Walking induced floor vibration design and control

Finally, a probabilistic vibration analysis of a floor with and without dampers is conducted with a large number of simulation cases considering the likely variations in loading within e

WALKING INDUCED FLOOR VIBRATION

DESIGN AND CONTROL

A dissertation submitted by

HUU ANH TUAN NGUYEN

for the award of

Doctor of Philosophy

Faculty of Engineering and Industrial Sciences Swinburne University of Technology

2013

Abstract

Disturbing walking-induced vibrations have been observed more frequently in recent times on long span lightweight floor systems as evidenced by the development of a number of new design guidelines for floor vibration assessment Constraining vibration levels to meet human comfort criteria is a vital serviceability requirement in the design

of floors The aim of the present research is to minimise the adverse vibrations from footfalls in new floors by providing better estimation of expected response and in existing floors by the use of a new configuration of tuned mass damper (TMD)

Current floor vibration guidelines are reviewed and modifications are proposed to enhance the accuracy of floor response prediction A significant finding is the development of an empirical expression for a unique factor that incorporates all effects

of the floor properties, pacing rate, resonant harmonic and non-resonant harmonic forcing components on the floor response The peak response due to a moving multi-harmonic force can now be easily computed as the multiplication of the proposed factor

by the steady state response due to a stationary single-harmonic force Also discussed are finite element (FE) and semi-FE approaches for predicting the worst-case response

of floors

One major achievement of the present research is the application of an innovative multi TMD system as an effective solution for floor vibration control A closed form solution for natural frequencies and steady state response of systems with multi TMDs is developed to facilitate preliminary design A custom made distributed multiple viscoelastic TMD system has been developed and successfully installed on a real office floor where disturbing walking-induced vibrations were observed Extensive FE investigations and various field tests performed on the real floor reveal that the response

level of the damper-retrofitted floor is suppressed by at least 40% to an acceptable limit for human comfort

Another contribution of the present research is the characterisation of human walking force based on the experimental footfall data obtained from an Australian biomechanics research program The descriptive statistics of basic gait parameters are determined and the intra- and inter-subject variability in gait parameters is examined from this footfall database Moreover, design values are proposed for the dynamic load coefficients corresponding to the first ten harmonics of walking

Finally, a probabilistic vibration analysis of a floor with and without dampers is conducted with a large number of simulation cases considering the likely variations in loading within each walk and between different walks and the possible changes in the dynamic properties of the floor and dampers This sensitivity analysis automatically covers the effect of damper off-tuning and further validates the effectiveness and reliability of the TMD method when demonstrating that the 90% and 95% fractile response levels of the floor can be reduced by about 43% using the dampers

Acknowledgments

It has been my great honour and real privilege to carry out this research under the supervision of Prof Emad Gad, Prof John Wilson and Prof Nicholas Haritos I am very grateful for their patient guidance, enthusiastic encouragement and constant support throughout my research journey I have benefited a lot from their valuable advice on the planning and development of this research work, research methodology, as well as academic writing and editing in English Besides academic aspects, I admire their generosity, professional and friendly attitude

My grateful thanks are also extended to Assoc Prof Noel Lythgo who kindly granted

me access to his experimental data on human gait measures and provided me with a brief tutorial on the Vicon Nexus software Moreover, I find myself fortunate to have worked alongside Mr Ibrahim Saidi and I thankfully acknowledge all his dedicated assistance with my experimental work I would like to offer my thanks to Ms Caroline Dean and Mr Sean Kinder for designing a detail of the damper as part of their undergraduate research project; to many friends and colleagues whose names are not mentioned here for their encouragement during my course of study

I am really grateful to Swinburne University of Technology for providing me with tuition fee waivers and the Vietnamese Ministry of Education and Training for awarding

me a stipend scholarship These supports opened an opportunity for me to be an international student for the first time

Most of all, I would like to express my deepest gratitude to all members of my family, especially my beloved parents and wife, for their unconditional support, understanding, abiding love and encouragement This thesis is dedicated to my family, to whom I am greatly indebted

Candidate’s Declaration

I hereby declare that this dissertation represents my own work and effort, except where otherwise acknowledged in the text I certify that this submission, to the best of my knowledge, contains no material previously published or written by another person except where due reference is made in the text I confirm that neither the submission nor the original work contained therein has been previously submitted to this university or any other institution for the award of a degree or other qualification

Candidate's signature: _

Huu Anh Tuan Nguyen

Table of Contents

Abstract i

Acknowledgments iii

Candidate’s Declaration iv

List of Figures xi

List of Tables xxi

Chapter 1 Introduction 1

1.1 Background and Motivation 1

1.2 Research Aim and Objectives 4

1.3 Research Methodology 5

1.4 Thesis Layout 6

Chapter 2 Literature Review 8

2.1 Introduction to Floor Dynamics 8

2.1.1 Human-induced loading 8

2.1.2 Floor dynamic properties and response 11

2.1.3 Factors affecting human comfort 17

2.2 Acceptance Criteria 21

2.2.1 Peak acceleration 23

2.2.2 RMS acceleration 23

2.2.3 Vibration dose value 25

2.2.4 RMS velocity 26

2.3 Predicting Dynamic Properties and Walking Response of Floors using Current Guidelines 27

2.3.1 AISC/CISC DG11 27

2.3.2 SCI P354 30

2.3.3 CCIP-016 36

2.3.4 EUR DG (HIVOSS) 38

2.4 Finite Element Modelling, Compared with Manual Methods and Physical Experiments 42

2.4.1 Introduction to floor vibration analysis using finite element 42

2.4.2 Analysis of composite floors 43

2.4.3 Analysis of concrete floors 50

2.5 Dynamic Testing of Floor Systems 51

2.5.1 Introduction 51

2.5.2 Unreferenced and instrumented heel drop tests 53

2.5.3 Extraction of natural frequency and damping 56

2.5.4 Measurement of mode shape and modal mass 62

2.5.5 Measurement of walking response 65

2.6 Rectification of Floor Vibrations 68

2.6.1 Structural and architectural modification 68

2.6.2 Specialist damping material 74

2.6.3 Tuned mass damper 78

2.6.4 Active control 89

2.6.5 Semi-active control 93

2.7 An Innovative Viscoelastic Tuned Mass Damper 98

2.7.1 Description of a new damper configuration 98

2.7.2 SDOF model for the sandwich beam TMD 100

2.8 Summary and Conclusions 103

Chapter 3 Improving a Prediction Method for Walking Induced Floor Vibration 107

3.1 Introduction 107

3.2 Determination of Steady State Factor 109

3.2.1 Methodology 109

3.2.2 Moving-walk forcing functions 110

3.2.3 Numerical integration method 112

3.2.4 Results 114

3.2.5 Discussion 116

3.3 Closed Form Expression for Steady State Factor 118

3.3.1 Response to stationary harmonic force 118

3.3.2 Development of empirical expressions for response to moving force 119

3.3.3 Literature proposals 123

3.3.4 Comparison and discussion 125

3.4 Prediction of Floor Response 128

3.4.1 Proposal for a simplified design formula 128

3.4.2 Discussion on required parameters 128

3.5 A Typical Generic Footbridge Worked Example 130

3.6 A Real Composite Floor 131

3.6.1 Description of case study floor 131

3.6.2 Preliminary FE analysis 133

3.6.3 Floor testing 135

3.6.4 Updated FE modal analysis and FE time history analysis 138

3.6.5 Prediction of floor response using the proposed method 140

3.6.6 Discussion 141

3.7 Summary and Conclusions 141

Chapter 4 Comparison of Methods for Walking Induced Floor Vibration 144

4.1 Introduction 144

4.2 Analysis of Floor Vibrations using Finite Elements 145

4.2.1 Resonant frequency and response in worst case 145

4.2.2 Effects of forcing functions based on different guidelines on floor response obtained from time history analysis 153

4.2.3 Calculation examples 157

4.3 Simplified method and Semi-FE method 166

4.3.1 Description of simplified methods 166

4.3.2 Description of Semi-FE method 167

4.3.3 Results and discussion 170

4.4 Summary and Conclusions 175

Chapter 5 Control of Walking Induced Floor Vibration 177

5.1 Introduction 177

5.2 Analysis of System with Multi and Distributed TMDs 178

5.2.1 Development of closed form solutions for natural frequencies and steady state response 178

5.2.2 Application 183

5.2.3 Validation against numerical results 184

5.3 Optimum Parameters for TMD 187

5.3.1 Commonly used formulae for optimum TMD 188

5.3.2 Proposal for new optimum parameters for TMD 190

5.3.3 Comparison and discussion 193

5.4 A Composite Floor Case Study 194

5.4.1 Description of the floor and remedial methods 194

5.4.2 Suppression of floor vibration by means of stiffening 196

5.4.3 Preliminary design of a damper system 201

5.4.4 Numerical study of performance of floor with distributed multiple identical dampers 203

5.4.5 Numerical study of various damper tuning strategies 210

5.4.6 Final design, manufacture and installation of dampers 217

5.4.7 Experimental study of dampers performance 222

5.4.8 Comment on rectifying the composite floor 227

5.5 A Modified Design for Sandwich-Beam Damper 228

5.6.1 Description of a prototype damper with sliding mass 228

5.6.2 Determination of damper's dynamic properties 230

5.6 Summary and Conclusions 233

Chapter 6 Characterisation of Human Walking Force 236

6.1 Introduction 236

6.2 Basic Gait Parameters 237

6.2.1 Description of walking test and participants 237

6.2.2 Determination of gait parameters 238

6.2.3 Intra-subject variability 240

6.2.4 Relationship between different gait parameters 243

6.2.5 Effects of footwear 246

6.3 Walking Force Function 249

6.3.1 Analysis procedure 249

6.3.2 Statistical analysis for design value of dynamic coefficient 253

6.3.3 Simplified design value 260

6.3.4 Comparison with current design guides 262

6.4 Summary and Conclusions 265

Chapter 7 Probabilistic Evaluation of Floor Vibration and TMD Performance

267

7.1 Introduction 267

7.2 Analysis Model 268

7.2.1 Simplified analysis model and equation of motion 268

7.2.2 Numerical integration for MDOF systems 269

7.3 Input Parameters for Random Vibration Analysis 270

7.3.1 Modal properties of the floor 270

7.3.2 Walking force function 271

7.3.3 Modal properties of the dampers 272

7.4 Probabilistic estimation of floor response 273

7.5 Assessment of Floor Acceptability 278

7.6 Multi Frequency Tuning and Optimum Damping Tuning 279

7.6.1 Multi-frequency tuning 279

7.6.2 Minimum damping and optimum damping for dampers 282

7.7 Summary and Conclusions 284

Chapter 8 Conclusions and Future Work 287

8.1 Introduction 287

8.2 Summary and Conclusions 287

8.2.1 Reduction factor and resonant build up factor 287

8.2.2 Methods for floor vibration analysis 289

8.2.3 Design formulae for tuned mass dampers 291

8.2.4 Reduction of footfall induced floor vibration 292

8.2.5 Characterisation of human walking force 294

8.2.6 Probabilistic analysis of floor vibration and evaluation of damper performance 296

8.2.7 Factors affecting performance of tuned mass dampers 297

8.3 Suggestions for Future Work 298

References 300

Publications Arising from the Present Research 315

List of Figures

Figure 1-1: Illustration of typical office fit-outs (Hewitt and Murray 2004) 2

Figure 2-1: Walking force (Bachmann and Ammann 1987) 9

Figure 2-2: Rhythmic excitations (Allen and Pernica 1998) 11

Figure 2-3: Model of a SDOF system 12

Figure 2-4: Possible types of floor response to footfall (Feldmann et al 2009) 16

Figure 2-5: The Reiher-Meister scale (Smith 1988) 18

Figure 2-6: Dieckmann’s human body model (Pavic and Reynolds 2002a) 18

Figure 2-7: Body postures and basicentric co-ordinate systems (ISO 2631-2 1989) 19

Figure 2-8: Z-axis base curve for acceleration (ISO 10137 2007) 22

Figure 2-9: AISC/CISC DG11 peak acceleration limit (Murray et al 2003) 22

Figure 2-10: Classification based on RMS velocity (European Commission 2008b) (RMS velocity in mm/sec) 26

Figure 2-11: Idealised footfall force pulse (Murray et al 2003) 30

Figure 2-12: Typical mode shapes for composite floor systems: (a) governed by secondary beam flexibility, (b) governed by primary beam (girder) flexibility (Hicks and Devine 2006) 31

Figure 2-13: Z-axis frequency weighting curve (Smith et al 2009) 33

Figure 2-14: Time history of a single footstep force 39

Figure 2-15: Classification for 3% damping (European Commission 2008b) 41

Figure 2-16: Modelling of composite floor components (Beavers 1998, Sladki 1999) 45

Figure 2-17: Eccentricity between joist members (Sladki 1999) 45

Figure 2-18: Schematic of test setup with shaker excitation (He and Fu 2001) 52

Figure 2-19: Measured and approximate heel drop impact (Murray 1975) 54

Figure 2-20: Instrumented heel drop test (Blakeborough and Williams 2003) 55

Figure 2-21: Measured time history of a heel impact (Blakeborough and Williams 2003) .55

Figure 2-22: Simple block diagram of an FRF (He and Fu 2001) 56

Figure 2-23: Free response of a damped SDOF system 57

Figure 2-24: Plot of magnitude versus frequency 59

Figure 2-25: RANDEC technique flow chart (He and Fu 2001) 60

Figure 2-26: Taking response segments (He and Fu 2001) 61

Figure 2-27: Averaging response segments (He and Fu 2001) 61

Figure 2-28: Measurement point grid (Cantieni and Biro 2005) 63

Figure 2-29: Multi-shaker floor testing (Reynolds 2008) 63

Figure 2-30: Application of transfer function (European Commission 2006) 67

Figure 2-31: Stiffening concrete beam (Bachmann and Ammann 1987) 69

Figure 2-32: Upgrading steel beam (Bachmann 1992b) 69

Figure 2-33: Cover plate and bottom chord reinforcement (Murray et al 2003) 70

Figure 2-34: Queen post hangers for steel joists and beams (Allen and Pernica 1998) 70

Figure 2-35: Retrofitting a laboratory floor (Alvis 2001) 72

Figure 2-36: Preloaded rod and concrete slot (Connor and Homen 2007) 72

Figure 2-37: Surface treatment with damping material(Sun and Lu 1995) 75

Figure 2-38: Constrained viscoelastic layer attached to floor beam (Nelson 1968) 75

Figure 2-39: Viscoelastic damper (Ljunggren and Agren 2002) 76

Figure 2-40: Partially composite beam with Resotec layer (Willford et al 2006) 77

Figure 2-41: Schematic of a primary structure and passive TMD 78

Figure 2-42: Plank TMD (Allen and Pernica 1984) 80

Figure 2-43: TMD attached to floor girder (Bachmann and Ammann 1987) 80

Figure 2-44: Details of TMD for ballroom floor (Webster and Vaicaitis 1992) 83

Figure 2-45: TMD for a footbridge (Bachmann 1992b) 83

Figure 2-46: Prototype liquid TMD (Shope and Murray 1995) 85

Figure 2-47: TMD designed by 3M Company (Rottmann and Murray 1997) 85

Figure 2-48: TMD produced by GERB Co (Collette 2002) 86

Figure 2-49: Nonlinear TMDs for a footbridge (Poovarodom et al 2003) 87

Figure 2-50: Pendulum tuned mass damper (Setareh et al 2006) 87

Figure 2-51: Cantilever tuned mass damper (Lamb 2011) 88

Figure 2-52: (a) A reaction mass actuator and (b) its theoretical model 90

Figure 2-53: Single-input/single-output control setup (Hanagan et al 2003b) 90

Figure 2-54: Magnetorheological damper (Koo 2003) 95

Figure 2-55: Force versus Velocity for sponge MR damper (Koo 2003) 95

Figure 2-56: Passive and semi-active damping (Koo et al 2004) 96

Figure 2-57: Semi-active pendulum tuned mass damper (Setareh et al 2007) 97

Figure 2-58: Schematic of a viscoelastic sandwich beam and its equivalent SDOF model .99

Figure 2-59: Testing of damper on a concrete beam (Saidi et al 2011) 100

Figure 3-1: Predicted versus measured floor frequency and response (Sladki 1999) 108

Figure 3-2: Simplified mode shape and SDOF model for a beam-like structure subjected to walking excitation 110

Figure 3-3: Typical walking force time history 112

Figure 3-4: Linear change of acceleration (Buchholdt 1997) 113

Figure 3-5: Typical acceleration response 115

Figure 3-6: Steady state factor for 6-Hz floors 115

Figure 3-7: Steady state factor for 7-Hz floors 116

Figure 3-8: Harmonic combination factor 117

Figure 3-9: Response to stationary harmonic force at resonance 118

Figure 3-10: R factor for single harmonic force 120

Figure 3-11: Numerical versus curve fitting results for R (single harmonic) 120

Figure 3-12: Cumulative probability of error when fitting R (single harmonic) 121

Figure 3-13: Numerical versus curve fitting results for R (multi harmonic) 122

Figure 3-14: Cumulative probability of error when fitting R (multi harmonics) 122

Figure 3-15: Various formulae for R factor for single harmonic force 126

Figure 3-16: Percentage error for closed form versus numerical solutions 126

Figure 3-17: Dynamic coefficients: (a) Murray’s formula, and (b) measured by Rainer and Pernica (1986) 129

Figure 3-18: A footbridge model: (a) cross section, and (b) fundamental mode shape 130 Figure 3-19: Framing details in plan view and fit-out of a real office floor 132

Figure 3-20: (a) Ceiling, and (b) false floor space 133

Figure 3-21: Resonant mode shape obtained from FE modal analysis 134

Figure 3-22: Testing setup 135

Figure 3-23: Floor response to a heel drop excitation 137

Figure 3-24: Measured floor response to walking 138

Figure 3-25: FE-predicted floor response to walking 140

Figure 4-1: Floor framing plan 146

Figure 4-2: First mode versus resonant mode 148

Figure 4-3: Simulated heel drop excitation: (a) forcing function, (b) floor response in the time domain, and (c) floor response in the frequency domain 149

Figure 4-4: Walking-response time history 151

Figure 4-5: Walking-response spectrum 152

Figure 4-6: Modelling of walking excitation 154

Figure 4-7: Floor response from time history FE analysis 156

Figure 4-8: (a) Floor framing plan and (b) 1st mode shape 158

Figure 4-9: Resonant modes for corner and interior bays 158

Figure 4-10: Floor response to heel drop 159

Figure 4-11: Walking-response time history, corner bay 160

Figure 4-12: Walking-response time history, interior bay 160

Figure 4-13: Floor framing plan and mode shapes 161

Figure 4-14: Response to heel drop 163

Figure 4-15: Response to different walking frequency 163

Figure 4-16: (a) Floor framing, and (b) 1st mode shape 164

Figure 4-17: Response to simulated heel drop excitation 164

Figure 4-18: (a) Floor framing, (b) heel drop response; and (c), (d) mode shapes 165

Figure 4-19: Idealised bay of a regular floor layout (used in hand-calculation approach) .167

Figure 4-20: Modes to be considered in semi-FE response calculations 168

Figure 4-21: Floor response to walking (On-site tests) 173

Figure 4-22: RMS acceleration predicted by Semi-FE method 174

Figure 4-23: Predicted floor response, normalised against experimental measurement .174

Figure 5-1: SDOF primary structure combined with MTMDs 178

Figure 5-2: Plan view of some sandwich-beam TMD configurations (not to scale) 184

Figure 5-3: Concept of equivalent single damper 184

Figure 5-4: Screen shots taken from SAP2000 186

Figure 5-5: Acceleration time history (SAP2000 solution) 187

Figure 5-6: A SDOF primary structure with a TMD 188

Figure 5-7: Different response spectra for µ = 0.01, ζs = 0.03, f = 0.9978 191

Figure 5-8: Response of structure with and without TMD 191

Figure 5-9: (a) Case study floor plan; (b) mode shape critical to the problematic bay.195 Figure 5-10: Remedial measures for floor vibration problem 195

Figure 5-11: Critical mode shapes resulted from different stiffening scenarios 197

Figure 5-12: Critical natural frequency of stiffened floor bay 198

Figure 5-13: Response to a critical walking 200

Figure 5-14: Maximum response of stiffened floor bay 200

Figure 5-15: (a) Available floor cavity, (b) Distributed multi TMDs 202

Figure 5-16: TMD-fitted floor response to simulated heel drop 204

Figure 5-17: Floor response to walking at resonance 205

Figure 5-18: Response spectrum for walking excitation 206

Figure 5-19: Effects of variation in floor frequency 207

Figure 5-20: Effects of variation in floor frequency and dampers frequency 209

Figure 5-21: Effects of variation in floor damping 210

Figure 5-22: Critical mode shapes 211

Figure 5-23: Response to heel drop: illustration of frequency-updated tuning 213

Figure 5-24: Floor response to walking at resonance (Mode 4 corresponds to 6.22 Hz) .215

Figure 5-25: Walking response spectrum (Mode 4 corresponds to 6.22 Hz and Mode 5

corresponds to 6.33 Hz) 216

Figure 5-26: Peak response of different floor bays 216

Figure 5-27: DMA machine 218

Figure 5-28: DMA results for rubber utilised in manufacturing the TMDs 218

Figure 5-29: A typical damper 220

Figure 5-30: Damper response to a pluck test 220

Figure 5-31: Dampers installed on the real floor 222

Figure 5-32: Floor response to shaker excitation 224

Figure 5-33: Peak response due to shaker excitations 225

Figure 5-34: Acceleration response time traces due to walking 226

Figure 5-35: Peak response due to walking 227

Figure 5-36: Schematic of damper with sliding mass 229

Figure 5-37: Details of guide rail 230

Figure 5-38: Frequency response of 450 mm damper 231

Figure 5-39: Relationship between frequency and length 232

Figure 6-1: Probability distribution of gait parameters for normal walk with shoes 238

Figure 6-2: Probability distribution of standard deviation of gait parameters 241

Figure 6-3: Cumulative probability of standard deviation of gait parameters 242

Figure 6-4: Relationship between gait parameters 245

Figure 6-5: Probability of gait parameters for walking barefoot at normal speed 246

Figure 6-6: Measured force time history for a single footstep 250

Figure 6-7: Generation of continuous footfall force 251

Figure 6-8: A typical FFT result 252

Figure 6-9: Measured dynamic coefficients 254

Figure 6-10: Cumulative probability of dynamic coefficient (fp = 1.8−2.2 Hz) 256

Figure 6-11: 90% fractile dynamic coefficients 257

Figure 6-12: 95% fractile dynamic coefficients 258

Figure 6-13: Cumulative probability distribution of dynamic coefficient 261

Figure 6-14: Comparison of methods for dynamic coefficient 263

Figure 7-1: Simplified analysis model of floor with multi TMDs 269

Figure 7-2: Example of step frequency and walking force for a walk activity 272

Figure 7-3: Typical response histories of floor with dampers 274

Figure 7-4: Frequency distribution of peak floor acceleration (500000 samples) 274

Figure 7-5: Cumulative probability of apeak (floor damping = 2–3%) 276

Figure 7-6: Cumulative probability of aRMS (floor damping = 2–3%) 276

Figure 7-7: Cumulative probability of vRMS (floor damping = 2–3%) 277

Figure 7-8: Cumulative probability of apeak (floor damping = 1–3%) 277

Figure 7-9: Cumulative probability of aRMS (floor damping = 1–3%) 277

Figure 7-10: Cumulative probability of vRMS (floor damping = 1–3%) 278

Figure 7-11: Floor with multi-frequency TMDs: cumulative probability of apeak 280

Figure 7-12: Floor with multi-frequency TMDs: cumulative probability of aRMS 280

Figure 7-13: Floor with multi-frequency TMDs: cumulative probability of vRMS 281

Figure 7-14: Cumulative probability of apeak: minimum vs optimum TMDs damping 283 Figure 7-15: Cumulative probability of aRMS: minimum vs optimum TMDs damping 283 Figure 7-16: Cumulative probability of vRMS: minimum vs optimum TMDs damping 284

List of Tables

Table 1-1: Loads and damping for different floor fit-outs (Hewitt and Murray 2004) 3

Table 2-5: Multiplying factors for low probability of adverse comment 24 Table 2-6: Multiplying factors recommended by SCI P354 (Smith et al 2009) 24 Table 2-7: Multiplying factors for offices recommended by SCI P076 (Wyatt 1989) 25 Table 2-8: Vibration Dose Value (m/s1.75) for various degrees of adverse comments 25

Table 2-11: K coefficients for walking force calculation (Feldmann et al 2009) 40

Table 4-5: Peak acceleration 172

Table 7-3: Influence of tuning frequencies on floor response 281 Table 7-4: Influence of TMDs' damping on floor response (floor damping = 1–3%) 284

Chapter 1 Introduction

1.1 Background and Motivation

Modern floor systems are being designed and constructed with longer spans owing to the need for larger column-free spaces in offices and commercial retail buildings This trend is supported by the use of advanced lightweight construction technology and high-strength materials The strength capacity of long-span composite floors and prestressed concrete floors could be assured without requirement for greater cross sections However, the longer the span, the greater the deflection and the lower the natural frequency would be These trends may result in modern floor constructions being more vibration-vulnerable regardless of the adequacy of their ultimate strength capacity Serviceability rather than strength is hence becoming the most critical design requirements for long-span floor systems

Floor systems can be subjected to dynamic forces generated by various human activities such as walking, running and dancing Resonant vibrations can arise in low frequency floors when natural frequencies of the floors closely match the excitation frequency or harmonics of human activities (Heins Jr and Yoo 1975) Examining more than 100 problematic composite floors, Murray found that the first natural frequency of the floors was in the 5–8 Hz range which can be in resonance with a harmonic of the footstep frequency (Favor 1997) As resonance can significantly amplify the vibration level, a comprehensive evaluation of vibration in low frequency floors should be considered For instance, the National Building Code of Canada requires dynamic analysis of floors with natural frequencies less than 6 Hz (NBC 1995)

Office fit-out would alter the inherent damping of a floor which comes from not only structural members but also non-structural elements Two floor systems having the same structural members and span lengths can still exhibit different levels of vibration response depending on the layout of architectural components including floor finishes, partitions, furnishings and mechanical ducts Unfortunately, changes in modern office

layouts associated with the removal of full height partitions, heavy filing cabinets, large bookshelves and other architectural elements result in a reduction of both floor mass and damping Figure 1-1 illustrates some typical types of office whilst actual loads and effective modal damping in accordance with various conditions of office fit-outs are given in Table 1-1 Hewitt and Murray (2004) have indicated quantitatively the differences in some basic parameters between a modern floor system and a traditional one A modern electronic office may typically have a bay length of 12 m with slab thickness of about 120 mm whilst the corresponding values for a traditional office are 7.5 m long bays and approximately 160 mm thick slabs The actual loading from modern office fit-out can be less than half of that arising from a traditional office fit-out Smaller amounts of damping, usually in the range of 2% to 3%, can be expected in an electronic office or open working area whilst damping levels of 5% or even greater can

be found in traditional floors with high density of partitions Reduction in damping could increase the floor response, which in turn affects the comfort of occupants

Figure 1-1: Illustration of typical office fit-outs (Hewitt and Murray 2004)

Table 1-1: Loads and damping for different floor fit-outs (Hewitt and Murray 2004)

(kPa)

Live load (kPa)

Damping Ratio

1 Traditional office:

full-height partitions running parallel to the

beam span, with or without suspended

ceiling and ductwork attached below the slab

2 Electronic office:

nearly no paperwork, limited numbers of file

cabinets, no full-height partitions, with

suspended ceilings and ductwork attached

below the slab

3 Open office plan:

cubicles and no full-height partitions, with

suspended ceiling and ductwork attached

below the slab

4 Office library:

full-height bookcases in heavily loaded

room, suspended ceiling and ductwork

attached below the slab

Floor vibration due to human activities is becoming a significant concern to designers and developers of long-span lightweight floor systems Disturbing floor vibrations caused by normal walking have been observed more frequently in recent times as evidenced by the development of new design guidelines (Murray et al 2003, Willford and Young 2006, Feldmann et al 2009, Smith et al 2009, Hicks and Smith 2011) Considering any potential vibration problems during the design stage is preferable to fixing an as-built floor, which would be more expensive and awkward (Sladki 1999) The thesis author hence found it essential to enhance the calculation methods for reasonably conservative prediction of floor vibrations

A variety of remedial measures to rectify floor vibration problems has been used with different levels of success (Nguyen et al 2012) When the thesis author started his PhD candidature, the concept of a new tuned mass damper (TMD) using viscoelastic material had been developed thanks to the dedicated work of a senior PhD candidate, Ibrahim Saidi (Saidi 2011) At that time, the performance of the new damper had been examined

on some simple laboratory beams only There was a huge motivation to explore the application of the new TMD on real floors with complex geometry Part of the research presented in this thesis was to join in the design of a distributed multi damper system for rectifying an in-service office floor This thesis therefore includes some shared experimental results that have been reported by Saidi (2011) However, independent analytical and numerical investigations are presented in this dissertation

1.2 Research Aim and Objectives

The aim of the research presented in this thesis is to minimise the adverse vibrations from human footfalls in new floors by providing better prediction of expected response and in existing floors by the use of a new configuration of tuned mass damper Specific objectives as presented below are to be accomplished to achieve the stated research aim

- Develop design charts and approximate formulae for preliminary estimation of floor response to walking The development is based on modification of the prediction method suggested by a widely used design guide

- Compare and contrast currently used procedures for prediction of floor vibration: manual, FE and semi-FE methods Some suggestions to enhance the accuracy of prediction using FE method are also investigated

- Make some contributions to TMD theory in general, not necessarily restricted to the context of TMD for floors, by developing design formulae and finding new optimum parameters for TMD

- Investigate two remedial methods, stiffening technique and TMD, for a real office floor subject to annoying vibrations due to footfall excitations

- Develop comprehensive FE models for vibration analysis of the original floor without dampers, the stiffened floor and the floor with dampers Different stiffening scenarios and various tuning policies for the dampers are investigated

- Design and build an innovative multiple TMD system for the case study floor Perform various field tests to assess the performance of the TMD system

- Propose a modified design for the current viscoelastic TMD to facilitate tuning

- Gain a better understanding of human walking force via statistically studying large experimental footfall data collected from a biomechanics research in Australia

- Perform a probability-based evaluation of the floor vibration and the damper performance, allowing for variations in both the excitations and the dynamic properties of the floor and damper

1.3 Research Methodology

It is first essential to conduct a comprehensive literature review on different methods for assessing footfall induced floor vibrations and remedial techniques to alleviate the annoying vibrations The stated research objectives are then achieved using both analytical and experimental investigations The analytical investigation is facilitated by

a number of computer programs that the author has developed, and by commercially available FE analysis packages and statistical software Standardised forcing functions introduced by current design guides have been used in most of the numerical investigations presented in the thesis However, findings related to the characterisation

of walking force are employed in the last chapter which discusses a probabilistic vibration analysis of floors with and without dampers The experimental study involves forced vibration testing of a real office floor where excessive vibrations are reported Various types of dynamic loadings are applied to the case study floor including heel drop, walking and shaker excitations FE model updating based on experimental data is also exercised to enhance the accuracy of FE modelling

Chapter 3 highlights some limitations of a currently widely used method for predicting floor response to walking, and proposes improvement to this method Design charts and empirical formulae for walking response are developed The accuracy of the proposed formulae and other proposals found in literature is examined Some worked examples are presented to illustrate the application of the proposed design formulae

Chapter 4 proposes an FE procedure to determine the most critical vibration mode of a floor bay from various modes obtained from an FE modal analysis The proposed procedure would enable the prediction of the worst-case vibration response of the floor This chapter also compares and contrasts procedures for assessment of walking induced floor vibration The presented material covers various manual methods suggested by different guidelines, time history FE analysis and semi-FE analysis methods

Chapter 5 starts with some contributions to general theory of TMD, which is not necessarily restricted to floor vibration context These include the development of a closed form solution for calculating the natural frequencies and steady state response of

a structure fitted with multiple TMDs, and a suggestion of optimum parameters for TMDs The study then focuses on application of the new viscoelastic TMD in a real

floor Various damper tuning strategies and their effects on the dampers’ performance are investigated Results from field testing of the real floor with and without dampers are reported Moreover, a traditional technique of suppressing floor vibration by means

of stiffening is examined numerically and compared with the TMD approach The chapter also introduces the design of a sliding-mass mechanism that can facilitate on-site tuning of the viscoelastic TMD

Chapter 6 discusses the characterisation of human walking force utilising experimental footfall data acquired from a biomechanics research program carried out in Australia The inter- and intra-subject diversity in gait parameters is examined, and the statistics of basic gait parameters such as walking speed, step frequency and step length are found for different walking conditions Furthermore, characteristic design values are proposed for the dynamic load factors that constitute mathematical models for walking excitations Comparison with current floor vibration design guides is also made

Chapter 7 examines a simple probability based analysis procedure to assess walking induced vibrations of floors with and without dampers The analysis takes into account the variability in waking force within each walk and between different walks, and the likely change in the dynamic characteristics of both the floor and dampers Some aspects of randomness in gait parameters found in Chapter 6 will be utilised The probability distribution of response levels for both the bare floor and damper-fitted floor will be computed from which the effectiveness of the dampers can be evaluated

Finally, the key findings and contributions of the present work are summarised in Chapter 8 Some proposals for future development of the research are also made

Chapter 2 Literature Review

This chapter presents a literature review on the design and control of human-induced floor vibrations Basic aspects of floor dynamics including human induced loading, floor response and human perception are firstly discussed This is followed by an overview of current floor vibration guidelines, which focuses on acceptance criteria and the methods to predict the floor dynamic properties and walking response A considerable amount of literature on finite element techniques for vibration analysis of composite floors and concrete floors is also discussed The next section covers basic procedures for floor testing and post processing techniques Subsequently, a variety of methods to rectify floor vibration problems is reviewed These remedial actions range from structural modifications to using high-damping materials and various types of passive and active dampers

2.1 Introduction to Floor Dynamics

A vibration problem can be characterised by three components, namely vibration source, transmission path and receiver (ISO 10137 2007) In the context of human-induced floor vibrations, examples of vibration source are people walking or running on

a floor which is considered as the transmission path, whilst the floor tenants are the receivers whose comfort is affected by the floor vibrations Some basic aspects of human loadings, floor response and human perception are presented in this section

2.1.1 Human-induced loading

Common types of dynamic forces generated on floor systems by human activities such

as walking, running and dancing can be considered as periodic loads Table 2-1 presents the average values of step frequency, forward speed and step length for common human activities (Bachmann and Ammann 1987) Figure 2-1 shows an example of the vertical loading produced by a person walking at a step frequency (pacing rate) of 2 Hz The

time dependent force F(t) was normalised against the static weight P of the walker

Continuous ground contact occurs during walking because at least one of the feet is always in contact with the ground As shown in Figure 2-1, the step period corresponding to a 2-Hz step frequency is 0.5 seconds whilst the loading duration of a single footfall is about 0.6 seconds Consequently, there are instances when both feet are

on the ground during which the individual footstep forces are overlapped Whilst ground contact is continuous for walking, it can be interrupted for the case of running with increasing step frequency

Table 2-1: Common step frequency, velocity and step length

(Bachmann and Ammann 1987)

(Hz)

Velocity (m/s)

Step length (m)

In developing mathematical models for describing the walking force, it is assumed that the excitation is perfectly periodic and a Fourier series with some harmonic components can be used to represent the loading function The step frequency and its multipliers constitute the frequencies of the harmonic components This walking force model has been suggested by a number of researchers and is adopted in current design guides (Rainer and Pernica 1986, Bachmann and Ammann 1987, Rainer et al 1988, Murray et

al 2003, Willford and Young 2006, ISO 10137 2007, Smith et al 2009) The general

form of the time dependent loading function F(t) for continuous walking can be

=

4 3 1

2sin1

)(

i

i p

i if t P

recommended by the North American guideline AISC/CISC DG11 are 0.5, 0.2, 0.1 and

0.05 for the first four harmonics (Murray et al 2003) Ellis (2000) even suggested that

lightly damped floors can be excited by up to the 8th harmonic and the dynamic coefficients α4 to α8can be taken as 0.1 A more detailed discussion on various walking force models associated with current design guides is presented in Section 2.3 There have also been investigations on the differences between walking force applied on floors and that applied on slender staircases (Bishop et al 1995, Kerr and Bishop 2001) Whilst office floors and footbridges are usually affected by walking excitation, floor systems of gymnasium and sport halls, concert halls and theatres can be subjected to vibrations due to rhythmic activities Figure 2-2(a) shows a simplified loading function for a group of people dancing The cyclic force is applied to the floor at the music beat frequency Resonance would occur when the floor frequency is sufficiently low to

match the first or second harmonics of step frequency of dancing which is normally in the range of 1.5−3 Hz In the case of jumping force produced by aerobics, the loading function can be approximately represented by a sequence of semi-sinusoidal pulses as shown in Figure 2-2(b) with discontinuous ground contact Not only the first but also the second and third harmonics of the step frequency contribute to the jumping force Resonance may hence occur at frequencies of up to 8.25 Hz, three times the expected maximum step frequency for jumping (Allen and Pernica 1998, Murray et al 2003)

Figure 2-2: Rhythmic excitations (Allen and Pernica 1998)

2.1.2 Floor dynamic properties and response

When a floor is idealised as a single degree of freedom (SDOF) system subjected to a

dynamic load F(t) as shown in Figure 2-3, its governing equation of motion can be

expressed as:

)

(t F kx x x

in which x, m, c, k are the displacement, modal mass, damping coefficient and stiffness

of the floor (Clough and Penzien 1993) The natural frequency fn of the floor depends on the ratio of the stiffness k to the mass m as given by Eq (2-3):

The main factors affecting the dynamic behaviour of a floor system are the mass, stiffness and damping of the floor The mass and stiffness can be estimated quite accurately from physical principles using information about the dimensions of different structural and non-structural members, the physical and mechanical characteristics of materials, the structural layouts, the estimated weight of the occupants, and so on However, the damping, which is considered as the key parameter influencing the floor response at resonance, cannot be predicted easily from physical principles Estimation

of damping, even by experimental modal analysis, may contain a multitude of errors (Reynolds et al 1998) Although damping would generally be nonlinear and amplitude dependent, it can reasonably be assumed as viscous damping at low amplitude It is common practice to convert all types of damping from different sources, structural and non-structural, to an overall equivalent viscous damping ratio which is a fraction of critical damping (Jeary 1996, Jeary 1997) The damping ratio ζ is related to the

parameters c, m, k by Eq (2-4) (Clough and Penzien 1993):

Figure 2-3: Model of a SDOF system

Stiffness and frequency

Increasing the stiffness and frequency can shift resonance to higher harmonics with lower dynamic coefficients Murray (2001) suggested that floors should not be designed with a natural frequency below 3 Hz Spanning the girders (primary beams) in the shorter direction of a bay could result in greater floor stiffness The provision of

sufficient composite action between the slab and its supporting members in a composite floor is necessary to ensure the stiffness of the whole system The stiffness of different structural members including deck, beams and girders, and the composite action between them determine the overall stiffness of the whole floor system On the other hand, non-structural components could be a source of additional stiffness and damping when they are appropriately provided Experimental investigations at the University of Oxford (Falati 1999) revealed that the addition of full-height partitions can increase the floor frequency and damping by about 150% and 77% respectively The additional stiffness achieved by adding full-height partitions to a floor normally well offsets the increase in the floor mass, thus enhancing the floor frequencies (Pernica 1987)

Damping and mass

Results from dynamic testing of floor systems demonstrated that mass and damping have a significant influence on human-induced floor vibrations (Haritos et al 2005) Increasing damping can reduce the vibration magnitude at resonant conditions, attenuate transmission of vibratory energy, enhance sound isolation, and accelerate the decay of free vibration (Ungar 1988) The inherent damping of a floor comes from not only structural members but also non-structural elements or architectural components Material type, construction type, connection details, slab thickness, fire protection, partitions, ceiling, desks, filing cabinets, bookcases, ductwork, plumbing, etc all affect the amount of damping of a completed floor system In many cases, the damping contributed by the bare structure is even less than that arising from non-structural components, furnishings, and occupants As a result, different floor systems built from the same material may have very different overall amount of damping (Murray 1975, Hewitt and Murray 2004)

The contribution of different sources of damping to the total damping can be evaluated using Table 2-2 as per the European technical report (Feldmann et al 2009) It can be seen that the type of structural material, the layouts of furniture and finishes are the key features determining the inherent damping of a floor For instance, an open plan composite floor with ceiling below the slab may have a damping ratio of 3% resulting

from 1% damping of the structural material, 1% damping contributed by furniture and 1% damping due to the presence of ceiling under the floor

Table 2-2: Damping estimation (Feldmann et al 2009)

Traditional office for 1 to 3 persons with

Total Damping D = D1 + D2 + D3

Current guidelines suggest damping values for various floor-type structures constructed from different material with different levels of furnishing Table 2-3 summarises the estimated damping for composite floors as suggested by the North American design guide AISC/CISC DG11 (Murray et al 2003) and UK guidelines CCIP 016 (Willford and Young 2006) and SCI P354 (Smith et al 2009) Comparing the damping values introduced by various methods, it would be reasonable to assume a damping ratio of 2%

to 3% for composite floors with typical fit-out including non-structural components, furnishings and demountable partitions However, the suggested damping values for preliminary design of bare floors can be as low as 1.3% for prestressed concrete construction or 1.8% for steel-concrete fully composite construction acceleration (ISO

10137 2007)

Table 2-3: Damping ratio for composite floors

DG11

Completed composite floors with low

fit-out, few non-structural components

Completed composite floors, fully

furnished with typical fit-out in normal

use

Completed composite floors with

extensive fit-out and full height

partitions between floors

Response to walking

In regard to response to walking excitations, floor systems are categorised into low frequency floors in which resonance may cause severe vibration amplification and high frequency floors where resonance becomes less important compared with transient response Figure 2-4(a) illustrates the build-up of resonant response under repeated footfall on low frequency floors When one of the harmonics of walking matches the floor frequency, a resonant condition occurs and the vibration level is significantly amplified On the other hand, if the floor frequency is sufficiently higher than the excitation frequency, transient response will be dominant and the response history would have the form of a series of impulses shown in Figure 2-4(b) The response to a footfall will have an initial peak followed by a decaying vibration before the next footfall It should be noted that the response histories shown in Figure 2-4 are for

stationary walk loadings, i.e due to a person walking on a single spot rather than moving along the floor

As harmonics beyond the fourth harmonic would have minimal contribution to the floor response, the cut-off frequency above which resonant build-up of response is not significant can be taken as four times the expected maximum step frequency (Willford

et al 2007) The threshold natural frequency to distinguish between low and high frequency floors is taken as 9−10 Hz as per the AISC/CISC DG11 (Murray et al 2003),

7 Hz as per the SCI P076 (Wyatt 1989), or 10 Hz as per the SCI P354 (Smith et al 2009) However, Živanovic and Pavic (2009) argued that the division into low- and high-frequency floors may lead to considerable errors in some circumstances Testing four beam-and-block floors with the fundamental natural frequency of 8.2−8.7 Hz, they found that the floors seemed not to belong to either of the two floor types The high frequency content above 10 Hz in the measured responses due to walking was much stronger than the low frequency one A floor vibration analysis treating these floors as low-frequency floors was found to significantly underestimate the response Therefore, both low and high frequency components of response need to be considered in order to gain a better response prediction, especially for floors with fundamental natural frequencies being close to the "cut-off" frequency

Figure 2-4: Possible types of floor response to footfall (Feldmann et al 2009)