27

Figure 5: Pendulum Tuned Mass Damper (PTMD) ...... 33

Figure 6: The Crystal Tower (Courtesy of Takashi Kassai) ...... 34

Figure 7: Bidirectional Tuned Mass Damper (BTMD). a) Three-dimensional view, ..... b) view in the xz plane, c) view in the yz plane ...... 37

Figure 8: PTMD and BTMD design procedure using tuning parameters proposed by

Sadek et al. (1997) as an example ...... 38

Figure 9: Five-story irregular 3D building structure with setbacks and an L-shape plan. 40

Figure 10: Plan view of 5-story building (+CR1=center of rigidity of the top two floors,

+CR2=center of rigidity of the bottom three floors, ●CM1=center of mass of the top two floors, ●CM2=center of mass of the bottom three floors) ...... 41

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Figure 11: Mode shapes of the 5-story 3D building with vertical and plan irregularity: (a)

Mode 1 (period = 0.81sec), (b) Mode 2 (period = 0.55sec), (c) Mode 3 (period = 0.50sec)

...... 43

Figure 12: Loma Prieta earthquake accelerogram (October 17, 1989, Magnitude 7.1) .... 44

Figure 13: Maximum absolute joint displacement (cm) for different earthquake incident angles for the 5-story building ...... 47

Figure 14: Maximum interstory displacement (drift) for 5-story building using different equations for optimum TMD tuning parameters ...... 48

Figure 15: Maximum absolute floor acceleration for 5-story building using different sets of equations for optimum TMD tuning parameters ...... 49

Figure 16: 15-story building reinforced concrete 3D building with plan irregularity

(Almazan et al. 2012). (a) Perspective view. b) Plan view. +CR= center of rigidity,

●CM= center of mass...... 53

Figure 17: Mode shapes of the 15-story building with plan irregularity (a) Mode 1 (period

= 2.93 sec), (b) Mode 2 (period = 2.88sec), (c) Mode 3 (period = 2.32sec) ...... 55

Figure 18: Maximum absolute displacement (cm) per earthquake ...... 57

Figure 19: 20-story building with plan irregularity (a) Perspective view, (b) Elevation and

Figure 20: Mode shapes of the 20-story building: (a) Mode 1 (period = 4.67 sec), ...... 60

Figure 21: Maximum absolute joint displacement (cm) per earthquake incident angles for the 20-story building ...... 61

Figure 22: Maximum displacement and acceleration of joint A6-5 and base shear of 5- story building with a PTMD system ...... 66

Figure 23: Maximum displacement and acceleration of joint A6-5 and base shear of 5- story building with a BTMD system ...... 67

Figure 24: Maximum displacement and acceleration of joint D1-15 and base shear of 15- story building with a PTMD system ...... 69

Figure 25: Maximum displacement and acceleration of joint D1-15 and base shear of 15- story building with a BTMD system ...... 70

Figure 26: Maximum displacement and acceleration joint C1-12 and base shear for 20- story building with a PTMD system ...... 71

Figure 27: Maximum displacement and acceleration joint C1-12 and base shear for 20- story building with a BTMD system ...... 72

Figure 28: Imperial Valley earthquake accelerogram, IV (November 15, 1979) ...... 79

Figure 29: San Fernando earthquake accelerogram, SF (February 09, 1971) ...... 79

Figure 30: Landers earthquake accelerogram, LD (June 28, 1992) ...... 79

Figure 31: Northridge earthquake accelerogram, NR (January 17, 1994)...... 80

Figure 32: ChiChi earthquake accelerogram, CC (September 20, 1999) ...... 80

Figure 33: 12-story irregular building structure with setbacks and T-shape plan (FEMA

2003). Left: Perspective and Elevation view. Right: Plan view of building at different story levels...... 84

Figure 34: Mode shapes of the 12-story building with vertical and plan irregularities in perspective view: (a) Mode 1 (period = 2.92sec), (b) Mode 2 (period = 2.26sec), (c)

Mode 3 (period = 2.14sec), (d) Mode 4 (period = 1.20sec) ...... 85

Figure 35: Maximum absolute displacement (m) per earthquake incident angles of 12- story with vertical and plan irregularities building ...... 86

Figure 36: Displacement (m) and acceleration (g) of joint C1-12 and base shear (kN) of the 12-story building with vertical and plan irregularities irregular building ...... 88

Figure 37: 12-story building with vertical setback, (a) Perspective views. (b) Plan view of

Levels 1-4, (c) Plan view of Levels 5-12. +CR= center of rigidity, ●CM= center of mass.

...... 91

Figure 38: Mode shapes of the 12-story irregular building with vertical setback: (a) Mode

1 & 2 (period 2.6 sec and 2.0 sec), (b) Mode 3 (period = 1.5 sec), (c) Mode 4 (period =

1.1 sec) ...... 92

Figure 39: Maximum absolute displacement (cm) per earthquake incident angles of 12- story building with vertical setback ...... 93

CHAPTER 1

INTRODUCTION

1.1 LITERATURE REVIEW

A review of the papers published on placement of passive, semi-active, active, and hybrid devices for vibration control of structures subjected to various dynamic loading such as earthquakes and wind is presented in this section.

1.1.1 Passive Control

Singh and Moreschi (2002) study the problem of optimum size and location of frequency-dependent and frequency-independent passive viscous and viscoelastic dampers for vibration control of linearly behaving building structures under seismic loading using genetic algorithm. They present examples of 6- and 24-story buildings with 3 degrees-of-freedom (DOF) per floor modeled for torsion.

Lopez Garcia and Soong (2002) study damper allocation distribution using

Simplified Sequential Search Algorithm (SSSA). The authors analyzed performance of regular building models with variance in height up to 20 stories, natural period varying

1 from 0.4 to 2.0 seconds, and damping level subjected to various seismic excitations.

Additionally, they compared damper locations obtained from the same seismic events with varying distances from the fault. The authors’ goal was to make the SSSA a simple and practical approach for implementation of passive viscous devices in a building; but by simplifying the methodology, the efficiency of the method in reducing the vibration weakened.

Bishop and Striz (2004) use genetic algorithm to obtain the minimum number of passive viscous dampers necessary to suppress structural vibrations in a space truss subjected to symmetric and asymmetric loadings. Their examples include 72- and 78- bar trusses where four dampers are found to be sufficient to yield the desired response.

Bhaskararao and Jagid (2006) study the structural response of two 2D 10- and 20- story tall frames rigidly connected by nonslip and slip mode friction dampers subjected to seismic loading. They conclude that using 5 dampers yield similar responses when compared with adding dampers on all floors.

Kokil and Shrikhande (2007) use a pattern search algorithm method to study placement of fluid viscous dampers for a single bay 3-D 10-story rigid-floor building by varying soil conditions, including symmetric and asymmetric examples with eccentricity varying from 0.15m to 0.225m from the center of mass. The authors conclude that efficacy of passive viscous dampers decreases as the plan irregularity increases.

Aydin et al. (2007) study placement of viscous dampers on a 2D 10-story, three- span planar steel frame subjected to seismic loading, using the steepest gradient search optimization method and various objective functions. They conclude that using top floor

2 displacement as an objective function decreases story displacements and inter-story drift but increases the base shear force.

Lavan et al. (2008) use a non-iterative optimization procedure steepest descent optimization technique that involves structural weakening and passive damping for inelastic shear-type 8-story 2D frame subjected to 100 ground motion records. They compared the approach of weakening and damping with nonlinear active sliding mode control and uncontrolled structure. The method reduced the inter-story drift and absolute acceleration by 70% and 60% respectively when compared with the uncontrolled case.

Liu et al. (2008) study the placement of viscous liquid dampers in a 3D 38-story building that has two towers and a 5-story podium subjected to seismic excitations. They studied three cases with different number of dampers and floor location arrangement and concluded implementing dampers in stories 28th and 36th floor yield a reduction of displacement and shear forces by 20-25% compared with the uncontrolled case.

Ameduri et al. (2009) use multi-objective genetic algorithm to determine number, placement and orientation of Shape Memory Alloys (SMA) wires embedded in a rectangular composite panel subjected to noise excitations. The authors compared their results from the genetic code to 50 case studies averaged non-genetic optimizations and determine efficiency loss. The final configuration resulted in optimal 5 SMA wires.

Apostolakis and Dargush (2010) discuss the topological optimal distribution and size of hysteretic passive devices such as yielding metallic buckling restrained braces

(BRB) and/or friction dampers in 2D 3- and 6-story steel moment-resisting frames

(MRFs) based on nonlinear time history analysis of 4 synthetic ground motions representing the west coast of the U.S. with 5% of probability of exceedance in 50 years.

They use genetic algorithm (Adeli and Cheng, 1993, 1994a&b; Hung and Adeli, 1994;

Adeli and Kumar, 1995a&b; Sarma and Adeli, 2000a&b, 2001, 2002; Kim and Adeli,

2001; Jiang and Adeli, 2008) to solve the resulting discrete optimization problem.

Optimization parameters are the position of the device, the device type, the yield/slip load, and the bracing stiffness. To evaluate the performance of each structure, a relative performance/fitness function was defined as a weighted function of the maximum inter- story drift, root mean squared (RMS) floor acceleration, and the maximum floor acceleration.

Estekanchi and Basim (2011) use endurance time method (ETM) and genetic algorithm to obtain optimal viscous damper coefficient and placement on a 3-story and a three-bay 8-story regular shear frames, and a 3-floor steel frame with vertical irregularity subjected to earthquake ground motions. The ETM approach decreases the number of time-history analysis requirement and its suggested as an alternative design procedure.

Mehrabian and Yousefi-Koma (2011) study optimal placement of piezoelectric actuators for vibration control of a flexible aluminum scaled model of the vertical tail fin of an F/A-18 fighter jet approximating the first two vibration modes of the full-scale fin.

Twelve piezoceramic actuators are bonded to each side of the aluminum plate fixed at the base but they consider the optimum location of a pair of them. They use neural networks to approximate the 3D surface for the frequency response function and GA to find the optimal placement of a pair of actuators.

Whittle et al. (2012) study implementation of passive linear viscous dampers on a

10-story steel moment resisting regular and vertically irregular buildings through five different placement techniques: Uniform damping and Stiffness Proportional damping,

Simplified Sequential Search Algorithm, Takewaki transfer function and Lavan fully- stressed Analysis and Redesign techniques. They concluded that the techniques that required more preparation not only meet the desired drift limit like the others but also improve drift reduction and distribution of viscous dampers. The authors also concluded that the advanced method’s performance is very similar, so when a designer’s decision between methods, the deciding choice should be in favor of the designer’s familiarity with such method.

Zhan et al. (2012) use genetic algorithm with matrix criterion optimization for parameter identification to obtain the number and location of sensors in a 2D 19-story concrete frame structure. The authors use modal assurance criterion that relates between mode shapes in a structure as an objective function to avoid placing sensors in low vibration areas. The final result obtained 6 sensor placements between 16 and 19 floors.

1.1.2 Semi-Active Control

Li et al. (2010) use genetic algorithm for placement of semi-active MR dampers in a 20-stories 3D benchmark building subjected to seismic loading. First, the authors optimized the active force of the MR dampers using 3 objective functions: inter-story drift, peak control force and evaluation index for effect of active control. Next, the authors studied placement of the dampers. They determined that for El Centro earthquake case the optimal number of dampers varied from 10-20 and for the Kobe earthquake case the optimal choice is between 20-30 dampers were required for first three bottom floors and top four floors and concluded that distribution of dampers can reduce the number of dampers required to yield desired inter-story drift.

Patil and Jangid (2011) study different arrangements of linear viscous dampers

(LVD) and semi-active variable friction dampers (SAVFD) for vibration control of 76- story, 306-m benchmark reinforced concrete (RC) building subjected to wind excitations

(Yang et al. 2004). They modeled the structure as a simple vertical cantilever Bernoulli-

Euler beam discretized as a 76-DOF (degree-of-freedom) system with one DOF per floor and considered three arrangements for dampers: a diagonal in every story (a total of 76 dampers), a diagonal in every story but connecting two stories (a total of 75 dampers), and a diagonal connecting every two stories (a total of 38 dampers). They report the latter two arrangements to be more effective than the first and the last to be the most economical. When only one LVD is used the authors conclude that a diagonal connection from 74th to 76th floor is the optimum location for the damper.

1.1.3 Active Control

Amini and Tavassoli (2005) apply two different optimization techniques to determine the number, placement and force of active dampers in 3- , 12- and 15-story shear frames subjected to six earthquake excitations. They used nonlinear optimization to minimize the absolute maximum value of actuator control force considering physical properties of the structure and artificial neural network to determine the number and location of active dampers. The 3-story example had optimal location at the 2nd floor. The

12-story frame had optimal locations at top 4 floors. The 15-story building had optimal locations on 5th and 6th story.

Agranovich and Ribakov (2010) propose a method for active damper placement on an 8-story reinforced concrete plane frame with stiff beams according to total energy

6 dissipation characteristics. They use a heuristic solution and the LQG control algorithm for active control of structures subjected to seismic loading.

Ribakov and Agranovich (2011) study placement of active dampers using LQR control algorithm with active damper force equation as performance index on regular 10- story reinforced concrete and 20-story 2D steel frames subjected to white-noise excitation and 3 earthquake accelerograms. The final optimum location of the active dampers in the

10-story example were at the floors 3, 5, 7, 8 and 9, while the 20-story example were at the floors 1, 5, 6, 7, 8, 14, 15 and 16.

Chakraborty et al. (2012) used a genetic algorithm to determine location and number of piezoelectric dampers in a smart fiber reinforced shell structure. They chose a fixed minimum and maximum number of actuators and optimized the location of 5 dampers after 50 generations.

1.1.4 Hybrid Control

Li et al. (2011) use a fuzzy logic-based control algorithm for nonlinear vibration suppression of a 20-story regular three-dimensional (3D) benchmark steel moment- resisting frame with a rectangular plan (measuring 30.48-m by 36.58 m in plan and

80.77-m in height) and equipped with an active mass damper (AMD) on the roof with a mass equal to 5% of the total weight of the structure, and passive viscous dampers on each floor (20 viscous dampers total). The authors’ note that in tall buildings controlled at the top floor by an AMD the inter-story drift can be amplified, an unintended and undesirable consequence. Use of a viscous damper on each floor will reduce the inter- story amplification phenomenon. They considered material nonlinearity only using a

7 bilinear hysteresis model and the resulting plastic hinges. Using El Centro and Northridge earthquake records, authors report a tuned mass damper (TMD) does not control the vibrations effectively because it is effective only in a very limited frequency range and linear model-based LQR controller is not effective in reducing the inter-story drift.

A summary of the papers reviewed in this section is presented in Table 1.

1.2 OBJECTIVES AND PLAN OF RESEARCH

The use of passive vibration control systems for seismic protection of structural systems has been an extensive field of study for the past three decades, yet the study of placement of these devices in irregular buildings has not been studied in depth.

Although active dampers have been proven to be more effective than passive dampers, active dampers are also more costly, more complex and need more maintenance. Additionally, passive dampers have proven to be a good practical option for retrofitting of historic buildings (Branco and Guerreiro 2011 and Nawrotzki 2006) and are easier to implement on existing buildings with relatively modest rehabilitation.

Nawrotzki (2008) points out adding TMDs create a dynamic upgrade in existing bridges and buildings that increases structural safety and improves serviceability and comfort conditions.

Table 1: Actuator Placement Review Control Structural Characteristics Conclusion/Comments/ Author Year Device Type Forces Method Type Type MDOF Important and Findings Chakraborty Total of 5 dampers after 50 2012 Active Piezoelectric Seismic Composite plane N/A Genetic Algorithm et al. generations Combination with matrix criterion Zhan et al. 2012 - Sensor Disturbance 2D shear frame 19 Genetic Algorithm optimization Comparison of 5 different Whittle et al. 2012 Passive Viscous Seismic 2D shear frame 6 5 different methods placement techniques Mehrabian Wind Neural Network, Total of 2 actuators out of 12 and Yousefi- 2011 Passive Piezoelectric dynamic Aircraft tail N/A Invasive Weed locations Koma vibration Optimization

Ribakov and Linear, RC and 2011 Active Actuator Seismic 10, 20 Optimization Control of Output of the systems Agranovich Steel

Algorithm 9

Regular and Estekanchi Endurance Time Obtain optimal placement and 2011 Passive Viscous Seismic Irregular shear 3, 8 Method damping coefficient in dampers. and Basim frames Amplitude Active Force Control Patil and Semi- Viscous & Sequential Set 2011 Wind Linear shear frame 76 obtained from Adapting Sliding Active Friction Procedure Jangid Mode Control method Distribution of dampers yield Semi- Li et al. 2011 MR Dampers Seismic Nonlinear MRF 20 Genetic Algorithm fewer dampers necessary for Active desired response Energy dissipation logic between Agranovich Active/Se Linear RC shear Optimization using 2010 LQG Control Seismic 8 undamped and 5th and 7th mi-active frame Heuristic solution and Ribakov comparison Apostolakis Hysteretic Damper configuration, Damper 2010 Passive Seismic MRF, BRB 2,4,6 Genetic Algorithm and Dargush (Friction Area, Damper yield/slip force Semi- Damper placement on bottom and Li et al. 2010 MR damper Seismic Nonlinear 20 Genetic Algorithm active top floors (continued)

Table 1: Continued Control Device Structural Characteristics Conclusion/Comments/ Author Year Forces Method Type Type Structure Type MDOF Important and Findings Shape Obtain number, placement and Ameduri et 2009 Active Metal Noise Rectangular panel - Genetic Algorithm al. orientation of SMAs Alloy wires Uniform, Takewaki and Stiffness Inelastic Nonlinear Lavan et al. 2008 Passive Viscous Seismic 8 3 optimization methods Proportional Damping with Shear weakening technique Modification of the Uniform Liu et al. 2008 Passive Viscous Seismic 3D Irregular Podium 38 Optimization targets Damper Distribution Method Fluid Single bay 3D shear Reduce maximum Base Shear and Kokil and 2007 Passive Viscous Seismic building; soil-structure 10 Optimization Shrikhande Interstory Drift Damper interaction 2D Regular shear Base Force Minimization of Base Force and

10 Aydin et al. 2007 Passive Viscous Seismic 10 frame Optimization Top Displacement Adjacent floors/ no optimization Bhaskararao Friction used/ Implementation of dampers 2006 Passive Seismic 2D shear Frame 10 Parametric study and Jangid Dampers based on maximum relative displacement on floors.

Amini and 2005 Active Controller Seismic Linear, 20 shear frame 3,12, 16 Optimization, Neural Obtain active control force Tavassoli Network Asymmetric Supress bending motion on an Bishop and 2004 Passive Viscous Linear, Space Trusses 72, 78 Genetic Algorithm Striz Loading aircraft tail Simplified Sequential Linear 4,8,12, Story displacement angles and Lopez Garcia 2002 Passive Seismic Linear 2D shear frame Search Algorithm and Soong Viscous 16,20 bottom shear setup (SSSA) Viscous/Vi Controlability Index obtained from Singh and 2002 Passive Seismic 3D Torsional 6, 24 Genetic Algorithm Moreschi scoelastic RMS interstory drift Optimality Criteria Unscontrained and constrained Takewaki 2000 Passive Viscous Seismic 2D linear shear frame 10 Method; Transfer eigenproblems Function

The purpose of this study is to provide an insight into the dynamic performance of dampers in buildings due to plan and setback irregularities subjected seismic loadings through a comparison of different techniques that have been proposed in the past for regular structures. It was noted by Villaverde (1985) that vibration absorbers yield a desired response subjected to earthquake loading only if the appropriate parameters of vibration absorber device are selected. Equations for parameters of vibration dampers have evolved over the years. Early equations were based on harmonic excitations acting directly on the mass which is different when earthquake excitation is acted at the base.

The parametric equations can also be for multi-degree of freedom structures and those with irregularities.

The objectives of this research are to:

1) Compare the performance of a conventional pendulum tuned mass damper

(PTMD) with Bidirectional Tuned Mass Damper (BTMD) using a 5-story

building with plan and elevation irregularity and a 15- and 20- story building

with plan irregularity.

2) Compare the vibration suppression performance of 8 different sets of

equations proposed for tuning the parameters of TMDs using a 5-story

building with plan and elevation irregularity, and a 15-story and a 20-story

building with plan irregularity.

3) Evaluate the effectiveness of BTMD using a 20-story building with plan

irregularity using various earthquake motions

4) Study the most appropriate locations of BTMD in five different structures

with plan and vertical irregularities

CHAPTER 2

STANDARD, PENDULUM, AND

BIDIRECTIONAL TUNED MASS DAMPERS

2.1 INTRODUCTION

The original concept of TMD goes back to an invention disclosed by Frahm in

1909 for a Dynamic Vibration Absorber (Frahm 1909).

One of the first applications of TMD was the 244m 60-story John Hancock building in Boston in 1975 (ENR 1975) in order to improve response to wind vibrations.

It has two 300-ton TMDs each consisting of a 5.2 m square by 1m depth lead-filled steel box riding on a 9m long steel plate. They are placed at the two ends of the 58th floor, 67- m apart and are tuned to a vibration frequency of 1.3 Hz (the estimated fundamental frequency of the structure).

Since then TMDS have been deployed in over 50 structures in several countries including U.S., Japan, China and Korea. A list of 31 Structures (13 Buildings, 9 towers, 6 chimneys and 3 bridges) where passive and active TMDs have been employed up to 1994 can be found in the web site: http://nisee.berkeley.edu/prosys/tuned.html. Holmes (1995) and Spencer and Nagarajaiah (2003) provide tables of structures including bridges where

12 passive and active damping devices were used. An updated version of these lists excluding the bridges is presented in Table 2. Other important recent examples include the 101-story Taipei 101 in Taipei (Figure 1), the Aspire Tower in Doha, Qatar (Figure 2) and Shanghai World Financial Center in Shanghai, China (Figure 3).

Kareem and Kline (1995) study effectiveness of multiple TMDs attached to the roof of a regular 186m-tall square (31m x 31m) and rectangular (31m x 155m) buildings subjected to wind and earthquake excitations. They determine that using multiple dampers can help the system to be effective over a range of frequencies. They conclude that using multiple dampers is more practical for implementation since multiple smaller size TMDs are used versus a much larger single TMD.

Figure 1: Taipei 101 high-rise building (http://goo.gl/TBuz8)

Figure 2: The Aspire Tower (Courtesy of Sonya&Travis)

Figure 3: Shanghai World Financial Center (http://goo.gl/leB2I)

Lin et al. (2010) use a translation-rotation coupled TMD for vibration reduction of a 2D frame and three 24m by15m eight-story asymmetric plan 3D frames (stiff, similarly stiff and flexible) modeled for torsion and subjected to earthquake loading. The authors use min-min-max optimization and take into account the first two modes of the primary structure for tuning of their damper device. Subsequently, Lin et al. (2012) used a bidirectional damper on top of an elastic and inelastic 20-story 3D asymmetric rectangular frame. They assert that calculating the parameters for the proposed device is less complicated compared with multiple tuned mass dampers.

Bekdas and Nigdeli (2011) use harmony search optimization to obtain optimum tuning parameters of a TMD system located on the roof of two 10-story shear frame structures subjected to earthquake loading.

Huang et al. (2011) use cost optimization to compare structural material redistribution and implementation of a tuned mass damper for a 60-story benchmark reinforced concrete structure with steel belt trusses subjected to wind loading. They conclude that a standard redesign of the building to satisfy interstory drift and peak acceleration would require an increase of 14% of the initial material cost, while using a tuned mass damper requires an increase of only 3.6% of the initial structural material cost.

Chung et al. (2012) use steepest descent nonlinear optimization procedure and sensitivity analysis of friction coefficient for design of a friction pendulum-type TMD using Taipei 101 model example subjected to white noise wind excitation. They conclude that using a friction pendulum TMD provides reduction in installation space.

Ikago et al. (2012) present a TMD enhanced by a rotational viscous damper applied to a SDOF and is compared with a conventional viscous damper and conventional TMD subjected to harmonic excitations on a shake table and numerical simulations. It was found that the proposed device is more effective compared to the other devices only when the system is subjected to synthesized ground motion.

Wong and Harris (2012) study the effectiveness of TMDs in reducing structural response of a six-story 3-bay moment-resisting steel frame subjected to 100 simulated non-stationary Gaussian ground motions, and conclude that: “a TMD can enhance the structure’s ability to dissipate energy at low levels of earthquake shaking, while less effective during moderate to strong earthquakes, which can cause a significant period shift associated with major structural damage. This ‘de-tuning’ effect suggests that an extremely sizable TMD is not effective in reducing damage of a structure.”

Moutinho (2012) proposes an alternative method over tuning parameter equations based on: “direct assignment of damping ratios of the building’s first two vibration modes by establishing the mass ratio corresponding to a specific structural solution.”

This method is used in a five-story steel frame with a large mass ratio rubber-bearing bracing system (acting as a TMD) on top floor subjected to seismic excitations. The author compared the proposed method with four TMD tuning parameter equations and concluded that in terms of significant reduction response, no method outperformed another greatly.

De Angelis et al. (2012) study the response of a TMD with a large mass based on

3 shake table tests of 3D 2-floor steel frame, one example having the additional mass as a high-damping rubber bearing positioned on the 2nd floor subjected to four earthquake

16 excitations. The authors claim that a TMD with a higher mass ratio is less frequency- dependent and more robust.

Almazan et al. (2012) study the performance and placement of one or more TMDs in asymmetric buildings using four different examples including a 15-story RC 3D structure with plan irregularity subjected to seismic loading with the goal of minimizing the inter-story displacement or drift. They use the Kanai-Tajimi spectrum to model the seismic excitation and place 1 or 2 TMDs on the roof. For asymmetric structures, they conclude that the optimum TMD frequency and location on the roof depend on structural lateral and torsional stiffness, eccentricity between centers of mass and rigidity and the frequency content of the ground motion, and the optimum location is near the geometric center of the plan. They also conclude: “if the uncertainty associated to the dynamic parameters of the main structure is ignored, a significant improvement would not be obtained by adding a second TMD.”

Table 2: Buildings and towers with TMD applications in chronological orders

Type of Structure No. Year Name Height Location TMD Type / Characteristics (Building/Tower) Floors completed 553 m CN Tower Tower - Toronto, Canada 1976 2 pendulum TMD Weight= 18ton

4 TMD John Hancock Building 241 m 60 Boston 1976 0.14 Hz Weight = 2x300ton

TMD Citycorp Center 0.16Hz Building 278 m 59 New York, USA 1978 Weight = 370 ton

Placement= 278m 19

2 TMD 0.10, 0.50 Hz Sydney Tower Tower 305 m - Sydney, Australia 1980 Weight = 220ton (180t/40t) Placement = 256m/165m

TMD Al Khobar 2 Chimneys 120 m - Saudi Arabia 1982 0.44Hz Weight = 7tons

TMD Ruwais Utilities Chimney - Abu Dhabi 1982 0.49 Hz Weight = 10tons

Nurnberg, TMD Deutsche Bundespost Tower 278 m - 1982 0.67Hz Germany Weight = 1.5tons (Continued)

Table 2 : Continued

TMD 0.43-0.44 Hz Chiba Port Tower Tower 125 m 4 Chiba, Japan 1986 Weight = 10, 15ton Placement = 125 m (top)

TMD Yanbu Cement Plant Chimney 81 m - Saudi Arabia 1984 0.49Hz Weight = 10tons

TMD Tiwest Rutile Plant Chimney 43 m - Cataby, Australia 1989 0.92 Hz

20 Weight =0.5 tons

2 TMD Fukuoka Tower Tower 234 m 2 Fukuoka, Japan 1989 0.31-0.33 Hz Weight =25-30 tons

Active TMD Higashiyama Sky Tower Tower 134 m 2 Nagoya, Japan 1989 0.49-0.55 Hz Weight= 20tons 2 TMD Kyobashi Center Building 33 m 11 Tokyo, Japan 1989 Weight=5 ton 1 TMD Fernsehturm Tower Television Tower 368 m - Berlin, Germany 1990 Weight= 1.5 tons 1 TMD 0.65 Hz Huis Ten Bosch Domtoren Tower 105 m - Nagasaki, Japan 1990 Weight= 7.8 tons Placement = 100m (top) (Continued)

Table 2: Continued

Pendulum TMD 0.24-0.28 Hz Crystal Tower Building 157 m 37 Osaka, Japan 1990 Weight= 540 ton

TMD Shimizu Tech Lab Building 30 m 7 Tokyo, Japan 1990 Weight =4.3 ton TMD BASF Chimney 100 m - Antwerp, Belgium 1992 0.34 Hz Weight= 8.5 tons TMD Frankfurt,

21 HKW Chimney 120 m - 1992 0.86 Hz Germany

Weight = 10 tons 2 ATMD ORC 2000 Symbol tower Building 188 m 50 Osaka, Japan 1992 0.21 Hz Weight= 200 ton

1TMD Applause Tower Building 162 m 34 Osaka, Japan 1992 Weight = 480 ton

2TMD Sendagaya INTES Building 58 m 11 Tokyo, Japan 1992 Weight= 72 ton

Active TMD Rokko Island Procter and Building 117 m 36 Kobe, Japan 1993 0.33-0.62 Hz Gamble Weight = 270 tons (Continued)

Table 2: Continued

2 active tuned mass Yokohama Landmark Yokohama City, dampers Building 296.3 m 73 1993 Tower Japan 0.185 Hz Weight = 340 ton 1 TMD Chifley Tower Building 209 m 53 Sydney, Australia 1993 Weight = 400ton Placement = 44th floor 1 TMD Al Taweelah Chimney 70 m - Abu Dhabi 1993 1.4 Hz

22 Weight= 1.35 ton

2 ATMD Kansai International Tower 86 m 7 Osaka, Japan 1993 0.8 Hz airport Weight= 10 ton AMD C Office Tower Building 130 m 32 Tokyo, Japan 1993 0.34 Hz Weight= 200 ton AMD KS Project Tower 121 m - Kanasawa, Japan 1993 Weight = 100 ton TMD and AMD Ando Nishikicho Building 68 m 14 Tokyo, Japan 1993 0.68Hz – 0.72Hz ATMD pendulum MKD8 Hikarigaoka Building 100 m 30 Tokyo, Japan 1993 0.44 Hz 3 TMD P&G Japan Headquarters Building 131 m 31 Kobe, Japan 1993 Weight = 270 tons

(Continued)

Table 2: Continued

1TMD Akita Tower Tower 112 m - Akita, Japan 1994 0.41 Hz 2 AMD Riverside Sumida Building 133 m 33 Tokyo, Japan 1994 0.29Hz Weight= 30 ton Active/Passive TMD Act City Building 213 m 45 Hamamatsa, Japan 1994 0.21 Hz Weight= 180 ton 23 Shinjuku Park Tower 3 ATMD Building 227 m 33 Tokyo, Japan 1994 Weight = 330 ton 2TMD Building M Building 30.4 m 9 Osaka, Japan 1994 1.33 Hz (x) 1.54 Hz (y) Sea Hawk Hotel and TMD Building 143 m 36 Fukuoka, Japan 1995 Resort Weight = 132tons 11 TMDs Hotel Burj-Al-Arab (7- Building 321 m 60 Dubai, U.A.E. 1997 Frequency= .8 – 2 Hz star) Weight= 11x5.00 ton 12 TMD (3 TMD per Kuala Lumpur, skybridge leg, 4 total) Petronas Twin Towers Building 451.9 m 88 1997 Malaysia 0.13, 0.17, 2.22 Hz Weight= 160lb each 1 TMD Itoyama Tower Building 89 m 18 Tokyo, Japan 1997 Weight = 48 ton (Continued)

Table 2: Continued

2 TMD TC Tower Building 348 m 85 Kau-Shon, Taiwan 1997 Weight= 100 ton Kaikyo-messe Dream 1 TMD Tower 153 m - Yamaguchi, Japan 1998 Tower Weight =10 ton Otis Shibayama Test 1 TMD Building 154 m 39 Chiba, Japan 1998 Tower Weight = 61ton 6 TMD 0.9 Hz Emirates Towers Tower 355m 54 Dubai, UAE 1999

24 Weight=1.2 tons each

(Mast) Pendulum TMD Steel Chimney Chimney 90 m - Bangkok, Thailand 1999 Frequency= 0.8 Hz Weight=4 ton 4 TMD Century Pak Tower Building 170 m 54 Tokyo, Japan 1999 Weight= 440 ton 1 AMD Nanjing Tower Tower 310 m - Nanjing, China 1999 Weight = 60 ton 2 TMD Shinagawa Intercity A Building 144 m 32 Tokyo, Japan 1999 Weight= 150 ton TMD Park Tower Building 252.2 m 67 Chicago, IL, USA 2000

7 TMD Stakis Metropole Hotel Building 60m 20 London, UK 2000 4.4 Hz Weight=4.5 tons (Continued)

Table 2: Continued

New York, NY, 1 TMD The Trump World Tower Tower 262.4 m 72 2001 USA Weight = 600 ton Cerulean Tower Tokyo 2 TMD Building 184 m 5 Tokyo, Japan 2001 Hotel Weight= 210 ton Triton Square office 4 ATMD Building Complex 195 m Tokyo, Japan 2001 complex Weight= 35 ton each Hotel Nikko Bayside 2 TMD Building 138 m 33 Osaka, Japan 2002 Osaka Weight =124 ton 4 TMD Dentsu New Headquarter Building 210 m 48 Tokyo, Japan 2002

25 Weight = 440 ton

2 ATMD&TMD Incheon International 0.71Hz Tower 100.4 m 22 Incheon, Korea 2001 Airport Control Tower Weight=11&13tons Placement = 19th floor

Spire of Dublin Monument 121.2 m - Dublin, Ireland 2003 TMD

4 TMD Refab2 Stack Brazil 2003 Weight= 55ton each

Highcliff Building 252.4 m 73 Hong Kong, China 2003 TMD

2 TMD at Masts 1.05 Hz Al Rostamani Tower Tower 67m 16 Dubai, UAE 2003 Weight=0.5 tons

(Continued)

Table 2: Continued

2 TMD Taipei 101 Building 449 m 101 Taipei, Taiwan 2004 0.15 Hz Weight = 730 ton/2x4.5ton TMD Bloomberg Tower Building 245.6 m 54 New York, USA 2004 Weight = 600 ton DoCoMo Telecommunications Tower 198.5 m 12 Osaka, Japan 2004 1 TMD Tower

Bright Start Tower TMD Mast 284m 60 Dubai, UAE 2005 0.95 Hz (Millennium Tower) 26 Weight=0.3 tons

Santiago de Chile, 4 TMD Parque Araucano Building 60m 20 2005 Chile Weight=170 tons Edinburgh, ATMD Air Traffic Control Tower Tower 57m - 2005 1.7-2.0 Hz Germany Weight=14 tons Meteorological Radar Catalunya Ring damper TMD Tower 50m - 2005 4.2 Hz Tower Province, Spain Weight=8 tons Pendulum TMD Aspire Tower Tower 300 m 36 Doha, Qatar 2007 0.22 Hz Weight= 140 ton Philadelphia, PA, TMD Comcast Center Building 297.1 m 57 2008 USA Weight= 1,300 tons Shanghai World ATMD Building 492 m 101 Shanghai, China 2008 Financial Center Placement at 90th floor (Continued)

Table 2: Continued

2 TMD Al Mas Tower Building 361m 68 Dubai, UAE 2008 1.6 & 3,2 Hz Weight=2 tons each (Masts) 2 TMD Villa Magura Magura Odobesti TV Tower 112m - 2008 2.2 Hz and 0.7 Hz Odobesti, Romania Weight= 0.3 and 2.85 tons 1 TMD Hangzhou Bay Bridge Tower Tower 130m - Jiaxing, China 2009 0.3 Hz Weight= 110tons 4 TMDs Lanxess, Chemical Plant Building 2 Ontario, Canada 2009 Weight= 3ton each ShenZhen WuTong Adaptive PTMD Tower 198 m - ShenZhen, China 2009 Mountain Tower ATMD & TMD Canton Tower

27 Tower 600 m 85 Guangzhou, China 2010 Weight=50ton &600ton (Guangzhou TV Tower) th

Placement at 85 floor 8 TMD Estela de la Luz Tower 104m - Mexico City, Mexico 2010 0.3 Hz Weight= 3 tons TMD Tokyo Skytree Tower 634.0 m - Tokyo, Japan 2012 Weight=100tons

cd kd

ms x

cs ks

Figure 4: SDOF system with a Tuned Mass Damper (TMD)

The equations representing a SDOF system with a tuned mass damper (Figure 4) are,

(1)

(2)

(3)

(4)

(5)

(6)

where represents the ratio of the TMD mass ( ) to structural mass ( ), ks and cs are the stiffness and damping coefficient of the structures, kd and cd are the stiffness and damping coefficient of the damper, is the damping ratio of the structure,

is the damping ratio of the TMD and γ is the ratio of the frequency of the TMD ( ) to the frequency of the structure ( ).

2.2 SELECTION OF OPTIMUM PARAMETERS VALUES FOR A TMD

Different researchers have presented equations for the optimum values of the parameters of the TMD using different criteria or approaches. A summary is provided in

Table 3. Most researchers present optimum values of the damping and frequency ratios based on a given mass ratio.

Den Hartog (1956) derived equations by minimizing the response of an undamped

SDOF system subjected to sinusoidal loading. Warburton (1982) derived equations considering harmonic and white noise random excitations directly on a SDOF system based on the notion that the average damping ratios of the resonant modes between the uncontrolled structure and the TMD is approximately equal to the effective damping ratio of a building with TMD.

Villaverde (1985) derived equations for a TMD with high damping and small mass ratios by minimizing the dynamic response of a 2D three-story frame and a 3D 10- story building subjected to seismic loading.

Fujino and Abe (1993) present four sets of equations for optimum tuning parameters of a 2DOF system for four different cases: free vibration, harmonic, self- excited, and random excitations based on an optimization procedure using the mean- square values of perturbation solutions of displacement responses assuming both mass and structural damping ratios of less than 0.02. The equations for the case of random excitations are included in Table 3 and used in this research.

Feng and Mita (1995) proposed two sets of equations, one for displacement and the other for acceleration, for the optimum values of frequency and damping ratios of a

TMD attached to a 2 DOF system by minimizing the dynamic response in terms of mean

28 square values of displacement and acceleration. They study the performance of their equations on a 200-m tall building 2D frame with a structural damping ratio of 0.02 subjected to earthquake excitations.

Sadek et al. (1997) modified the equations proposed by Villaverde (1985) and use curve fitting to find optimum parameters of the TMD system such that the first two modes of the controlled structure have the same damping ratio with a value equal to greater than the average damping ratio of TMD and structure alone.

Rana and Soong (1998) performed a parametric study of tuning the first mode of

SDOF and 3-DOF systems subjected to harmonic and seismic excitations using the min- max numerical optimization. The authors also studied the out-of-tune effect of TMD, dubbed detuning, on frequency and damping ratios. They noted that frequency ratio is more sensitive to detuning effect than damping ratio and that increasing mass and damping ratios reduces the severity of detuning effect on structure.

They report deterioration in the response when the SDOF and 3-DOF systems are subjected to earthquake excitations compared with harmonic excitation. The authors also use multiple TMDs for the 3-DOF system subjected to two different seismic accelerograms and report improvement in the response of the structure since multiple

TMDs are tuned for multiple mode shapes.

Table 3: Equations for finding the optimum tuning parameters of a TMD Year of Frequency ratio Damping ratio Author Comments Publication First equations for optimum Den tuning parameters of a TMD 1956 and an undamped SDOF √ Hartog system based on a harmonic excitation Based on random white-noise √ Warburton 1982 √ acceleration excitation and an undamped SDOF system

Fujino and √ √ Based on random excitations 31 1993 Abe for undamped 2DOF systems

(i) √ Based on white-noise Feng and √ excitation of 2DOF system for 1995 Mita displacement (i) and (ii) √ acceleration (ii)

(i) (i) √ Based on earthquake Sadek et excitation for both undamped 1997 al. (i) and damped (ii) 30 SDOF (ii) [ √ ] (ii) √ systems Based on white noise excitation on undamped and Rudinger 2006 damped linear and nonlinear √ √ SDOF systems with linear damping =1 Based on force and white Krenk and noise acceleration excitation 2008 √ Hogsberg on undamped and damped 2DOF systems (Continued)

Table 3: Continued

Year of Frequency ratio Damping ratio Author Comments Publication

√ √ (i) (i) Based on seismic excitation of a damped SDOF for ranges of

( ) ground frequency ratio of Hoang et 2008 √( ) (ii) √ / : al. ( ) (ii) (i)

(ii) √ (iii) √ (iii) (iii)

Rudinger (2006) used statistical linearization to derive TMD optimum tuning parameter equations for a damped SDOF system subjected to white noise excitation. The author concludes TMD optimum tuning parameters are the same for undamped and damped systems that behave linearly. On the contrary, optimal parameters for a nonlinear system with TMD are dependent on structural damping and excitation intensity (vibration amplitude). Krenk and Hogsberg (2008) present optimum tuning equations for undamped and damped 2DOF systems subjected to force and acceleration excitations independent of the structural damping ratio.

Hoang et al. (2008) present equations for optimum parameters of a TMD for a

SDOF system subjected to seismic excitation for different ranges of the ratio of the dominant ground frequency ( to structural frequency ( , , using a nonlinear programming technique. The idea is to have site-specific parameters thus taking into account the type of soil on which a structure is built. The authors presented a numerical example where they studied the effect of large mass ratios for TMD in a bridge subjected to earthquake loading. They noted that the: “optimal TMD has lower tuning frequency and higher damping ratio with increasing mass ratio.” They also concluded that using a large mass ratio can help against parameter uncertainties that occur between theoretical model and the actual practical implementation.

The values of and obtained through equations presented in Table 3 determine the TMD stiffness and damping parameters (Den Hartog 1956, Rana and

Soong 1998, Connor 2003):

(7)

(8)

2.3 PENDULUM TUNED MASS DAMPER

Conventional mass & spring TMD damper requires a large mass and a large space for installation thus creating architectural constraints (Nagase 2000). One alternative solution is using a pendulum TMD (PTMD) configuration consisting of a mass and a cable shown in Figure 5. When a building is subjected to an earthquake motion the

PTMD will create a force in opposite direction of the floor movement. PTMD has been used in a number of high-rise buildings such as the 37-story 157m-high Crystal Tower located in Osaka, Japan, built in 1990, and shown in Figure 6 to reduce wind-induced displacements by 50% (Nagase and Hisatoku 1992). The fundamental periods of the structure in two principal directions are 4.7 and 4.3 seconds. The building holds 2

PTMDs of weight 180 tons and 360 tons on the top story.

(a) (b)

Figure 5: Pendulum Tuned Mass Damper (PTMD)

Figure 6: The Crystal Tower (Courtesy of Takashi Kassai)

The following equations are used in the design of a PTMD (Connor 2003),

(9) where T is the tension in the cable, x is displacement of the structure, and xd is the displacement of the PTMD. Assuming θ is small the following approximations are made:

(10)

(11)

Substituting Eqs. 10 and 11 into Eq. 9:

(12)

Then, the spring stiffness is calculated as

(13)

The natural frequency becomes,

(14)

and the natural period for the pendulum TMD is,

√ (15)

As such, the tuning parameters of a PTMD are the mass md and the length L.

2.4 BIDIRECTIONAL TUNED MASS DAMPER

Almazan et al. (2007) proposed a bidirectional TMD (BTMD) device and applied it to control vibrations of a 25-story reinforced concrete 3D regular frame and an 80-m by

3-m diameter thin-walled cylindrical steel structure subjected to seismic loading. This device is a combination of two cables forming a Y-shape connecting to the mass at the middle, and a friction damper connected to the mass from the bottom (Figure 7).

The tuning parameters of a BTMD are the mass md and the lengths and of the cables from the mass to the fix support and to Y-intersection (Figure 7) which are determined as follows:

(16)

(17)

̃ (18)

where and stand for the fundamental natural frequencies of the structure in the two principal directions considered. These frequencies are determined based on the

̃ dominance of the modal mass participation factor. The optimal slip force is obtained based on the equivalent damping ratio, (which is set equal to . is the viscous damping ratio of the friction damper located at the connection and is the remaining distance from the intersection to the floor. The optimum tuning parameters can be the same values used for a standard TMD presented in Table 3 and Figure 8.

A mass ratio ) within the range of 3-5% has been recommended in the literature

(Connor 2003). A mass ratio of is used in this research. A structural damping

( ) of 0.05 is recommended for steel structures (ASCE, 2005) and used in this research.

The design steps for modeling PTMD and BTMD are presented in the flowchart of Figure 8 where the tuning parameter equations proposed by Sadek et al. 1997 are used as an example to demonstrate the procedure. A MATLAB R2012b script file and command window that is used to calculate the optimum tuning parameters is located in

Appendix B. (MATLAB is a numerical computing environment and programming language developed by MathWorks, Inc.)

(c)

Figure 7: Bidirectional Tuned Mass Damper (BTMD). a) Three-dimensional view, b) view in the xz plane, c) view in the yz plane

Obtain response information Determine dominant modal from uncontrolled structure frequencies and mass

Calculate frequency ratio: Select mass ratio and compute 휇 휸풐풑풕 [ 휉 √ ] TMD mass 휇 푠 휇 푚 휇 푑 푚푠

Calculate damping ratio

√

Determine PTMD/BTMD

frequencies

Determine PTMD/BTMD parameters

Figure 8: PTMD and BTMD design procedure using tuning parameters proposed by Sadek et al. (1997) as an example

CHAPTER 3

COMPARISON OF EQUATIONS FOR OPTIMUM TUNING PARAMETERS

3.1 INTRODUCTION

In this chapter, eight different sets of equations for finding the optimum BTMD tuning parameters summarized in Table 3 are compared using three different irregular structures.

Example 1: Five-story 3D Ordinary Moment Resisting Steel Frame with Plan and

Elevation Irregularity

The 5-story 3D ordinary moment resisting steel frame with setbacks and an L- shape plan is shown in Figures 9 and 10. This example was originally created by Young and Adeli (2013). Roof and floor framing systems consist of steel beams and 6-in thick lightweight concrete over steel metal deck working as a rigid diaphragm. The loads on the structure include dead load from self-weight for steel members and concrete floors, additional dead load for roofing, partitions, furniture and other structure items (15psf), cladding (25psf), live load (50psf) and wind load corresponding to 85mph, gust factor

0.85 and exposure category C. It is designed for a design acceleration spectrum of 0.28g,

39 site class D and seismic category B (ASCE, 2010). The beam and column sizes are summarized in Table 4. Centers of mass (CM) and centers of rigidity (CR) are identified in Figure 10.

Figure 9: Five-story irregular 3D building structure with setbacks and an L-shape

plan.

Table 4: Beam and column designs for the 5-story irregular building

Story Column L Beam Section L Beam Section Weight of (i) Section (in) – Interior (in) – Perimeter floor (kN) 5 W14X34 120 W14X26 180 W14X22 236.2 4 W14X34 120 W14X26 180 W14X22 236.2 3 W14X34 120 W14X26 180 W14X22 1222.6 2 W14X61 120 W14X26 180 W14X22 1222.6 1 W14X61 144 W14X26 180 W14X22 1250.4 Total Weight 4168.1

CM + 1 CR1

CM2

+ CR2

Figure 10: Plan view of 5-story building (+CR1=center of rigidity of the top two floors, +CR2=center of rigidity of the bottom three floors, ●CM1=center of mass of the top two floors, ●CM2=center of mass of the bottom three floors)

Table 5 presents the modal response data for the first 12 modes of the uncontrolled structure. The more irregular the structure, the larger number of modes is needed for accurate determination of the dynamic response of the structure (Liang et al.

2012). In this example, the dominant modes, that is, the modes with the largest modal participation factors, are found to be the first two modes (their participation factors are identified in shaded boxes in Table 5).The first three mode shapes are shown in Figure

11.

Table 5: Free vibration modal response of the 5-story irregular building

Modal Participation Factors Period Mode X Y Z (sec) Translation Translation Torsion 1 0.807 0.001 0.470 0.036 2 0.553 0.480 0.071 0.770 3 0.502 0.150 0.360 0.021 4 0.444 0.180 0.027 0.003 5 0.279 0.052 0.000 0.016 6 0.240 0.018 0.004 0.027 7 0.183 0.047 0.027 0.023 8 0.165 0.024 0.039 0.068 9 0.082 0.032 0.000 0.004 10 0.055 1.11E-03 5.98E-08 8.90E-05 11 0.035 7.52E-05 5.34E-06 2.18E-05 12 0.014 3.24E-03 2.22E-05 1.88E-06 Total Sum = 0.989 0.998 0.970

The most dominant mode turns out to be the second mode of vibrations. As such, the parameters of PTMD are determined for the second mode of vibrations and BTMD for the first two modes of vibrations. The latter is used for the comparative study in this chapter. A BTMD is placed at the CR of the roof location (B-5 in Figure 10).

(a) (b)

(c)

Figure 11: Mode shapes of the 5-story 3D building with vertical and plan irregularity: (a) Mode 1 (period = 0.81sec), (b) Mode 2 (period = 0.55sec), (c) Mode 3 (period = 0.50sec)

The uncontrolled structure is subjected to Loma Prieta, Northern California, earthquake of October 17, 1989, with Magnitude 7.1 (Station 58378) displayed in Figure

12 using twenty four different incident angles with increments of 15 degrees in the horizontal plane.

0.3 0.2 0.1 0 -0.1 -0.2 -0.3

-0.4 Acceleration (g) Acceleration -0.5 -0.512

0 5 10 15 20 25 30 35 40 Time (sec)

Figure 12: Loma Prieta earthquake accelerogram (October 17, 1989, Magnitude 7.1)

The maximum absolute displacement for each incident angle and the node at which this occurs are presented in Table 6. Each node is identified with a letter

(indicating the grid in the N-S direction) followed by a number (indicating the grid in the

E-W direction), a dash, and a second number (indicating the floor number). For example,

A6-5 indicates the node at the intersection of grid lines A and 6 on the 5th floor.

The maximum absolute displacement was obtained for the incident angle of 75 degrees with respect to the X-axis shown in Figure 10. Next, the incident angle resolution was reduced to 5 degrees around the incident angle of 75 degrees between 60 and 90 degrees. The maximum absolute displacement was obtained for the incident angle of 80 degrees at joint A6-5. The maximum absolute joint displacements in cm for different earthquake incident angles are depicted in Figure 13.

The critical angle of incidence of 80 degrees is used to compare the 8 sets of equations for PTMD/BTMD tuning parameters presented in Table 3. The results of the comparisons for maximum absolute displacements and accelerations are presented in

Table 7. The detailed values of the optimum tuning parameters for the eight sets of equations are included in Tables 26 and 31 of Appendix A.

Based on Table 7 the formulas suggested by Hoang et al. (2008) using δ=1 present the most effective control in decreasing the maximum displacement by 41.3%, maximum acceleration by 35.3%, while formulas suggested by Sadek et al. (1997) result in the highest decrease in absolute base shear by 25.7%.

Results for the maximum inter-story drifts for the eight sets of equations are presented in Table 8 and Figure 14 and for the maximum floor accelerations are presented in Table 8 and Figure 15. Equations proposed by Hoang et al. (2008) provide the most effective inter-story displacement control. Equations proposed by Fujino and

Abe (1993), Feng and Mita (1995), Sadek et al. (1997), Krenk and Hogsberg (2008) and

Hoang et al. (2008) yield similar maximum floor acceleration results and are more effective than the rest of the equations.

Table 6: Maximum absolute displacement for 28 different earthquake incident angles for the 5-story irregular building

Incident Angle Max Displacement Joint (°) (cm) (Grid-Floor) 0 5.12 C6-5 15 6.33 C6-5 30 7.47 C6-5 45 8.37 C6-5 60 8.84 C6-5 70 8.88 A6-5 75 8.98 A6-5 80 9.02 A6-5 85 8.99 A6-5 90 8.90 A4-5 105 8.27 A6-5 120 7.72 A6-5 135 7.37 A6-5 150 6.57 A6-5 165 5.58 C6-5 180 5.12 C6-5 195 6.33 C6-5 210 7.47 C6-5 225 8.37 C6-5 240 8.84 C6-5 255 8.98 A6-5 270 8.90 A4-5 285 8.27 A6-5 300 7.72 A6-5 315 7.37 A6-5 330 6.57 A6-5 345 5.58 C6-5 360 5.12 C6-5

Figure 13: Maximum absolute joint displacement (cm) for different earthquake incident angles for the 5-story building

Table 7: Optimum tuning parameters and maximum absolute displacement (cm) and acceleration (g) and base shear (kN) for the 5-story building Maximum Response of the Controlled Optimal Parameters Structure Method Frequency Damping Absolute Absolute Absolute ratio ratio Displacement Acceleration Base Shear (cm) (g) (kN) Hoang et al. (δ=1) 0.858 0.083 5.44 0.460 802.7 Sadek et al. 0.963 0.219 6.37 0.494 720.2 Feng and Mita 0.964 0.083 6.44 0.502 764.0 Hoang et al. (δ=3) 0.928 0.086 6.49 0.504 763.9 Krenk and 0.971 0.086 6.67 0.511 754.5 Hogsberg Warburton 0.964 0.0860 6.95 0.523 745.4 Den Hartog 0.971 0.101 6.96 0.520 764.2 Fujino and Abe 0.964 0.0857 7.03 0.527 743.2 Rudinger 0.971 0.005 7.82 0.583 788.9 Uncontrolled - - 9.27 0.710 969.6

4 Den Hartog Warburton Fujino and Abe 3

Feng and Mita 49

Sadek et al.

Floor No. Floor 2 Rudinger Hoang et al. (δ=1) Hoang et al. (δ=3) 1 Krenk and Hogsberg Uncontrolled

0 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 Maximum absolute inter-story displacement (m)

Figure 14: Maximum interstory displacement (drift) for 5-story building using different equations for optimum TMD tuning parameters

4 Den Hartog Warburton

Fujino and Abe 3

50 Feng and Mita

Sadek et al.

Floor No. Floor 2 Rudinger

Hoang et al. (δ=1) Hoang et al. (δ=3) 1 Krenk and Hogsberg Uncontrolled

0 0.105 0.205 0.305 0.405 0.505 0.605 0.705 0.805 Maximum absolute acceleration (g)

Figure 15: Maximum absolute floor acceleration for 5-story building using different sets of equations for optimum TMD tuning parameters

Table 8: Maximum absolute floor acceleration and interstory displacement (drift) for 5-story building using different equations for optimum TMD tuning parameters Floor No. Fujino and Feng Sadek Hoang et al. Hoang et Krenk and Den Hartog Warburton Rudinger Uncontrolled \Method Abe and Mita et al. (δ=1) al. (δ=3) Hogsberg Inter-story displacement (drift)

5 0.0258 0.026 0.026 0.025 0.025 0.024 0.023 0.025 0.025 0.029

4 0.0213 0.021 0.022 0.016 0.020 0.021 0.010 0.016 0.018 0.024

3 0.0045 0.005 0.004 0.005 0.005 0.005 0.005 0.005 0.005 0.009

2 0.0079 0.008 0.008 0.008 0.008 0.008 0.009 0.008 0.008 0.012

1 0.0101 0.010 0.010 0.011 0.010 0.011 0.011 0.011 0.010 0.015

51 0 0.0000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Acceleration (g)

5 0.520 0.523 0.527 0.502 0.494 0.542 0.461 0.504 0.511 0.703

4 0.440 0.351 0.349 0.355 0.334 0.431 0.367 0.356 0.351 0.467

3 0.320 0.310 0.313 0.295 0.310 0.294 0.303 0.295 0.297 0.319

2 0.280 0.279 0.284 0.283 0.273 0.250 0.285 0.283 0.281 0.266

1 0.160 0.143 0.162 0.165 0.154 0.151 0.170 0.166 0.164 0.157

0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Example 2: Fifteen-story reinforced concrete Moment Resisting Frame with plan irregularity

This example is a 15-story irregular reinforced concrete moment-resisting frame with an L-shape plan shown in Figure 16. This building was created originally by

Almazan et al. (2012) according to the Chilean code for seismic design of structures and industrial installation which is based the American Concrete Institute (ACI) 318-05 code and the 2006 International Building Code (IBC). Valarezo et al. (2007) compared designs of reinforced concrete frames using 8 different seismic codes including the Chilean seismic code and the IBC code and concluded that the two codes have very similar design requirements. The beams have a 0.3m by 0.7m cross-sectional area. All columns have a square 0.5m by 0.5m cross-sectional area, except columns along grid line 4 which have a

1. m by 0.4m cross-sectional area and grid location C-3 which have a 0.4m by 1.m cross- sectional area. The slab is 0.15m thick and is modeled as a rigid diaphragm. This model was checked using the SAP2000 verification code feature for ACI-318-05 and IBC-2006.

Table 9 presents the free vibrations response data for the uncontrolled structure.

Figure 17 shows the first three mode shapes of the structure. This structure is subjected to the same Loma Prieta earthquake accelerogram used in the previous 5-story example

(Figure 12) and studied using 24 different incident angles with increments of 15 degrees in the horizontal plane. The maximum absolute displacement for each incident angle and the node at which this occurs are presented in Table 10.

The maximum absolute displacement was obtained for the incident angle of 45 degrees with respect to the X-axis (Figure 16). Next, the incident angle resolution was reduced to 5 degrees around the incident angle of 45 degrees between 45 and 60 degrees.

The maximum absolute displacement was obtained for the incident angle of 50 degrees at joint D1-15. The maximum absolute joint displacements in cm for different earthquake incident angles are also plotted in Figure 18. This angle of incidence is used to compare the eight sets of equations for TMD tuning parameters presented in Table 3.

The results of the comparisons for maximum absolute displacements and accelerations are presented in Table 11. The detailed values of the optimum tuning parameters for the eight sets of equations are included in Table 27 and Table 32.

Equations proposed by Sadek et al. (1997) are the most effective in reducing the structural response, shaded in Table 11. Equations proposed by Hoang et al. (2008) and

Krenk and Hogsberg (2008) yield the next best results. Equations presented by Sadek et al.

(1997) decrease the maximum absolute displacement by 49.76%, maximum acceleration by 32.14%, and absolute base shear (kN) by 42.22%.

The results of the comparisons for maximum absolute displacements and accelerations are presented in Table 11.

CM CR 18m Y

(b)

(a)

Figure 16: 15-story building reinforced concrete 3D building with plan irregularity (Almazan et al. 2012). (a) Perspective view. b) Plan view. +CR= center of rigidity, ●CM= center of mass.

Table 9: Free vibration modal response of the 15-story irregular building

Modal Mass Participation Factors Period Mode X Y Z (sec) Translation Translation Torsion 1 2.933 0.006 0.781 0.187 2 2.875 0.774 0.006 0.296 3 2.317 0.001 0.000 0.289 4 0.946 0.000 0.099 0.028 5 0.918 0.099 0.000 0.029 6 0.739 0.002 0.000 0.044 7 0.533 0.000 0.037 0.013 8 0.513 0.036 0.000 0.007 9 0.407 0.002 0.000 0.021 10 0.377 0.000 0.000 0.000 11 0.368 0.000 0.000 0.000 12 0.362 0.000 0.000 0.000 Total Sum = 0.921 0.925 0.915

Equations proposed by Sadek et al. (1997) are the most effective in reducing the structural response, shaded in Table 11. Equations proposed by Hoang et al. (2008) and

Krenk and Hogsberg (2008) yield the next best results. Equations presented by Sadek et al.

(1997) decrease the maximum absolute displacement by 49.76%, maximum acceleration by 32.14%, and absolute base shear (kN) by 42.22%.

(a) (b)

(c)

Figure 17: Mode shapes of the 15-story building with plan irregularity (a) Mode 1 (period = 2.93 sec), (b) Mode 2 (period = 2.88sec), (c) Mode 3 (period = 2.32sec)

Table 10: Maximum absolute displacement (cm) for different earthquake incident angles of 15-story building Incident Angle Max Displacement Joint (°) (cm) (grid-floor) 0 22.25 D1-15 15 22.60 D1-15 30 22.90 D1-15 45 23.07 D1-15 50 23.09 D1-15 55 23.08 D1-15 60 23.05 D1-15 75 22.90 D1-15 90 22.50 D1-15 105 22.10 D2-15 120 22.00 A1-15 135 21.99 A1-15 150 22.20 A1-15 165 22.20 A1-15 180 22.30 D1-15 195 22.60 D1-15 210 22.90 D1-15 225 23.07 D1-15 240 23.05 D1-15 255 22.90 D1-15 270 22.50 D1-15 285 22.10 D1-15 300 22.00 D1-15 315 21.99 D2-15 330 22.20 A1-15 345 22.20 A1-15 360 22.30 A1-15

Figure 18: Maximum absolute displacement (cm) per earthquake incident angles of 15-story building

Table 11: Optimum tuning parameters and maximum absolute displacement (cm), acceleration (g) and base shear (kN) response for the 15-story building

Maximum Response of the Controlled Optimal Parameters Structure Method Frequency Damping Absolute Absolute Absolute ratio ratio Displacement Acceleration Base Shear (cm) (g) (kN) Den Hartog 0.971 0.101 14.7 0.128 1011 Warburton 0.964 0.086 13.5 0.128 922.6 Fujino and Abe 0.964 0.086 13.5 0.128 922.6 Feng and Mita 0.964 0.083 13.56 0.129 925.5 Sadek et al. 0.963 0.219 11.31 0.124 792.9 Rudinger 0.971 0.005 14.89 0.128 1003 Hoang et al. 0.858 0.083 14.09 1022 (δ=1) 0.124 Hoang et al. 0.928 0.086 13.68 930.3 (δ=3) 0.130 Krenk and 0.971 0.085 13.58 920.0 Hogsberg 0.129 Uncontrolled - - 22.52 0.228 1372.2

Example 3 Twenty-story Steel Moment Resisting Frame with plan irregularity

This example is a 20-story steel moment resisting frame structure with plan irregularity created originally by Liew et al. (2001) and used by Jiang et al. (2002) for plastic analysis of steel 3D buildings. This example is also used by Jiang and Adeli

(2005) in their dynamic wavelet neural network model for structural system identification, Long and Hung (2008) for local buckling determination during plastic analysis, and Chiorean (2009) for development of a computer program for nonlinear inelastic analysis.

The perspective, elevation and plan view of this building are shown in Figure 19.

Roof and floor framing systems consist of steel beams and lightweight concrete with a slab thickness of 0.15m over steel metal deck working as a rigid diaphragm (Figure 19c).

The building is designed for a combination of 4.8 kN/m2 static gravity load and 0.96 kN/m2 wind load acting on the y-direction. Table 12 presents the free vibrations response data for the uncontrolled structure including the mass modal participation factors. Figure

20 shows the first three mode shapes of the structure. This structure is also subjected to the same Loma Prieta earthquake accelerogram displayed in Figure 12. The maximum displacement was found at joint D3-20 on the roof.

The maximum absolute joint displacements in cm for different earthquake incident angles are also plotted in Figure 21. The critical angle of earthquake incidence for the uncontrolled structure was determined to be 5 degrees with respect to the X axis

(Figure 19c).

W8x31

)

W10x60 3.658m

W12x8

W12x106

W14x132 height(story 73.16= m

W14x145

Z W14x159

W14x176 (b) X (b) (a)

G1= W12x26

G2= W16x36 G2 G2 G3= W21x57 G1 G1 G1

(c)

14.63m

●+

G3 G2 Y G3 G1 G1 G1 X 21.95m

Figure 19: 20-story building with plan irregularity (a) Perspective view, (b) Elevation and (c) Plan view. +CR= center of rigidity, ●CM= center of mass.

(a) (b) (c)

Figure 20: Mode shapes of the 20-story building: (a) Mode 1 (period = 4.67 sec), (b) Mode 2 (period = 4.26 sec), (c) Mode 3 (period = 3.66 sec)

This angle of incidence is used to compare the eight sets of equations for TMD tuning parameters presented in Table 3. The results of the comparisons for maximum absolute displacements and accelerations are presented in Table 14. The detailed values of the optimum tuning parameters for the eight sets of equations are included in Table 28 and Table 33 in Appendix A. This building was modeled using ETABS and SAP2000 software packages.

Table 12: Free vibration modal response of the 20-story irregular building Modal Participation Factors Period Mode X Y Z (sec) Translation Translation Torsion 1 4.666 0.756 0.001 0.121 2 4.256 0.001 0.718 0.444 3 3.659 0.002 0.000 0.166 4 1.707 0.000 0.127 0.088 5 1.647 0.101 0.000 0.010 6 1.396 0.000 0.000 0.021 7 1.043 0.000 0.054 0.037 8 0.981 0.040 0.000 0.005 9 0.853 0.000 0.000 0.009 10 0.728 0.000 0.025 0.017 11 0.679 0.023 0.000 0.003 Total Sum = 0.924 0.926 0.926

Figure 21: Maximum absolute joint displacement (cm) per earthquake incident angles for the 20-story building

Table 13: Maximum absolute displacement per earthquake incident angles of 20-story building

Incident angle Maximum absolute Joint (°) displacement (cm) (grid-floor) 0 25.526 D3-20 5 25.533 D3-20 8 25.531 D3-20 15 25.49 D3-20 30 25.10 D3-20 45 24.37 D3-20 60 23.65 C3-20 75 23.20 A2-20 90 23.20 A1-20 105 23.30 A2-20 120 23.10 A2-20 135 23.10 A2-20 150 24.30 D3-20 165 25.20 D3-20 180 25.53 D3-20 195 25.49 D3-20 210 25.10 D3-20 225 24.37 D3-20 240 23.65 C3-20 255 23.20 A2-20 270 23.20 A1-20 285 23.30 A2-20 300 23.10 A2-20 315 23.10 A2-20 330 24.30 D3-20 345 25.20 D3-20 355 25.47 D3-20 360 25.526 D3-20

Table 14: Optimum tuning parameters and maximum absolute displacement (cm), acceleration (g) and Base Shear (kN) response for the 20-story building

Maximum Response of the Controlled Optimal Parameters Structure Method Frequency Damping Absolute Absolute Absolute ratio ratio Displacement Acceleration Base Shear (cm) (g) (kN) Den Hartog 0.971 0.101 18.77 0.188 378.2 Warburton 0.964 0.086 16.11 0.188 378.1 Fujino and Abe 0.964 0.086 16.11 0.188 378.1 Feng and Mita 0.964 0.083 16.15 0.177 378.5 Sadek et al. 0.963 0.219 14.45 0.167 349.5 Rudinger 0.971 0.005 18.31 0.179 395.9 Hoang et al. (δ=1) 0.858 0.083 17.55 0.179 393.8 Hoang et al. (δ=3) 0.928 0.086 17.25 0.179 389.5 Krenk and 0.971 0.085 16.10 0.187 377.8 Hogsberg Uncontrolled - - 25.53 0.287 466.2

Similar to Example 2, equations proposed by Sadek et al. (1997) yield the most effective vibration reductions for the 20-story irregular structure; they decrease the maximum absolute displacement by 43.4%, maximum acceleration by 41.8%, and absolute base shear (kN) by 25.3%.

3.2 CONCLUDING REMARKS

Equations proposed by Hoang et al. (2008) yield the best result for the 5-th story irregular structure and those proposed by Sadek et al. (1997) yield the best results for the

15th and the 20th story irregular buildings. In Tables 7, 11, and 14 it is observed that

Sadek et al. (1997) yields a relatively large value for the optimum damping ratio, 0.219,

63 which shows the impact of this ratio in the TMD tuning equations for flexible structures with a fundamental period of vibrations of, say, greater than 1 second (T > 1 sec).

By examining the displacement, acceleration and base shear results, it is observed the performance of the BTMD in reducing the vibration responses is affected by the rigidity of the structure; it is more effective for taller and more flexible structures. It is recommended that Hoang et al. (2008) equations be used for irregular structures with a fundamental period of vibrations of less than one second and Sadek et al. (1997) equations be used for structures with a fundamental period of vibrations greater than one second. These recommendations will be used in the following two chapters.

CHAPTER 4

COMPARISON OF PTMD AND BTMD

4.1 INTRODUCTION

In this chapter, the vibration control and reduction performances of BTMD and

PTMD are compared using the same three examples presented in Chapter 3. These structures are subjected to the same Loma Prieta earthquake accelerogram shown in

Figure 12. In all examples, they are placed at the top floor and at the center of rigidity

(CR) depicted in Figures 10, 16 and 19.

Example 1 Five-story 3D Ordinary Moment Resisting Steel Frame with Plan and

Elevation Irregularity

The time-history results for maximum displacement, acceleration and base shear for PTMD and BTMD using Hoang et al. (2008) equations for the tuning parameters are shown in Figure 22 and 23. Figure 22 show the time-histories of displacement and acceleration of joint A6-5 in the Y direction and the time-history for the base shear in the

Y direction were the solid line is the uncontrolled case and the broken line is that of the

PTMD or BMTD case.

10 Uncontrolled PTMD 5 5.63

-5 Displacement (cm) Displacement -9.24 -10 0 5 10 15 20 25 30 35 40 Time (s)

0.8 0.694 Uncontrolled 0.6 PTMD

0.4

0.2

-0.2 Acceleration (g) Acceleration -0.4 -0.442 -0.6 0 5 10 15 20 25 30 35 40 Time (s)

1000 Uncontrolled PTMD 500

-500

Base Shear (kN) Shear Base -716

-1000 -973

0 5 10 15 20 25 30 35 40 Time (s)

Figure 22: Maximum displacement and acceleration of joint A6-5 and base shear of 5- story building with a PTMD system

10 Uncontrolled BTMD 5 3.85

-5 Displacement (cm) Displacement -9.24 -10 0 5 10 15 20 25 30 35 40 Time (s)

0.8 0.694 Uncontrolled 0.6 BTMD

0.4 0.316 0.2

-0.2

Acceleration (g) Acceleration -0.4

-0.6 0 5 10 15 20 25 30 35 40 Time (s)

1000 Uncontrolled BTMD 500

-500

-617 Base Shear (kN) Shear Base

-1000 -973

0 5 10 15 20 25 30 35 40 Time (s) Figure 23: Maximum displacement and acceleration of joint A6-5 and base shear of 5- story building with a BTMD system

These figures show that BTMD is more effective compared with PTMD to the tune of 31.6%, 28.5%, and 15.21% for maximum displacement, acceleration and base shear, respectively.

Example 2: Fifteen-story reinforced concrete Moment Resisting Frame with plan irregularity

The time-history results for maximum displacement, acceleration and base shear for PTMD and BTMD using Sadek et al. (1997) equations for the tuning parameters are shown in Figure 24 and 25. This figures show that BTMD is more effective compared with PTMD to the tune of 31.4%, 33.2%, and 27.6% for maximum displacement, acceleration and base shear respectively.

Example 3 Twenty-story Steel Moment Resisting Frame with plan irregularity

The time-history results for maximum displacement, acceleration and base shear for PTMD and BTMD using Sadek et al. (1997) equations for the tuning parameters are shown in Figure 26 and 27. These figures show that BTMD is more effective compared with PTMD to the tune of 11.72%, 13.9%, and 23.1% for maximum displacement, acceleration and base shear, respectively.

4.2 SUMMARY OF COMPARISON RESULTS BETWEEN PTMD AND BTMD

Based on the results obtained from examples presented in this section, it can be concluded that BTMD performs consistently better than PTMD. Therefore, BTMD will be the chosen passive control device in the next chapter for the investigation of optimal placement.

30 Uncontrolled 20 PTMD

-10 -16.9 -20 Displacement (cm) Displacement -22.5 -30 0 5 10 15 20 25 30 35 40 Time (s)

0.2 Uncontrolled PTMD 0.1

-0.1

-0.176 -0.2 Acceleration (g) Acceleration -0.228

-0.3 0 5 10 15 20 25 30 35 40 Time (s)

1500 Uncontrolled 1000 PTMD

500

-500

-1000

Base Shear (kN) Shear Base -1072.9

-1500 -1425.6

0 5 10 15 20 25 30 35 40 Time (s)

Figure 24: Maximum displacement and acceleration of joint D1-15 and base shear of 15- story building with a PTMD system

30 Uncontrolled 20 BTMD

10 11.6

-10

-20 Displacement (cm) Displacement -22.5 -30 0 5 10 15 20 25 30 35 40 Time (s)

0.2 Uncontrolled 0.131 BTMD 0.1

-0.1

-0.2 Acceleration (g) Acceleration -0.228

-0.3 0 5 10 15 20 25 30 35 40 Time (s)

1500 Uncontrolled 1000 BTMD 777 500

-500

-1000 Base Shear (kN) Shear Base

-1500 -1425.6

0 5 10 15 20 25 30 35 40 Time (s)

Figure 25: Maximum displacement and acceleration of joint D1-15 and base shear of 15- story building with a BTMD system

30 Uncontrolled 20 20.2 PTMD 15.4 10

-10

-20 Displacement (m) Displacement

-30 0 5 10 15 20 25 30 35 40 Time (s)

0.3 Uncontrolled 0.22028 0.2 PTMD 0.17991

0.1

-0.1

Acceleration (g) Acceleration -0.2

0 5 10 15 20 25 30 35 40 Time (s)

600 Uncontrolled 466.202 PTMD 400 294.9632 200

-200 Base Shear (kN) Shear Base

-400 0 5 10 15 20 25 30 35 40 Time (s)

Figure 26: Maximum displacement and acceleration joint C1-12 and base shear for 20- story building with a PTMD system

30 Uncontrolled 20 20.2 BTMD

10 11.7

-10

-20 Displacement (cm) Displacement

-30 0 5 10 15 20 25 30 35 40 Time (s)

0.3 Uncontrolled 0.2 0.22 BTMD 0.155 0.1

-0.1

Acceleration (g) Acceleration -0.2

0 5 10 15 20 25 30 35 40 Time (s)

600 Uncontrolled 466 PTMD 400

200 227

-200 Base Shear (kN) Shear Base

-400 0 5 10 15 20 25 30 35 40 Time (s)

Figure 27: Maximum displacement and acceleration joint C1-12 and base shear for 20- story building with a BTMD system

CHAPTER 5

BTMD PLACEMENT

5.1 INTRODUCTION

In this chapter, the best location for placing a single BTMD is investigated for 4 different irregular multi-story building structures. Each uncontrolled structure is subjected to the same Loma Prieta accelerogram shown in Figure 12 using twenty four different incident angles with increments of 15 degrees in the horizontal plane.

Subsequently, the angle of incident is refined near the most critical angle of incident among the 24. The direction that results in the maximum displacement anywhere in the structure is determined and used for the subsequent BTMD placement investigation.

A single BTMD is placed on every beam one at a time in a selected number of top floors within internal walls of the structure and the maximum absolute joint displacement and acceleration as well the maximum base shear are obtained. External beams are excluded based on practical and architectural considerations.

Example 1: 5-Story Building Structure

This example is the same structure shown in Figures 9 and 10. One BTMD is placed on every beam, one at a time, in the top three floors within internal walls of the structure according to the grid shown in Figure 10. In a given floor, the BTMD location designation BC-5 (Figure 9), for example, indicates a beam between grid lines B and C on grid line 5, and B-56 indicates a beam on grid line B between grid lines 5 and 6. The critical angle of earthquake incidence for the uncontrolled structure was determined to be

80 degrees with respect to the X axis (Figure 10). BTMD placement results are shown in

Table 15 for the joint that resulted in maximum response. The best location for placement of a BTMD to minimize displacement and acceleration is on the top 5th (top) floor at locations B-56 and B-45 in Figure 10 with a reduction response of 48% and 56.1% respectively. The best location to minimize the base shear is obtained at location B-56 on the 4th floor with a reduction of 36.6%.

Example 2: 15-story building structure

This example is the same structure shown Figure 16. A BTMD is placed on every beam in the top three floors within internal walls of the structure according to the grid shown in Figure 16(b). The critical angle of earthquake incidence for the uncontrolled structure was determined to be 50 degrees with respect to the X axis (Figure 16). The

BTMD placement results are shown in Table 16 for the joint that resulted in maximum response. The best location for placement of a BTMD to minimize displacement, acceleration, and base shear is on the top floor at BC-2 near the center of rigidity with a reduction response of 48.4%, 42.5%, and 45.5% respectively.

Table 15: Maximum displacement and acceleration response and the base shear for the 5- story building for different PTMD placement

Maximum PTMD\BTMD Maximum Absolute Maximum Absolute Base Shear location Displacement (cm) Acceleration (g) (kN) 5th Story BC-5 6.390 0.498 942.8 AB-5 6.069 0.491 965 B-45 4.820 0.312 656.2 B-56 4.820 0.316 653.5 4th Story BC-5 7.32 0.582 928.5 AB-5 7.28 0.575 935.8 B-45 6.61 0.440 622.1 B-56 6.60 0.440 616.8 3rd Story B-56 7.47 0.632 715.1 B-45 7.52 0.632 715.5 B-34 8.45 0.633 917.7 B-23 8.47 0.637 918.8 B-12 9.44 0.641 922.6 C-12 9.46 0.661 972.8 C-23 9.47 0.659 970.8 C-34 9.45 0.655 970 C-45 8.35 0.643 816.2 C-56 8.36 0.642 816.4 D-56 9.36 0.688 1014 D-45 9.37 0.678 1014 D-34 9.38 0.679 1015 E-34 9.41 0.674 1038 E-45 9.41 0.673 1039 Uncontrolled 9.27 0.710 973.3

Table 16: Maximum displacement and acceleration response and the base shear for the 15-story building for different BTMD placement Max. Absolute Maximum Absolute Maximum BTMD Displacement (cm) Acceleration (g) Absolute Base Location Joint. Joint Joint No. Joint No. Shear (kN) Displacement Acceleration 15th Floor AB-3 D1-15 12.46 D1-15 0.150 796.3 AB-2 D1-15 12.32 D1-15 0.150 793.0 BC-3 A1-15 11.89 A1-15 0.149 785.3 BC-2 A1-15 11.79 A1-15 0.149 779.2 B-34 D1-15 12.05 D1-15 0.148 770.8 B-23 D1-15 12.03 D1-15 0.148 773.4 B-12 D2-15 12.01 D1-15 0.148 771.4 C-12 A1-15 12.29 A1-15 0.150 790.0 14th Floor AB-3 D1-15 12.52 D1-15 0.152 805.2 AB-2 D1-15 12.35 D1-15 0.152 801.8 BC-3 A1-15 11.83 A1-15 0.151 788.8 BC-2 A1-15 11.93 A1-15 0.151 796.3 B-34 D1-15 12.06 D1-15 0.148 778.8 B-23 D1-15 12.04 C4-15 0.148 780.7 B-12 D2-15 12.02 A1-15 0.148 779.0 C-12 A1-15 12.29 A1-15 0.151 796.9 13th Floor AB-3 D1-15 12.59 D1-15 0.154 815.5 AB-2 D1-15 12.40 D1-15 0.153 811.9 BC-3 A1-15 11.91 A1-15 0.153 799.7 BC-2 A1-15 12.01 A1-15 0.153 806.7 B-34 D1-15 12.12 D1-15 0.151 789.1 B-23 D1-15 12.09 A1-15 0.151 790.5 B-12 D2-15 12.07 A1-15 0.151 788.9 C-12 A4-15 12.31 A4-15 0.153 806.1

Example 3: 20-Story steel moment resisting frame with plan irregularity

This example is the same structure shown Figure 19. A BTMD is placed on every beam in the top two floors within internal walls of the structure according to the grid shown in Figure 19. The critical angle of earthquake incidence for the uncontrolled structure was determined to be 5 degrees with respect to the X axis (Figure 19). The

BTMD placement results are shown in Table 17 for the joint that resulted in maximum response. The best location for placement of a BTMD to minimize displacement, acceleration, and base shear is at location BC-12, where a beam is added between grid lines B and C and halfway between grid lines 1 and 2 on the top floor (identified by a dashed line in Figure 18). Use of BTMD resulted in a response reduction of 43.4% for displacement, 41.8% for maximum acceleration, and 25.3% for maximum base shear.

A building equipped with a BTMD device is designed for the first two dominant modes of the structure. When the building is subjected to an earthquake accelerogram, the seismic record contains a range of frequencies that affect the level of effectiveness of the damping device. This example structure is subjected to five additional seismic records shown in Table 18 and depicted in Figures 28 to 32 acting at the critical angle found previously: the Imperial Valley (1979), San Fernando (1971), Landers (1992), Northridge

(1994) and ChiChi (1999) earthquake respectively. The comparison results are summarized in Table 19. Installation of the BTMD reduces the structural maximum absolute displacement, acceleration, and base shear at least 21.8%, 20.7% and 11.3% respectively.

Table 17: Maximum absolute displacement, absolute acceleration and base shear responses for the 20-story building for different BTMD placement

Maximum Absolute Maximum Absolute Maximum BTMD Displacement (cm) Acceleration (g) Base Shear Location Joint Joint. Joint Joint No. (kN) No. Displacement Acceleration 20th Floor AB-12 D3-20 15.98 D3-20 0.184 415.8 B-12 D3-20 16.10 D3.20 0.178 418.0 C-12 C3-20 16.16 C3-20 0.174 417.3 CD-12 C3-20 16.09 C3-20 0.183 414.5 CD-2 D3-20 16.07 C3-20 0.187 416.7 BC-12 C3-20 15.97 C3-20 0.167 411.2 19th Floor B-12 D3-20 16.98 D3-20 0.177 418.8 C-12 D3-20 17.10 D3.20 0.174 419.0 CD-2 C3-20 17.16 C3-20 0.178 419.3 18th Floor B-12 D3-20 17.89 D3-20 0.194 415.8 C-12 D3-20 18.01 D3.20 0.198 418.0 CD-2 C3-20 17.92 C3-20 0.194 417.3 Uncontrolled D3-20 25.53 D3-20 0.287 466.2

Table 18: Earthquake ground motions

Earthquake Magnitude Station Component PGA Date Name (Ms) Number (deg) (g) Nov. 15, 1979 Imperial Valley 6.58 5051 315 0.20 Feb. 09, 1971 San Fernando 6.61 80053 90 0.11 Nov. 17, 1989 Loma Prieta 6.93 58378 0 0.51 June 28, 1992 Landers 7.28 12149 0 0.17 Jan. 17, 1994 Northridge 6.69 24278 360 0.51 Sept 20, 1999 ChiChi 7.61 TCU 0 0.18

0.25 0.2 0.204

0.15

0.1

0.05

-0.05 Acceleration (g) Acceleration -0.1

-0.15 0 5 10 15 20 25 30 35 40 Time (sec)

Figure 28: Imperial Valley earthquake accelerogram, IV (November 15, 1979)

0.15

0.1 0.11

0.05

-0.05 Acceleration (g) Acceleration

-0.1 0 5 10 15 20 25 30 Time (sec)

Figure 29: San Fernando earthquake accelerogram, SF (February 09, 1971)

0.2 0.171 0.15

0.1

0.05

-0.05

Acceleration (g) Acceleration -0.1

-0.15 0 5 10 15 20 25 30 35 40 45 50 Time (sec)

Figure 30: Landers earthquake accelerogram, LD (June 28, 1992)

0.6 0.514 0.4

0.2

-0.2 Acceleration (g) Acceleration

-0.4 0 5 10 15 20 25 30 35 40 Time (sec)

Figure 31: Northridge earthquake accelerogram, NR (January 17, 1994)

0.2 0.183 0.15 0.1

0.05 0

-0.05

-0.1 Acceleration (g) Acceleration -0.15 -0.2 0 10 20 30 40 50 60 Time (sec)

Figure 32: ChiChi earthquake accelerogram, CC (September 20, 1999)

Table 19: Comparison results of 20-story building equipped with BTMD at BC-12 roof location using 6 seismic records.

Maximum absolute displacement (cm) Maximum absolute acceleration (g) Maximum absolute base shear (kN) Ground Motion Uncontrolled Controlled % Uncontrolled Controlled % Uncontrolled Controlled % Joint Result Joint Result Reduction Joint Result Joint Result Reduction Results Results Reduction

A1- C1-20 D3-20 C3-20 IV 20 30.6 14.9 51.3% 0.252 0.149 40.8% 463.9 418.9 19.7%

82 SF

A1- A1-20 A1-20 C1-20 20 17.6 10.8 38.6% 0.195 0.127 34.8% 348.2 313.2 20.1%

D3- C1-20 D3-20 C1-20 LP 20 25.5 15.9 43.4% 0.254 0.167 41.8% 435.1 411.2 24.2%

D3- A1-20 D3-20 C1-20 LD 20 12.4 8.2 28.1% 0.261 0.207 20.7% 306.9 284.9 11.3%

D1- C1-20 D3-20 B2-20 NR 20 39.2 28.2 41.9% 0.909 0.603 33.7% 1507.5 1000.0 33.7%

A1- D1-20 A1-20 C3-20 CC 20 63.5 32.1 49.5% 0.379 0.254 32.9% 985.5 874.4 17.2%

Example 4: Twelve story moment resisting frame irregular building.

This example is a 12-story irregular steel moment resisting frame building presented in the Federal Emergency Management Agency publication (FEMA, 2003) shown in Figure 33. The plan consists of seven 30-ft bays along the x-direction and seven

25ft bays along the y-direction. All stories have a height of 12.5-ft with the exception of the first story and the basement having a height of 18ft. Columns range in size from roof level W24x146 to W24x229 at ground level. Girders vary from W30x108 to W30x132.

Steel members have a yield stress of 36ksi. In the basement, the columns are embedded to pilaster cast into reinforced concrete walls. Concrete caps over piles support the interior steel columns. Reinforced concrete grade beams connect the tie beams and pile caps.

Roof and floor framing systems consist of steel beams and lightweight concrete over steel metal deck working as a rigid diaphragm. Typical slab has 4in thickness; the non-typical floors levels G, 5 and 9 have 6-in thickness. The building has been designed based on the National Earthquake Hazard Reduction Program (NEHRP) recommended provisions for seismic regulations for new buildings and other structures (FEMA, 2003).

In addition to the dead load of the structure the following loads are considered: a live load of 50psf, cladding loading of 15psf, and a partition load of 10psf, roofing load of 15psf, and special loadings of 60 psf to account for excessive live load on terraces. The building total height is 155ft.

Seismic design is based on the equivalent lateral procedure (EFL) procedure. The building is located in Stockton, California. The structure is modeled as a Special Moment

Resisting Frame (SMRF) with a response modification coefficient of R=8 and deflection

82 amplification coefficient of CD = 5.5. Following the provisions of ASCE 7-10 (ASCE,

2010), following spectral response acceleration values are used from the U.S. Geological

Survey probabilistic seismic hazard maps: Ss= 1.25 and S1=0.40. A damping ratio of 0.05 is used for steel structures. The design acceleration spectra of 0.373g and soil type C are used. Analysis is performed using ETABS and SAP2000 (developed by Computers and

Structures, Inc., Berkeley, California).

Table 20 presents the free vibrations response data for the uncontrolled structure including the mass modal participation factors. Figure 34 shows the first three mode shapes of the structure. The critical angle of earthquake incidence for the uncontrolled structure was determined to be 0 degrees with respect to the X axis (Figure 33). The maximum absolute joint displacements in cm for different earthquake incident angles are also plotted in Figure 35.

The BTMD placement results are shown in Table 22 for the joint that resulted in maximum response. The detailed values of the optimum tuning parameters for the BTMD using Sadek et al. 1997 equations are included in Table 29 of Appendix A.

The best location for the BTMD placement is at D4 or D3 near the center of rigidity on the 12th (top) floor of the structure. The maximum displacement was found at joint C1-12 on the roof, the time-history results are plotted in Figure 36. The corresponding vibration reductions for maximum displacement, acceleration and base shear are 50.8%, 46.9%, 41.2% respectively.

(a) Levels 10- Roof

155.50 m

(b) Levels 6-9

(c) Levels G-5 Figure 33: 12-story irregular building structure with setbacks and T-shape plan (FEMA 2003). Left: Perspective and Elevation view. Right: Plan view of building at different story levels.

(a) (b)

Figure 34: Mode shapes of the 12-story building with vertical and plan irregularities in perspective view: (a) Mode 1 (period = 2.92sec), (b) Mode 2 (period = 2.26sec), (c) Mode 3 (period = 2.14sec), (d) Mode 4 (period = 1.20sec)

Table 20: Free vibration modal response of the 12-story FEMA irregular building Modal Mass Participation Factors Period Mode X Y Z (sec) Translation Translation Torsion 1 2.92 0.82 0.00 0.19 2 2.26 0.00 0.83 0.34 3 2.14 0.01 0.00 0.32 4 1.20 0.12 0.00 0.03 5 0.96 0.00 0.03 0.01 6 0.91 0.00 0.09 0.05 7 0.71 0.02 0.00 0.01 8 0.59 0.00 0.00 0.01 9 0.55 0.00 0.02 0.01 10 0.49 0.00 0.00 0.00 Total Sum 0.98 0.98 0.97

Figure 35: Maximum absolute displacement (m) per earthquake incident angles of 12- story with vertical and plan irregularities building

Table 21: Maximum absolute displacement (m) per earthquake incident angles of 12- story with vertical and plan irregularities irregular building

Incident Angle Max Displacement Joint (°) (m) (grid-floor) 0 0.242 C1-12 5 0.241 C1-12 8 0.239 C1-12 15 0.236 C1-12 30 0.210 C1-12 45 0.175 C1-12 60 0.132 C1-12 75 0.099 C8-12 90 0.087 C1-12 105 0.105 F1-12 120 0.146 F1-12 135 0.185 F1-12 150 0.216 F1-12 165 0.236 C1-12 180 0.242 C1-12 195 0.236 C1-12 210 0.210 C1-12 225 0.175 C1-12 240 0.132 C1-12 255 0.099 C8-12 270 0.087 C1-12 285 0.105 F1-12 300 0.146 F1-12 315 0.185 F1-12 330 0.216 F1-12 345 0.236 C1-12 355 0.241 C1-12 360 0.242 C1-12

30 Uncontrolled 20 BTMD

-10 -11.9

-20 Displacement(cm) -24.2 -30 0 5 10 15 20 25 30 35 40 Time (s)

0.3 Uncontrolled 0.2 BTMD

0.1

-0.1 -0.103

Acceleration (g) Acceleration -0.2 -0.194

0 5 10 15 20 25 30 35 40 Time (s) 4 x 10 3 Uncontrolled 2 BTMD

-1 -11288

Base Shear (kN) Shear Base -2 -20177

-3 0 5 10 15 20 25 30 35 40 Time (s)

Figure 36: Displacement (m) and acceleration (g) of joint C1-12 and base shear (kN) of the 12-story building with vertical and plan irregularities irregular building

Table 22: Maximum displacement and acceleration response and the base shear for the 12-story building with vertical and plan irregularities for different BTMD placement Maximum Absolute Maximum Absolute Maximum BTMD Displacement (cm) Acceleration (g) Base Shear Location Joint. Joint Joint No. Joint No. (kN) Displacement Acceleration Uncontrolle C1-12 25.2 C1-12 0.204 18521.8 d 12th Floor D4 C8-12 12.80 C8-12 0.149 11898.3 D3 C1-12 13.02 C1-12 0.149 11865.0 D2 C1-12 13.31 C1-12 0.150 11893.3 D5 C1-12 13.72 C1-12 0.151 12024.0 D6 C1-12 14.25 C1-12 0.152 12257.8 D7 C1-12 14.89 C1-12 0.153 12578.1 E4 C1-12 12.80 C1-12 0.148 11882.3 E3 C1-12 13.02 C1-12 0.149 11858.2 E2 C1-12 13.32 C1-12 0.150 11889.1 E5 C1-12 13.73 C1-12 0.150 12023.9 E6 C1-12 14.27 C1-12 0.151 12261.7 E7 C1-12 14.91 C1-12 0.156 12581.5 11th Floor D2 C8-12 13.19 C8-12 0.153 12171.9 D3 C1-12 13.43 C1-12 0.152 12170.7 D4 C1-12 13.74 C1-12 0.153 12244.4 D5 C1-12 14.12 C1-12 0.153 12285.8 D6 C1-12 14.61 C1-12 0.154 12615.8 D7 C1-12 15.17 C1-12 0.155 12909.1 E2 C8-12 13.17 C1-12 0.153 12163.8 E3 C1-12 13.43 C1-12 0.152 12159.9 E4 C1-12 13.74 C1-12 0.153 12238.7 E5 C1-12 14.13 C1-12 0.151 12384.6 E6 C1-12 14.56 C1-12 0.154 12635.2 E7 C1-12 15.11 C1-12 0.155 12920.7 10th Floor D2 C8-12 13.64 C8-12 0.160 12412.2 D3 C1-12 13.93 C1-12 0.160 12478.8 D4 C1-12 14.24 C1-12 0.161 12590.8 D5 C1-12 14.59 C1-12 0.162 12735.1 D6 C1-12 15.00 C1-12 0.162 12949.5 D7 C1-12 15.47 C1-12 0.163 13206.8 E2 C8-12 13.62 C8-12 0.160 12388.0 E3 C1-12 13.93 C1-12 0.160 12465.7 E4 C1-12 14.25 C1-12 0.161 12583.3 E5 C1-12 14.59 C1-12 0.162 12730.6

Example 5: Podium with setback irregularity

The following example is an irregular twelve-story three-dimensional steel building with vertical setbacks (Figure 37). Saleh and Adeli (1998) created this example subjected to 3 different types of dynamic loading to study placement optimization according to 4 different placement arrangements and actuator force optimization using high-performance parallel computing. They concluded that optimal arrangement of controllers depends on the height and aspect ratio of the structure. Adeli and Kim (2005) study this example subjected to earthquake ground motions using a wavelet-based algorithm to obtain optimal forces in a hybrid control system using passive dampers and two semi-active tuned liquid column damper (TLCD) system.

Jiang and Adeli (2005) study this example subjected to earthquake ground motions for health monitoring using dynamic wavelet neural network for structural nonlinear system identification. Table 23 presents the free vibrations response data for the uncontrolled structure including the mass modal participation factors. Figure 38 shows the first three mode shapes of the structure. The critical angle of earthquake incidence for the uncontrolled structure was determined to be 0 degrees with respect to the X axis (Figure 37c).

The maximum displacement was found at joint C1-12 on the roof for earthquake.

The maximum absolute joint displacements in cm for different earthquake incident angles are also plotted in Figure 39. The detailed values of the optimum tuning parameters for the BTMD for Sadek et al. 1997 equations are included in Table 30. The BTMD placement results are shown in Table 25 for the joint that resulted in maximum response.

It can be seen that the influence of the higher modes of vibration in this structure have an influence in the placement of the structure.

Therefore, the best location for the BTMD is at the 8th floor in AB-12 or AB-23 locations. The corresponding vibration reductions for maximum displacement, acceleration and base shear are 57.6%, 47.1%, 23.6% respectively

W14 61 W14 x + CR CM

(b)

W14 90 W14 x 12 x 4.5m = 54m = 54m 4.5m 12 x

W14 68 W14 x

+ CR CM Y X

(a)

(c)

Figure 37: 12-story building with vertical setback, (a) Perspective views. (b) Plan view of Levels 1-4, (c) Plan view of Levels 5-12. +CR= center of rigidity, ●CM= center of mass. (Adeli and Saleh 1998, Adeli and Kim 2005, Adeli and Jiang 2005)

(a) (b) (c)

Figure 38: Mode shapes of the 12-story irregular building with vertical setback: (a) Mode 1 & 2 (period 2.6 sec and 2.0 sec), (b) Mode 3 (period = 1.5 sec), (c) Mode 4 (period = 1.1 sec)

Table 23: Free vibration modal response of the 12-story building with vertical setback

Period Modal Mass Participation Factors Mode (sec) X Translation Y Translation Z Torsion 1 2.603 0.001 0.667 0.189 2 2.008 0.621 0.001 0.205 3 1.481 0.000 0.000 0.241 4 1.094 0.000 0.199 0.054 5 0.770 0.236 0.000 0.074 6 0.688 0.000 0.006 0.121 7 0.683 0.000 0.054 0.000 8 0.471 0.057 0.000 0.018 9 0.447 0.000 0.000 0.024 10 0.441 0.000 0.018 0.002 11 0.359 0.000 0.019 0.006 12 0.302 0.016 0.003 0.010 Total Sum 0.931 0.968 0.944

Figure 39: Maximum absolute displacement (cm) per earthquake incident angles of 12- story building with vertical setback

Table 24: Maximum absolute displacement (cm) per earthquake incident angles of 12- story building with vertical setback Incident Angle (°) Max Displacement (cm) Joint (grid-floor) 0 8.27 B3-12 15 9.09 B3-12 30 11.26 C4-12 45 13.85 C4-12 60 15.96 C4-12 75 17.28 C4-12 80 17.51 C4-12 85 17.625 C4-12 88 17.642 C4-12 90 17.630 C4-12 100 17.33 C4-12 105 17.03 C4-12 120 15.57 C4-12 135 13.45 C4-12 150 11.02 C4-12 165 8.96 C4-12 180 8.27 C4-12 195 9.09 B3-12 210 11.26 C4-12 225 13.85 C4-12 240 15.96 C4-12 255 17.28 C4-12 270 17.63 C4-12 285 17.03 C4-12 300 15.57 C4-12 315 13.45 C4-12 330 11.02 C4-12 345 8.96 C4-12 360 8.27 B3-12

Table 25: Maximum displacement and acceleration response and the base shear for the 12-story building for different BTMD placement

Maximum Absolute Maximum Absolute Maximum BTMD Displacement (cm) Acceleration (g) Base Shear Location Joint Joint. Joint Joint (kN) No. Displacement No. Acceleration Uncontrolle C2-12 17.64 C2-12 0.314 181.72 d 12th Floor AB-12 A2-12 8.29 A2-12 0.132 126.8 AB-23 A2-12 8.30 A2-12 0.132 126.8 BC-12 C2-12 8.42 C2-12 0.140 125.7 BC-23 C2-12 8.41 C2-12 0.141 125.8 B-2 C2-12 8.34 C2-12 0.145 126.2 11th Floor AB-12 A2-12 8.15 A2-12 0.155 130.9 AB-23 A2-12 8.17 A2-12 0.156 131.2 BC-12 C2-12 8.28 C2-12 0.160 130.5 BC-23 C2-12 8.26 C2-12 0.162 130.6 B-2 C2-12 8.19 C2-12 0.166 130.6 10th Floor AB-12 A2-12 7.95 A2-12 0.157 139.7 AB-23 A2-12 7.96 A2-12 0.157 139.7 BC-12 C2-12 8.08 C2-12 0.157 139.3 BC-23 C2-12 8.07 C2-12 0.166 139.3 B-2 C2-12 7.92 C2-12 0.162 138.2 9th Floor AB-12 C2-12 7.54 C2-12 0.167 141.4 AB-23 C2-12 7.54 C2-12 0.168 141.4 BC-12 C2-12 7.60 C2-12 0.159 141.5 BC-23 C2-12 7.59 C2-12 0.164 141.5 B-2 C2-12 7.52 C2-12 0.165 141.8 8th Floor AB-12 C2-12 7.64 C2-12 0.17 143.2 AB-23 C2-12 7.64 C2-12 0.171 143.3 BC-12 C2-12 7.74 C2-12 0.167 143.3 BC-23 C2-12 7.73 C2-12 0.167 143.3 B-2 C2-12 7.59 C2-12 0.264 142.7

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH

6.1 SUMMARY OF CONCLUSIONS

In this research, eight different sets of equations for finding the optimum BTMD tuning parameters were compared using three different three-dimensional structures with vertical and plan irregularities. It was found that equations proposed by Hoang et al.

(2008) yield the best result for the 5-th story irregular structure and those proposed by

Sadek et al. (1997) yield the best results for the 15th and the 20th story irregular buildings.

Equations proposed by Hoang et al. (2008) yield the best result for the 5-th story irregular structure, because it improved maximum displacement by 14.6% compared to

Sadek et al. (1997) which was the second best option and 41.3% compared to uncontrolled case. Improved maximum acceleration by 6.9% compared to Sadek et al.

(1997) which were the second best option and 35.2% compared to uncontrolled case.

However, for maximum base shear Sadek et al. (1997) was the best option by 10.3% compared to the best second option and 25.7% compared to uncontrolled case.

Equations proposed by Sadek et al. (1997) yield the best results for the 15th story irregular building. Improved maximum displacement by 16.5% compared to Fujino and

Abe (1993) which was the second best option and 49.8% compared to uncontrolled case.

Improved maximum acceleration by 7.3% compared to Fujino and Abe (1993) which was the second best option and 45.5% compared to uncontrolled case. Improved maximum base shear by 13.8% compared to Fujino and Abe (1993) which was the second best option and 42.2% compared to uncontrolled case.

Equations proposed by Sadek et al. (1997) yield the best results for the 20th story irregular building. Improved maximum displacement by 16.2% compared to Fujino and

Abe (1993) which was the second best option and 43.4% compared to uncontrolled case.

Improved maximum acceleration by 6.7% compared to Fujino and Abe (1993) which was the second best option and 41.8% compared to uncontrolled case. Improved maximum base shear by 7.5% compared to Fujino and Abe (1993) which was the second best option and 25.3% compared to uncontrolled case.

The performance of the bidirectional tuned massed dampers was compared with that of the pendulum tuned mass dampers using three different structures with plan and vertical irregularities ranging in height from 5 to 20 stories and dominant frequencies ranging from 0.55 sec to 4.25 sec subjected to Loma Prieta earthquake. It was concluded that BTMD performs consistently better than PTMD. BTMD is advantageous over

PTMD because it can be tuned for two modes of vibration which means it can be used as an alternative to using two TMDs and increasing the space availability in a structure.

BTMD outperforms PTMD in all cases,

For the 5-story with plan and vertical irregularities, the BTMD is more effective to the tune of 31.6%, 28.5%, and 15.21% for maximum displacement, acceleration and base shear, respectively.

For the 15-story concrete building with plan irregularity, the BTMD is more effective compared with PTMD to the tune of 31.4%, 33.2%, and 27.6% for maximum displacement, acceleration and base shear respectively.

For the 20-story steel building with plan irregularity, the BTMD is more effective compared with PTMD to the tune of11.72%, 13.9%, and 23.1% for maximum displacement, acceleration and base shear respectively.

Installation of the BTMD in highly flexible 20-story building subjected to six different high impact earthquake accelerograms reduces the structural maximum absolute displacement, acceleration, and base shear at least 21.8%, 20.7% and 11.3% respectively.

Depending on the irregularity in the structure, the best optimal placement of the

BTMD can vary. The setback location in vertical irregularity plays an important role in the distribution of stiffness in the structure and the placement of the device should be

98 near the setback irregularity as seen in example 1. It was also discovered the influence of higher modes of vibration in the structure with respect to the placement of the damper along the height of the structure.

For the 5-story with plan and vertical irregularities, the best location for placement of a BTMD to minimize displacement and acceleration is on the top 5th (top) floor (B-56 and B-45) with a reduction response of 48% and 56.1% respectively. The best location to minimize the base shear is obtained at location B-56 on the 4th floor with a reduction of 36.6%.

For the 15-story concrete building with plan irregularity, the best location for placement of a BTMD to minimize displacement, acceleration, and base shear is on the top floor at BC-2 near the center of rigidity with a reduction response of 48.4%, 42.5%, and 45.5% respectively.

For 20-story steel building with plan irregularity, the best location for placement of a BTMD to minimize displacement, acceleration, and base shear is on the top floor at

BC-12 resulted in a response reduction of 43.4%, 41.8% and 25.3% respectively.

For the 12-story building with setback and plan irregularity, the best location for placement of a BTMD to minimize displacement, acceleration, and base shear is on the top floor at D4 near the CR with a reduction response of 50.8%, 46.9%, 41.2% respectively.

For the 12-story building with vertical irregularity, the best location for the

BTMD is at the 9th floor in B-2 locations. The corresponding vibration reductions for maximum displacement, acceleration and base shear are 57.6%, 47.1%, 23.6% respectively.

The BTMD system can be combined with a semi-active control mechanism to improve its performance against uncertainties. If there was power failure, the BTMD can work as a back-up system in case of external power sources needed for the active device is not available.

6.2 RECOMMENDATIONS FOR FUTURE RESEARCH

As more measured data becomes available, further comparison should be conducted with refinement of the proposed equations as needed. Therefore, it is recommended that the study of the tuned mass dampers using more buildings structures under other strong nature hazards.

Therefore, further research is recommended to optimize the placement using evolutionary optimization techniques such as genetic algorithms. Furthermore, study of semi-active and active controlled of irregular structure is still a field for exploration for structures subjected to strong ground motions.

Recently, researchers have used higher mass ratio (higher TMD mass) in order to overcome the tuning uncertainties that occur in practical applications. Hoang et al. (2008) assert that with higher mass ratio, it requires higher damping ratio and a lower frequency ratio. They advocate the use of higher mass ratio for robust performance against system parameters uncertainties and also excitation frequency uncertainty.

When designing buildings in seismically active areas, the dominant frequency of the building – which depends on the building materials, the number of stories and the subsoil -must also be taken into account for the design of BTMD and PTMD. It is

100 important to mention that special consideration has been taken in terms of modeling true pendulum behavior using SAP2000 software package. If the designer is interested in implementing BTMD as depicted in Almazan et al. (2007), true pendulum behavior using cables requires nonlinear static and direct-integration time-history analysis and the nonlinearity homogenous behavior for the friction device is currently not available in the software package and should be modeled as an approximation. Therefore, a future research study would be to include nonlinearity homogenous behavior for the friction device as part of the investigation.

101

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APPENDIX A

BTMD AND PTMD DETAILED PARAMETERS FOR THE IRREGULAR STRUCTURES

Table 26: Design parameter information for the 5th story building

Variable Value Unit Description

µ 0.03 - Mass ratio (ms/md)

Ts 0.81 sec Structural period

ωs 1.23 1/s Structural frequency

ms 4168.0 kN Structural mass 2 md 12.7 kN-sec /m TMD mass

Table 27: Design parameter information for the 15th story building Variable Value Unit Description

µ 0.03 - Mass ratio (ms/md)

Ts 2.93 sec Structural period

ωs 0.34 1/s Structural frequency

ms 23028 kN Structural mass 2 md 70.42 kN-sec /m TMD mass

117

Table 28: Design parameter information for the 20th story building Variable Value Unit Description

µ 0.03 unitless Mass ratio (ms/md)

Ts 4.67 sec Structural period

ωs 0.21 1/s Structural frequency

ms 14,859 kN Structural mass 2 md 45.44 kN-sec /m TMD mass

Table 29: Design parameter information for the 12th story FEMA building Variable Value Unit Description

µ 0.03 unitless Mass ratio (ms/md)

Ts 2.92 sec Structural period

ωs 0.34 1/s Structural frequency

ms 101,985 kN Structural mass 2 md 311.9 kN-sec /m TMD mass

0.963 unitless Frequency ratio (ωs / ωd)

ξopt 0.219 unitless Damping ratio of TMD

ωd 0.33 1/s TMD frequency

Td 3.03 sec TMD period

kd 1,239.8 kN/m TMD stiffness

cd 272.6 kN-sec/m TMD damping

kh 43732.8 kN/m Hanger stiffness

Lx 2.29 meters Pendulum length Lx

Ly 1.07 meters Pendulum length Ly

118

Table 30: Design parameter information for the 12th story podium building

Variable Value Unit Description

µ 0.03 unitless Mass ratio (ms/md)

Ts 2.6 sec Structural period

ωs 0.38 1/s Structural frequency

ms 3,036.9 kN Structural mass

2 md 9.29 kN-sec /m TMD mass

0.963 unitless Frequency ratio (ωs / ωd)

ξopt 0.219 unitless Damping ratio of TMD

ωd 0.37 1/s TMD frequency

Td 2.7 sec TMD period

kd 46.6 kN/m TMD stiffness

cd 9.12 kN-sec/m TMD damping

kh 55,160.2 kN/m Hanger stiffness

Lx 1.81 meters Pendulum length for ωx

Ly 1.07 meters Pendulum length for ωy

119

Table 31: PTMD/BTMD tuning parameters for 5th story irregular building

Hoang et Hoang et Variable Den Fujino and Feng and Sadek at Krenk & Units Description Waburton Rudinger al. al. name Hartog Abe Mita el. Hogsberg δ=1 δ= 3 Frequency

unitless ratio (ωs / 0.971 0.964 0.964 0.964 0.963 0.978 0.971 0.858 0.928 ωd) Damping unitless ξopt ratio of TMD 0.101 0.086 0.086 0.083 0.219 0.005 0.085 0.083 0.086 TMD kN/m kd Stiffness 728.3 717.4 728.3 661.1 715.9 739.0 686.9 564.6 660.5

121 TMD 1/s

ωd frequency 1.20 1.19 1.20 1.19 1.19 1.21 1.20 1.15 1.15

cd TMD 18.9 15.8 16.0 15.2 33.8 0.95 15.9 13.1 14.5 kN-sec/m Damping 0.83 0.84 0.83 0.84 0.84 0.83 0.83 0.87 0.87 Td sec TMD period

Pendulum 0.17 0.17 0.17 0.18 0.17 0.17 0.17 0.19 0.19 meters L (Lx) Length

120

Table 32: PTMD/BTMD tuning parameters for 15th story irregular building

Variable Fujino and Feng and Krenk & Hoang et al. Hoang et al. Den Hartog Waburton Sadek at el. Rudinger name Units Description Abe Mita Hogsberg delta=1 delta =3

Frequency unitless 0.971 0.964 0.964 0.964 0.963 0.978 0.971 0.858 0.928 ratio (ωs / ωd)

Damping ξopt unitless 0.101 0.086 0.086 0.083 0.219 0.005 0.085 0.083 0.086 ratio of TMD

TMD 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.31 0.33 ω 1/s d frequency

122 TMD 305.3 300.7 301.1 300.7 300.1 309.8 305.3 238.4 278.9 kd kN/m Stiffness

kN- TMD 30.7 24.93 25.04 24.02 63.73 1.45 24.30 18.48 20.7 c d sec/m Damping

Hanger 44,186 43,522 43,562 43,523 43,435 44,837 44,186 34,509 34,509 k kN/m h stiffness 3.02 3.04 3.04 3.04 3.04 3.00 3.02 3.41 3.16 Td sec TMD period

Pendulum 2.26 2.30 2.30 2.30 2.30 2.23 2.26 2.90 2.48 L (L ) meters x Length

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Table 33: PTMD/BTMD tuning parameters for 20th story irregular building

Variable Fujino and Feng and Krenk & Hoang et al. Hoang et al. Units Description Den Hartog Waburton Sadek at el. Rudinger name Abe Mita Hogsberg delta=1 delta= 3 Frequency

unitless ratio (ωs / 0.971 0.964 0.964 0.964 0.963 0.978 0.971 0.858 0.928 ωd) Damping unitless ξopt ratio of TMD 0.101 0.086 0.086 0.083 0.219 0.005 0.085 0.083 0.086

TMD 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.18 0.20 123 1/s

ωd frequency

kd TMD 76.44 76.37 76.22 78.68 77.53 44.58 52.2 kN/m Stiffness 77.53 76.37

TMD 12.05 10.1 10.14 9.72 25.80 0.59 9.83 7.48 8.38 kN-sec/m cd Damping

Hanger 17,393 17,132 17,148 17,132 17,098 17,650 17,393 13,584 15,892 kN/m kh stiffness

4.81 4.85 4.84 4.85 4.85 4.78 4.81 5.44 5.03 Td sec TMD period

Pendulum 5.75 5.84 5.83 5.84 5.85 5.67 5.75 7.36 6.29 meters L (Lx) Length

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APPENDIX B

MATLAB CODE ROUTINE FOR OBTAINING OPTIMAL TMD TUNING PARAMETERS

Script File clear all clc format compact u=input('Please enter mass ratio = '); e_s=input('Please enter structural damping ratio (e.g ASCE 7-05 for steel structures= 0.05)= '); d=input('Please enter ground frequency from soil profile (used in Hoang et al. 2008 equation)= '); disp('======') fprintf('Optimum tuning parameters of a TMD for: \nMass ratio = %5.2f \nStructural damping ratio = %5.2f\n',u,e_s) fprintf('======\n') Den_Hartog_freq_ratio=1/(1+u); Den_Hartog_damping_ratio=sqrt(3*u/(8*(1+u)^3)); fprintf('Den Hartog equations give %5.3f frequency ratio\n\t\t\t\t\t and %5.3f damping ratio\n',... Den_Hartog_freq_ratio,Den_Hartog_damping_ratio) disp('***************************************************************** ***********') Waburton_freq_ratio=sqrt(1-u/2)/(1+u); Waburton_damping_ratio=sqrt((u*(1-u/4))/(4*(1+u)*(1-u/2))); fprintf('Waburton equations give %5.3f frequency ratio\n\t\t\t\t and %5.3f damping ratio\n',... Waburton_freq_ratio,Waburton_damping_ratio) disp('***************************************************************** ***********') Fujino_freq_ratio=sqrt(1-u/2)/(1+u); Fujino_damping_ratio=1/2*sqrt((u*(1+3*u/4))/((1+u)*(1+u/2))); fprintf('Fujino and Abe equations give %5.3f frequency ratio\n\t\t\t\t\t\t and %5.3f damping ratio\n',... Fujino_freq_ratio,Fujino_damping_ratio) disp('***************************************************************** ***********')

123

Feng_freq_ratio=sqrt(1-u/2)/(1+u); Y=Feng_freq_ratio; Feng_damping_ratio_disp=1/2*sqrt((1+u)*Y^4+Y^2+(1- 3*(1+u)^2*Y^2)/((1+u)^3)); Feng_damping_ratio_acc=Y/2*sqrt((1+u)*Y^2+1-1/(1+u)); fprintf('Feng and Mita equation (i) give %5.3f frequency ratio\n\t\t\t\t\t\t and %5.3f damping ratio\n',... Feng_freq_ratio,Feng_damping_ratio_disp) fprintf('Feng and Mita equation (ii) give %5.3f frequency ratio\n\t\t\t\t\t\t and %5.3f damping ratio\n',... Feng_freq_ratio,Feng_damping_ratio_acc) disp('***************************************************************** ***********') Sadek_freq_ratio=1/(1+u)*(1-e_s*sqrt(u/(1+u))); Sadek_damping_ratio=e_s/(1+u)+sqrt(u/(1+u)); fprintf('Sadek et al. equations give %5.3f frequency ratio\n \t\t\t\t\t and %5.3f damping ratio\n',... Sadek_freq_ratio,Sadek_damping_ratio) disp('***************************************************************** ***********') Rudinger_stiff_ratio=(u*(2+u))/((2*(1+u)^2)); Rudinger_freq_ratio=sqrt(Rudinger_stiff_ratio/u); Rudinger_damping_ratio=sqrt((3*u^4+4*u^3)/(4*(1+u)^3)); fprintf('Rudinger equations give %5.3f frequency ratio\n\t\t\t\t\tand %5.3f damping ratio\n',... Rudinger_freq_ratio,Rudinger_damping_ratio) disp('***************************************************************** ***********') switch d case 1 Hoang_d1_freq_ratio=sqrt((1-6*u)*(1+u^2)/(1+u))-0.7*e_s; Hoang_d1_damping_ratio= sqrt((u*(1+2.5*u+2*u^2)))/(2*(1+2.7*u)); fprintf('Hoang et al. equations with ground frequency = %1i result in \n %5.3f frequency ratio and %5.3f damping ratio\n',... d,Hoang_d1_freq_ratio,Hoang_d1_damping_ratio) case 3 Hoang_d3_freq_ratio=sqrt(1-u/2)/(1+u)-0.7*e_s/(1-u/2); Hoang_d3_damping_ratio=sqrt((u*(1-u/4))/(4*(1+u)*(1-u/2)))+0.25*u*e_s; fprintf('Hoang et al. equations with ground frequency %1i result in \n %5.3f frequency ratio and %5.3f damping ratio\n'... ,d,Hoang_d3_freq_ratio,Hoang_d3_damping_ratio) end disp('***************************************************************** ***********') Krenk_freq_ratio=1/(1+u); Krenk_damping_ratio=1/2*sqrt(u/((1+u))); fprintf('Krenk and Hogsberg equations give %5.3f frequency ratio\n\t\t\t\t\t\t\t and %5.3f damping ratio\n',... Krenk_freq_ratio,Krenk_damping_ratio) disp('***************************************************************** ***********')

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Command Window

Please enter mass ratio = 0.03

Please enter structural damping ratio (e.g ASCE 7-05 for steel structures= 0.05)= 0.05

Please enter ground frequency from soil profile (used in Hoang et al.

2008 equation)= 1

======

Optimum tuning parameters of a TMD for:

Mass ratio = 0.03

Structural damping ratio = 0.05

======

Den Hartog equations give 0.971 frequency ratio

and 0.101 damping ratio

***********************************************************************

Waburton equations give 0.964 frequency ratio

and 0.086 damping ratio

***********************************************************************

Fujino and Abe equations give 0.964 frequency ratio

and 0.086 damping ratio

***********************************************************************

Feng and Mita equation (i) give 0.964 frequency ratio

and 0.083 damping ratio

Feng and Mita equation (ii) give 0.964 frequency ratio

and 0.478 damping ratio

***********************************************************************

Sadek et al. equations give 0.963 frequency ratio

and 0.219 damping ratio

***********************************************************************

125

Rudinger equations give 0.978 frequency ratio

and 0.005 damping ratio

***********************************************************************

Hoang et al. equations with ground frequency = 1 result in

0.858 frequency ratio and 0.083 damping ratio

***********************************************************************

Krenk and Hogsberg equations give 0.971 frequency ratio

and 0.085 damping ratio

***********************************************************************

126