Vibrações de Estruturas Induzidas Pelo Vento

Wind Effects on Structures Modern Structural Design for Wind Emil Simiu DongHun Yeo WILEY Blackwell 4th Edition Wind Effects on Structures k k k k Wind Effects on Structures Modern Structural Design for Wind Emil Simiu PE PhD NIST Fellow National Institute of Standards and Technology USA DongHun Yeo PE PhD Research Structural Engineer National Institute of Standards and Technology USA Fourth Edition k k k k This edition first published 2019 2019 John Wiley Sons Ltd Edition History John Wiley Sons 1e 1978 John Wiley Sons 2e 1986 John Wiley Sons 3e 1996 All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise except as permitted by law Advice on how to obtain permission to reuse material from this title is available at httpwwwwileycomgopermissions The right of Emil Simiu and DongHun Yeo to be identified as the authors of this work has been asserted in 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should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages including but not limited to special incidental consequential or other damages Library of Congress CataloginginPublication Data Names Simiu Emil author Yeo DongHun author Title Wind effects on structures modern structural design for wind Emil Simiu PE PhD NIST Fellow National Institute of Standards and Technology DongHun Yeo PE PhD Research Structural Engineer National Institute of Standards and Technology Description Fourth edition Hoboken NJ John Wiley Sons 2019 Includes bibliographical references and index Identifiers LCCN 2018038948 print LCCN 2018040522 ebook ISBN 9781119375906 Adobe PDF ISBN 9781119375937 ePub ISBN 9781119375883 hardcover Subjects LCSH Windpressure BuildingsAerodynamics Wind resistant design Classification LCC TA6545 ebook LCC TA6545 S55 2019 print DDC 624175 dc23 LC record available at httpslccnlocgov2018038948 Cover Design Wiley Cover Image Jackal Pan Getty Images Set in 1012pt WarnockPro by SPi Global Chennai India 10 9 8 7 6 5 4 3 2 1 k k k k For Devra SueYeun Zohar Nitzan Abigail and Arin k k k k vii Contents Preface to the Fourth Edition xix Introduction xxi Part I Atmospheric Flows Extreme Wind Speeds Bluff Body Aerodynamics 1 1 Atmospheric Circulations 3 11 Atmospheric Thermodynamics 3 111 Temperature of the Atmosphere 3 112 Radiation in the Atmosphere 4 113 Compression and Expansion Atmospheric Stratification 5 114 Molecular and Eddy Conduction 6 115 Condensation of Water Vapor 7 12 Atmospheric Hydrodynamics 7 13 Windstorms 10 131 LargeScale Storms 10 132 Local Storms 10 References 16 2 The Atmospheric Boundary Layer 17 21 Wind Speeds and Averaging Times 17 22 Equations of Mean Motion in the ABL 19 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 20 231 The Ekman Spiral 21 232 Neutrally Stratified ABL Asymptotic Approach 22 233 BruntVäisäla Frequency Types of Neutrally Stratified ABLs 24 234 The Logarithmic Mean Wind Profile 27 235 Power Law Description of ABL Wind Speed Profiles 30 236 ABL Flows in Different Surface Roughness Regimes 31 237 Relation Between Wind Speeds with Different Averaging Times 33 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces 35 241 Turbulence Intensities 35 k k k k viii Contents 242 Integral Turbulence Scales 36 243 Spectra of Turbulent Wind Speed Fluctuations 38 244 Crossspectral Density Functions 44 25 Horizontally NonHomogeneous Flows 45 251 Flow Near a Change in Surface Roughness Fetch and Terrain Exposure 45 252 Wind Profiles over Escarpments 46 253 Hurricane and Thunderstorm Winds 48 References 51 3 Extreme Wind Speeds 55 31 Cumulative Distributions Exceedance Probabilities Mean Recurrence Intervals 55 311 Probability of Exceedance and Mean Recurrence Intervals 55 3111 A Case Study The Fair Die 55 3112 Extension to Extreme Wind Speeds 56 312 Mixed Wind Climates 56 32 Wind Speed Data 57 321 Meteorological and Micrometeorological Homogeneity of the Data 57 322 Directional and NonDirectional Wind Speeds 58 323 Wind Speed Data Sets 58 3231 Data in the Public Domain 58 3232 Data Available Commercially 60 33 Nyear Speed Estimation from Measured Wind Speeds 61 331 Epochal Versus PeaksOverThreshold Approach to Estimation of Extremes 61 332 Extreme Value Distributions and Their Use in Wind Climatology 62 333 Wind Speed Estimation by the Epochal Approach 63 3331 Method of Moments 63 334 Sampling Errors in the Estimation of Extreme Speeds 64 335 Wind Speed Estimation by the PeaksOverThreshold Approach 65 336 Spatial Smoothing 66 337 Development of Large Wind Speed Datasets 66 34 Tornado Characterization and Climatology 66 341 Tornado Flow Modeling 66 342 Summary of NUREGCR4461 Rev 2 Report 17 67 343 DesignBasis Tornado for Nuclear Power Plants 68 References 70 4 Bluff Body Aerodynamics 73 41 Governing Equations 74 411 Equations of Motion and Continuity 74 412 The NavierStokes Equation 75 413 Bernoullis Equation 76 42 Flow in a Curved Path Vortex Flow 77 43 Boundary Layers and Separation 78 44 Wake and Vortex Formations in TwoDimensional Flow 82 k k k k Contents ix 45 Pressure Lift Drag and Moment Effects on TwoDimensional Structural Forms 89 46 Representative Flow Effects in Three Dimensions 93 461 Cases Retaining TwoDimensional Features 96 462 Structures in ThreeDimensional Flows Case Studies 99 References 102 5 Aerodynamic Testing 105 51 Introduction 105 52 Basic Similarity Requirements 105 521 Dimensional Analysis 105 522 Basic Scaling Considerations 107 53 Aerodynamic Testing Facilities 108 54 Wind Tunnel Simulation of Atmospheric Boundary Layers 120 541 Effect of Type of Spires and Floor Roughness Elements 120 542 Effect of Integral Scale and Turbulence Intensity 122 543 Effects of Reynolds Number Similarity Violation 123 544 Comparisons of Wind Tunnel and FullScale Pressure Measurements 125 55 Blockage Effects 127 56 The HighFrequency Force Balance 128 57 Simultaneous Pressure Measurements at Multiple Taps 129 References 132 6 Computational Wind Engineering 135 61 Introduction 135 62 Governing Equations 136 63 Discretization Methods and Grid Types 136 64 Initial and Boundary Conditions 137 641 Initial Conditions 137 642 Boundary Conditions 138 65 Solving Equations 139 66 Stability 139 67 Turbulent Flow Simulations 140 671 Resolved and Modeled Turbulence 140 672 Direct Numerical Simulation DNS 140 673 Large Eddy Simulations LES 142 674 ReynoldsAveraged NavierStokes Simulation RANS 144 675 Hybrid RANSLES Simulation 146 676 Performance of Turbulence Models 148 68 Verification and Validation Uncertainty Quantification 148 681 Sources of Inaccuracy in CWE Simulations 149 682 Verification and Validation 150 683 Quantification of Errors and Uncertainties 151 69 CWE versus Wind Tunnel Testing 151 610 Best Practice Guidelines 152 References 152 k k k k x Contents 7 Uncertainties in Wind Engineering Data 157 71 Introduction 157 72 Statistical Framework for Estimating Uncertainties in the Wind Loads 157 73 Individual and Overall Uncertainties 159 731 Uncertainties in the Estimation of Extreme Wind Speeds 159 732 Uncertainties in the Estimation of Exposure Factors 160 733 Uncertainties in the Estimation of Pressure Coefficients 161 734 Uncertainties in Directionality Factors 164 References 164 Part II Design of Buildings 167 8 Structural Design for Wind 169 81 Modern Structural Design for Wind A Brief History 169 82 DatabaseAssisted Design 171 83 Equivalent Static Wind Loads 174 84 DAD versus ESWL 176 References 176 9 Stiffness Matrices SecondOrder Effects and Influence Coefficients 179 91 Stiffness Matrices 179 92 SecondOrder Effects 180 93 Influence Coefficients 181 References 181 10 Aerodynamic Loads 183 101 Introduction 183 102 Pressure Tap Placement Patterns and Tributary Areas 183 103 Aerodynamic Loading for DatabaseAssisted Design 184 104 Peaks of Spatially Averaged Pressure Coefficients for Use in Code Provisions 186 1041 Pressures Within an Area A Contained in a Specified Pressure Zone 186 1042 Identifying Areas A Within a Specified Pressure Zone 187 105 Aerodynamic Pressures and WindDriven Rain 193 References 193 11 Dynamic and Effective WindInduced Loads 195 111 Introduction 195 112 The SingleDegreeofFreedom Linear System 196 113 TimeDomain Solutions for 3D Response of MultiDegreeofFreedom Systems 197 1131 Natural Frequencies and Modes of Vibration 198 1132 Solutions of Equations of Motion of Forced System 199 114 Simultaneous Pressure Measurements and Effective Windinduced Loads 200 Reference 201 k k k k Contents xi 12 Wind Load Factors and Design Mean Recurrence Intervals 203 121 Introduction 203 122 Uncertainties in the Dynamic Response 204 123 Wind Load Factors Definition and Calibration 205 124 Wind Load Factors vs Individual Uncertainties 206 1241 Effect of Wind Speed Record Length 206 1242 Effect of Aerodynamic Interpolation Errors 206 1243 Number of Pressure Taps Installed on Building Models 207 1244 Effect of Reducing Uncertainty in the Terrain Exposure Factor 207 1245 Flexible Buildings 207 1246 Notes 207 125 Wind Load Factors and Design Mean Recurrence Intervals 208 References 209 13 Wind Effects with Specified MRIs DCIs InterStory Drift and Accelerations 211 131 Introduction 211 132 Directional Wind Speeds and Response Surfaces 212 133 Transformation of Wind Speed Matrix into Vectors of Largest Wind Effects 213 1331 Matrix of Largest Directional Wind Speeds 213 1332 Transformation of Matrix Uij into Matrix of DemandtoCapacity Indexes DCIpk m Uij 214 1333 Vector DCImi maxjDCIpk m Uij 214 134 Estimation of Directional Wind Effects with Specified MRIs 214 135 NonDirectional Wind Speeds Wind Directionality Reduction Factors 215 136 DemandtoCapacity Indexes 217 137 InterStory Drift and Floor Accelerations 217 References 218 14 Equivalent Static Wind Loads 219 141 Introduction 219 142 Estimation of Equivalent Static Wind Loads 219 References 224 15 WindInduced Discomfort in and Around Buildings 225 151 Introduction 225 152 Occupant WindInduced Discomfort in Tall Buildings 226 1521 Human Response to WindInduced Vibrations 226 153 Comfort Criteria for Pedestrian Areas Within a Built Environment 227 1531 Wind Speeds Pedestrian Discomfort and Comfort Criteria 228 154 Zones of High Surface Winds Within a Built Environment 229 1541 Wind Effects Near Tall Buildings 229 1542 Wind Speeds at Pedestrian Level in a Basic Reference Case 8 232 1543 Case Studies 235 155 Frequencies of Ocurrence of Unpleasant Winds 242 k k k k xii Contents 1551 Detailed Estimation Procedure 242 1552 Simplified Estimation Procedure 246 References 248 16 Mitigation of Building Motions 251 161 Introduction 251 162 SingleDegreeofFreedom Systems 252 163 TMDs for MultipleDegreeofFreedom Systems 255 References 256 17 Rigid Portal Frames 259 171 Introduction 259 172 Aerodynamic and Wind Climatological Databases 260 173 Structural System 261 174 Overview of the Design Procedure 262 175 Interpolation Methods 263 176 Comparisons Between Results Based on DAD and on ASCE 7 Standard 264 1761 Buildings with Various Eave Heights 264 1762 Buildings with Various Roof Slopes 265 References 265 18 Tall Buildings 267 181 Introduction 267 182 Preliminary Design and Design Iterations 267 183 Wind Engineering Contribution to the Design Process 268 184 Using the DADESWL Software 268 1841 Accessing the DADESWL Software 269 1842 Project Directory and its Contents 269 1843 Software Activation Graphical User Interface 270 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 271 1851 Building Description 271 1852 Using the DAD and the ESWL Options 274 References 280 Part III Aeroelastic Effects 283 19 VortexInduced Vibrations 287 191 LockIn as an Aeroelastic Phenomenon 287 192 VortexInduced Oscillations of Circular Cylinders 287 193 AcrossWind Response of Chimneys and Towers with Circular Cross Section 292 References 296 20 Galloping and Torsional Divergence 297 201 Galloping Motions 297 k k k k Contents xiii 2011 GlauertDen Hartog Necessary Condition for Galloping Motion 297 2012 Modeling of Galloping Motion 300 2013 Galloping of Two Elastically Coupled Square Cylinders 300 202 Torsional Divergence 303 References 304 21 Flutter 305 211 Formulation of the TwoDimensional Bridge Flutter Problem in Smooth Flow 306 212 Aeroelastic Lift and Moment Acting on Airfoils 307 213 Aeroelastic Lift Drag And Moment Acting on Bridge Decks 308 214 Solution of the Flutter Equations for Bridges 311 215 TwoDimensional Bridge Deck Response to Turbulent Wind in the Presence of Aeroelastic Effects 311 References 312 22 Slender Chimneys and Towers 315 221 Slender Chimneys with Circular Cross Section 315 2211 Slender Chimneys Assumed to be Rigid 315 2212 Flexible Slender Chimneys 318 2213 Approximate Expressions for the AcrossWind Response 318 222 Aeroelastic Response of Slender Structures with Square and Rectangular Cross Section 321 223 Alleviation of VortexInduced Oscillations 325 References 327 23 SuspendedSpan Bridges 331 231 Introduction 331 232 Wind Tunnel Testing 331 233 Response to Vortex Shedding 335 234 Flutter and Buffeting of the FullSpan Bridge 338 2341 Theory 338 2342 Example Critical Flutter Velocity and Buffeting Response of Golden Gate Bridge 341 235 Stay Cable Vibrations 344 2351 Cable Vibration Characteristics 344 2352 Mitigation Approaches 344 References 345 Part IV Other Structures and Special Topics 347 24 Trussed Frameworks and Plate Girders 349 241 Single Trusses and Girders 350 242 Pairs of Trusses and of Plate Girders 352 2421 Trusses Normal to Wind 353 2422 Trusses Skewed with Respect to Wind Direction 353 k k k k xiv Contents 2423 Pairs of Solid Plates and Girders 355 243 Multiple Frame Arrays 357 244 Square and Triangular Towers 361 2441 Aerodynamic Data for Square and Triangular Towers 361 References 366 25 Offshore Structures 367 251 Wind Loading on Offshore Structures 367 2511 Wind Loads on Semisubmersible Units 368 2512 Wind Loads on a Guyed Tower Platform 371 252 Dynamic Wind Effects on Compliant Offshore Structures 376 2521 Turbulent Wind Effects on Tension Leg Platform Surge 376 References 382 26 Tensile Membrane Structures 385 References 386 27 Tornado Wind and Atmospheric Pressure Change Effects 389 271 Introduction 389 272 Wind Pressures 390 273 Atmospheric Pressure Change Loading 393 274 Experimental Modeling of TornadoLike Wind Flows 396 References 397 28 Tornado and HurricaneBorne Missile Speeds 399 281 Introduction 399 282 TornadoBorne Missile Speeds 399 2821 Deterministic Modeling of DesignBasis Missile Speeds 400 2822 Probabilistic Modeling of DesignBasis Missile Speeds 401 283 HurricaneBorne Missile Speeds 403 2831 Basic Assumptions 403 2832 Numerical Solutions 405 2833 Simplified Flow Field Closed Form Solutions 406 References 408 Appendices 409 Appendix A Elements of Probability and Statistics 411 A1 Introduction 411 A11 Definition and Purpose of Probability Theory 411 A12 Statistical Estimation 411 A2 Fundamental Relations 412 A21 Addition of Probabilities 412 A22 Compound and Conditional Probabilities The Multiplication Rule 412 A23 Total Probabilities 413 A24 Bayes Rule 413 A25 Independence 414 k k k k Contents xv A3 Random Variables and Probability Distributions 415 A31 Random Variables Definition 415 A32 Histograms Probability Density Functions Cumulative Distribution Functions 415 A33 Changes of Variable 417 A34 Joint Probability Distributions 417 A4 Descriptors of Random Variable Behavior 419 A41 Mean Value Median Mode Standard Deviation Coefficient of Variation and Correlation Coefficient 419 A5 Geometric Poisson Normal and Lognormal Distributions 420 A51 The Geometric Distribution 420 A52 The Poisson Distribution 421 A53 Normal and Lognormal Distributions 421 A6 Extreme Value Distributions 422 A61 Extreme Value Distribution Types 422 A611 Extreme Value Type I Distribution 422 A612 Extreme Value Type II Distribution 423 A613 Extreme Value Type III Distribution 423 A62 Generalized Extreme Value GEV Distribution 423 A63 Generalized Pareto Distribution GPD 423 A64 Mean Recurrence Intervals MRIs for Epochal and PeaksoverThreshold POT Approaches 424 A7 Statistical Estimates 425 A71 Goodness of Fit Confidence Intervals Estimator Efficiency 425 A72 Parameter Estimation for Extreme Wind Speed Distributions 426 A8 Monte Carlo Methods 427 A9 NonParametric Statistical Estimates 428 A91 Single Hazards 428 A92 Multiple Hazards 428 References 430 Appendix B Random Processes 433 B1 Fourier Series and Fourier Integrals 433 B2 Parsevals Equality 435 B3 Spectral Density Function of a Random Stationary Signal 435 B4 Autocorrelation Function of a Random Stationary Signal 437 B5 CrossCovariance Function CoSpectrum Quadrature Spectrum Coherence 438 B6 Mean Upcrossing and Outcrossing Rate for a Gaussian Process 439 B7 Probability Distribution of the Peak Value of a Random Signal with Gaussian Marginal Distribution 441 Reference 442 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 443 C1 Introduction 443 C2 Peak Estimation by PeaksOverThreshold PoissonProcess Procedure 444 k k k k xvi Contents C3 Dependence of Peak Estimates by BLUE Upon Number of Partitions 451 C4 Summary 451 References 452 Appendix D Structural Dynamics 455 D1 Introduction 455 D2 The SingleDegreeofFreedom Linear System 455 D21 Response to a Harmonic Load 456 D22 Response to an Arbitrary Load 456 D23 Response to a Stationary Random Load 457 D3 Continuously Distributed Linear Systems 458 D31 Normal Modes and Frequencies Generalized Coordinates Mass and Force 458 D311 Modal Equations of Motion 458 D32 Response to a Concentrated Harmonic Load 459 D33 Response to a Concentrated Stationary Random Load 460 D34 Response to Two Concentrated Stationary Random Loads 460 D35 Effect of the Correlation of the Loads upon the Magnitude of the Response 462 D36 Distributed Stationary Random Loads 462 D4 Example AlongWind Response 463 References 466 Appendix E Structural Reliability 467 E1 Introduction 467 E2 The Basic Problem of Structural Safety 468 E3 FirstOrder SecondMoment Approach Load and Resistance Factors 469 E31 Failure Region Safe Region and Failure Boundary 469 E32 Safety Indexes 470 E33 Reliability Indexes and Failure Probabilities 473 E34 Partial Safety Factors Load and Resistance Factor Design 474 E35 Calibration of Safety Index 𝛽 Wind Directionality and Mean Recurrence Intervals of Wind Effects 475 E4 Structural Strength Reserve 475 E41 Portal Frame Ultimate Capacity Under Wind with Specified Direction 476 E42 Portal Frame Ultimate Capacity Estimates Based on MultiDirectional Wind Speeds 476 E43 Nonlinear Analysis of Tall Buildings Under Wind Loads 477 E5 Design Criteria for MultiHazard Regions 477 E51 Strong Winds and Earthquakes 477 E52 Winds and Storm Surge 478 References 478 Appendix F World Trade Center Response to Wind 481 F1 Overview 481 F11 Project Overview 481 F12 Report Overview 481 k k k k Contents xvii F2 NISTSupplied Documents 482 F21 Rowan Williams Davies Irwin RWDI Wind Tunnel Reports 482 F22 Cermak Peterka Petersen Inc CPP Wind Tunnel Report 482 F23 Correspondence 482 F24 NIST Report Estimates of Wind Loads on the WTC Towers Emil Simiu and Fahim Sadek April 7 2004 482 F3 Discussion and Comments 482 F31 General 482 F32 Wind Tunnel Reports and Wind Engineering 483 F321 CPP Wind Tunnel Report 483 F322 RWDI Wind Tunnel Report 484 F323 Building Period used in Wind Tunnel Reports 484 F324 NYCBC Wind Speed 484 F325 Incorporating Wind Tunnel Results in Structural Evaluations 485 F326 Summary 485 F33 NIST Recommended Wind Loads 485 Index 487 k k k k xix Preface to the Fourth Edition The quarter of a century that elapsed since the publication of the third edition of Wind Effects on Structures has seen a number of significant developments in micrometeo rology extreme wind climatology aerodynamic pressure measurement technology uncertainty quantification the optimal integration of wind and structural engineering tasks and the use of big data for determining and combining effectively multiple directionalitydependent time series of wind effects of interest Also following a 2004 landmark report by Skidmore Owings and Merrill LLP on large differences between independent estimates of wind effects on the World Trade Center towers it has increasingly been recognized that transparency and traceability are essential to the credibility of structural designs for wind A main objective of the fourth edition of Wind Effects on Structures is to reflect these developments and their consequences from a design viewpoint Progress in the developing Computational Wind Engineering field is also reflected in the book Modern pressure measurements by scanners and the recording and use of aerody namic pressure time series have brought about a significant shift in the division of tasks between wind and structural engineers In particular the practice of splitting the dynamic analysis task between wind and structural engineers has become obsolete performing dynamic analyses is henceforth a task assigned exclusively to the structural engineering analyst as has long been the case in seismic design This eliminates the unwieldy timeconsuming backandforth between wind and structural engineers which typically discourages the beneficial practice of iterative design The book provides the full details of the wind and structural engineers tasks in the design process and uptodate userfriendly software developed for practical use in structural design offices In addition new material in the book concerns the determination of wind load factors or of design mean recurrence intervals of wind effects determined by accounting for wind directionality The first author contributed Chapters 13 portions of Chapter 4 Chapters 5 7 and 8 Sections 91 and 93 Chapters 1012 and 15 portions of Chapter 17 and Part III Part IV and Appendices A B D and E The second author contributed Chapter 6 Section 92 and Section 235 The authors jointly contributed Chapters 13 14 16 and 18 They reviewed and are responsible for the entire book Professor Robert H Scanlan contributed parts of Chapter 4 and of Part III Appendix F authored by Skidmore Owings and Merrill LLP is part of the National Institute of Standards and Technology World Trade Center investigation Chapter 17 is based on a doctoral thesis by Dr F Habte supervised by the first author and Professor A Gan Chowdhury k k k k xx Preface to the Fourth Edition Dr Sejun Park made major contributions to Chapters 14 and 18 and developed the attendant software Appendix C is based on a paper by A L Pintar D Duthinh and E Simiu We wish to pay a warm tribute to the memory of Professor Robert H Scanlan 19142001 and Dr Richard D Marshall 19342001 whose contributions to aeroelasticity and building aerodynamics have profoundly influenced these fields The authors have learned much over the years from Dr Nicholas Isyumovs work an example of competence and integrity We are grateful to Professor B Blocken of the Eindhoven University of Technology and KU Leuven Dr A Ricci of the Eindhoven University of Technology and Dr T Nandi of the National Institute of Standards and Technology for their thorough and most helpful reviews of Chapter 6 We thank Professor D Zuo of Texas Tech University for useful comments on cablestayedbridge cable vibrations We are indebted to many other colleagues and institutions for their permission to reproduce materials included in the book The references to the authors National Institute of Standards and Technology affiliation are for purposes of identification only The book is not a US Government publication and the views expressed therein do not necessarily represent those of the US Government or any of its agencies Rockville Maryland Emil Simiu DongHun Yeo k k k k xxi Introduction The design of buildings and structures for wind depends upon the wind environment the aerodynamic effects induced by the wind environment in the structural system the response of the structural system to those effects and safety requirements based on uncertainty analyses and expressed in terms of wind load factors or design mean recur rence intervals of the response For certain types of flexible structure slender structures suspendedspan bridges aeroelastic effects must be considered in design I1 The Wind Environment and Its Aerodynamic Effects For structural design purposes the wind environment must be described i in meteo rological terms by specifying the type or types of storm in the region of interest eg largescale extratropical storms hurricanes thunderstorms tornadoes ii in microm eteorological terms ie dependence of wind speeds upon averaging time dependence of wind speeds and turbulent flow fluctuations on surface roughness and height above the surface and in extreme wind climatological terms directional extreme wind speed data at the structures site probabilistic modeling based on such data Such descriptions are provided in Chapters 13 respectively The description of the wind flows micrometeorological features is needed for three main reasons First those features directly affect the structures aerodynamic and dynamic response For example the fact that wind speeds increase with height above the surface means that wind loads are larger at higher elevations than near the ground Second turbulent flow fluctuations strongly influence aerodynamic pressures and produce in flexible structures fluctuating motions that may be amplified by resonance effects Third micrometeorological considerations are required to transform measured or simulated wind speed data at meteorological stations or other reference sites into wind speed data at the site of interest Micrometeorological features are explicitly considered by the structural designer if wind pressures or forces acting on the structure are determined by formulas specified in code provisions However for designs based on windtunnel testing this is no longer the case Rather the structural designer makes use of records of nondimensional aero dynamic pressure data and of measured or simulated directional extreme wind speeds at the site of interest in the development of which micrometeorological features were taken into account by the wind engineer and are implicit in those records However the k k k k xxii Introduction integrity of the design process requires that the relevant micrometeorological features on which those records are based be fully documented and accounted for To perform a design based on aerodynamic data obtained in windtunnel tests or in numerical simulations the structural engineer needs the following three products 1 Time series of pressures at large numbers of taps nondimensionalized with respect to the wind tunnel or numerical simulation mean wind speed at the reference height commonly the elevation of the building roof Chapters 46 2 Matrices of directional mean wind speeds at the site of interest at the prototype reference height 3 Estimates of uncertainties in items 1 and 2 Chapter 7 These products and the supporting documentation consistent with Building Infor mation Modeling BIM requirements to allow effective scrutiny must be delivered by the wind engineering laboratory to the structural engineer in charge of the design The wind engineers involvement in the structural design process ends once those products are delivered The design is then fully controlled by the structural engineer In particular as was noted in the Preface dynamic analyses need no longer be performed partly by the structural engineer and partly by the wind engineer but are performed solely and more effectively by the structural engineer This eliminates unwieldy timeconsuming backandforth between the wind engineering laboratory and the structural design office which typically discourages the beneficial practice of iterative design Chapters 17 constitute Part I of the book I2 Structural Response to Aerodynamic Excitation The structural designer uses software that transforms the wind engineering data into applied aerodynamic loads This transformation entails simple weighted summations performed automatically by using a software subroutine Given a preliminary design the structural engineer performs the requisite dynamic analyses to obtain the inertial forces produced by the applied aerodynamic loads The effective wind loads ie the sums of applied aerodynamic and inertial loads are then used to calculate demandtocapacity indexes DCIs interstory drift and building accelerations with specified mean recur rence intervals This is achieved by accounting rigorously and transparently for i direc tionality effects ii combinations of gravity effects and wind effects along the prin cipal axes of the structure and in torsion and iii combinations of weighted bending moments and axial forces inherent in DCI expressions Typically to yield a satisfactory design eg one in which the DCIs are not significantly different from unity successive iterations are required All iterations use the same applied aerodynamic loads but differ ent structural members sizes Part II of the book presents details on of the operations just described software for performing them and examples of its use supported by a detailed users manual and a tutorial Also included in Part II is a critique of the highfrequency force balance technique commonly used in wind engineering laboratories before the development of multichannel pressure scanners material on windinduced discom fort in and around buildings tuned mass dampers and requisite wind load factors and design mean recurrence intervals of wind effects k k k k Introduction xxiii Part III presents fundamentals and applications related to aeroelastic phenomena vortexinduced vibrations galloping torsional divergence flutter and aeroelastic response of slender towers chimneys and suspendedspan bridges Part IV contains material on trussed frameworks and plate girders offshore structures tensile mem brane structures tornado wind and atmospheric pressure change effects and tornado and hurricaneborne missile speeds Appendices AE present elements of probability and statistics elements of the the ory of random processes the description of a modern peaksoverthreshold procedure that yields estimates of stationary time series peaks and confidence bounds for those estimates elements of structural dynamics based on a frequencydomain approach still used in suspendedspan bridge applications and elements of structural reliability that provide an engineering perspective on the extent to which the theory is or is not useful in practice The final Appendix F is a highly instructive Skidmore Owings and Merrill report on the estimation of the World Trade Center towers response to wind loads k k k k 1 Part I Atmospheric Flows Extreme Wind Speeds Bluff Body Aerodynamics k k k k 3 1 Atmospheric Circulations Wind or the motion of air with respect to the surface of the Earth is fundamentally caused by variable solar heating of the Earths atmosphere It is initiated in a more immediate sense by differences of pressure between points of equal elevation Such differences may be brought about by thermodynamic and mechanical phenomena that occur in the atmosphere both in time and space The energy required for the occurrence of these phenomena is provided by the sun in the form of radiated heat While the sun is the original source the source of energy most directly influential upon the atmosphere is the surface of the Earth Indeed the atmo sphere is to a large extent transparent to the solar radiation incident upon the Earth much in the same way as the glass roof of a greenhouse That portion of the solar radi ation that is not reflected or scattered back into space may therefore be assumed to be absorbed entirely by the Earth The Earth upon being heated will emit energy in the form of terrestrial radiation the characteristic wavelengths of which are long in the order of 10 𝜇 compared to those of heat radiated by the sun The atmosphere which is largely transparent to solar but not to terrestrial radiation absorbs the heat radiated by the Earth and reemits some of it toward the ground 11 Atmospheric Thermodynamics 111 Temperature of the Atmosphere To illustrate the role of the temperature distribution in the atmosphere in the production of winds a simplified version of model circulation will be presented In this model the vertical variation of air temperature of the humidity of the air of the rotation of the Earth and of friction are ignored and the surface of the Earth is assumed to be uniform and smooth The axis of rotation of the Earth is inclined at approximately 66 30 to the plane of its orbit around the sun Therefore the average annual intensity of solar radiation and consequently the intensity of terrestrial radiation is higher in the equatorial than in the polar regions To explain the circulation pattern as a result of this temperature differ ence Humphreys 1 proposed the following ideal experiment Figure 11 Assume that the tanks A and B are filled with fluid of uniform temperature up to level a and that tubes 1 and 2 are closed If the temperature of the fluid in A is raised while the temperature in B is maintained constant the fluid in A will expand and reach the Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 4 1 Atmospheric Circulations B A 2 1 a b c Figure 11 Circulation pattern due to temperature difference between two columns of fluid Source From Ref 1 Copyright 1929 1940 by W J Humphreys level b The expansion entails no change in the total weight of the fluid contained in A The pressure at c therefore remains unchanged and if tube 2 were opened there would be no flow between A and B If tube 1 is opened however fluid will flow from A to B on account of the difference of head b a Consequently at level c the pressure in A will decrease while the pressure in B will increase Upon opening tube 2 fluid will now flow through it from B to A The circulation thus developed will continue as long as the temperature difference between A and B is maintained If tanks A and B are replaced conceptually by the column of air above the equator and above the pole in the absence of other effects an atmospheric circulation will develop that could be represented as in Figure 12 In reality the circulation of the atmosphere is vastly complicated by the factors neglected in this model The effect of these factors will be discussed later in this chapter The temperature of the atmosphere is determined by the following processes Solar and terrestrial radiation as discussed previously Radiation in the atmosphere Compression or expansion of the air Molecular and eddy conduction Evaporation and condensation of water vapor 112 Radiation in the Atmosphere As a conceptual aid consider the action of the following model The heat radiated by the surface of the Earth is absorbed by the layer of air immediately above the ground or the k k k k 11 Atmospheric Thermodynamics 5 Warm air Cold air Equator North pole ω Figure 12 Simplified model of atmospheric circulation Figure 13 Transport of heat through radiation in the atmosphere Heat radiated into outer space surface of the ocean and reradiated by this layer in two parts one going downward and one going upward The latter is absorbed by the next higher layer of air and again reradi ated downward and upward The transport of heat through radiation in the atmosphere according to this conceptual model is represented in Figure 13 113 Compression and Expansion Atmospheric Stratification Atmospheric pressure is produced by the weight of the overlying air A small mass or particle of dry air moving vertically thus experiences a change of pressure to which there corresponds a change of temperature in accordance with the Poisson dry adiabatic equation T T0 p p0 0288 11 A familiar example of the effect of pressure on the temperature is the heating of com pressed air in tire pump If in the atmosphere the vertical motion of an air particle is sufficiently rapid the heat exchange of that parcel with its environment may be considered to be negligible that is the process being considered is adiabatic It then follows from Poissons equation that since ascending air experiences a pressure decrease its temperature will also decrease k k k k 6 1 Atmospheric Circulations I Lapse rate prevailing in the atmosphere II Adiabatic lapse rate h I II h2 2h2 T2 1h1 T1 T T1 h1 T2 T2 Figure 14 Lapse rates The temperature drop of adiabatically ascending dry air is known as the dry adiabatic lapse rate and is approximately 1C100 m in the Earths atmosphere Consider a small mass of dry air at position 1 Figure 14 Its elevation and temper ature are denoted by h1 and T1 respectively If the particle moves vertically upward sufficiently rapidly its temperature change will effectively be adiabatic regardless of the lapse rate temperature variation with height above ground prevailing in the atmo sphere At position 2 while the temperature of the ambient air is T2 the temperature of the element of air mass is T 2 T1 h2 h1 𝛾a where 𝛾a is the adiabatic lapse rate Since the pressure of the element and of the ambient air will be the same it follows from the equation of state that to the difference T 2 T2 there corresponds a difference of density between the element of air and the ambient air This generates a buoyancy force that if T2 T 2 acts upwards and thus moves the element farther away from its initial position superadiabatic lapse rate as in Figure 14 or if T2 T 2 acts downwards thus tending to return the particle to its initial position The stratification of the atmosphere is said to be unstable in the first case and stable in the second If T2 T 2 that is if the lapse rate prevailing in the atmosphere is adiabatic the stratification is said to be neutral A simple example of the stable stratification of fluids is provided by a layer of water underlying a layer of oil while the opposite unstable case would have the water above the oil 114 Molecular and Eddy Conduction Molecular conduction is a diffusion process that effects a transfer of heat It is achieved through the motion of individual molecules and is negligible in atmospheric processes Eddy heat conduction involves the transfer of heat by actual movement of air in which heat is stored k k k k 12 Atmospheric Hydrodynamics 7 115 Condensation of Water Vapor In the case of unsaturated moist air as an element of air ascends and its temperature decreases at an elevation where the temperature is sufficiently low condensation will occur and heat of condensation will be released This is equal to the heat originally required to change the phase of water from liquid to vapor that is the latent heat of vaporization stored in the vapor The temperature drop in the saturated adiabatically ascending element is therefore slower than for dry air or moist unsaturated air 12 Atmospheric Hydrodynamics The motion of an elementary air mass is determined by forces that include a vertical buoyancy force Depending upon the temperature difference between the air mass and the ambient air the buoyancy force acts upwards causing an updraft downwards or is zero These three cases correspond to unstable stable or neutral atmospheric strat ification respectively It is shown in Section 233 that depending upon the absence or a presence of a stably stratified air layer above the top of the atmospheric boundary layer called capping inversion neutrally stratified flows can be classified into truly and conventionally neutral flows The horizontal motion of air is determined by the following forces 1 The horizontal pressure gradient force per unit of mass which is due to the spatial variation of the horizontal pressures This force is normal to the lines of constant pressure called isobars that is it is directed from highpressure to lowpressure regions Figure 15 Let the unit vector normal to the isobars be denoted by n and consider an elemental volume of air with dimensions dn dy dz where the coordi nates n y z are mutually orthogonal The net force per unit mass exerted by the horizontal pressure gradient along the direction of the vector n is dy dz p p p ndn dn dy dz 𝜌 1 𝜌 p n 12 where p denotes the pressure and 𝜌 is the air density 2 The deviating force due to the Earths rotation If defined with respect to an absolute frame of reference the motion of a particle not subjected to the action of an external force will follow a straight line To an observer on the rotating Earth however the path described by the particle will appear curved The deviation of the particle with Figure 15 Direction of pressure gradient force n Direction of pressure gradient force High pressure Low pressure Isobar k k k k 8 1 Atmospheric Circulations Vgr Gradient wind level Free atmosphere δ Boundary layer depth Figure 16 The atmospheric boundary layer respect to a straight line fixed with respect to the rotating Earth may be attributed to an apparent force the Coriolis force Fc m f v 13 where m is the mass of the particle f 2𝜔 sin 𝜑 is the Coriolis parameter 𝜔 07292 104 s1 is the angular velocity vector of the Earth 𝜑 is the angle of latitude and v is the velocity vector of the particle referenced to a coordinate system fixed with respect to the Earth The force Fc is normal to the direction of the particles motion and is directed according to the vector multiplication rule 3 The friction force The surface of the Earth exerts upon the moving air a horizontal drag force that retards the flow This force decreases with height and becomes neg ligible above a height 𝛿 known as gradient height The atmospheric layer between the Earths surface and the gradient height is called the atmospheric boundary layer see Chapter 2 The wind velocity speed at height 𝛿 is called the gradient velocity1 and the atmosphere above this height is called the free atmosphere Figure 16 In the free atmosphere an elementary mass of air will initially move in the direction of the pressure gradient force the driving force for the air motion in a direction normal to the isobar The Coriolis force will be normal to that incipient motion that is it will be tangent to the isobar The resultant of these two forces and the consequent motion of the particle will no longer be normal to the isobar so the Coriolis force which is perpendicular to the particle motion will change direction and will therefore no longer be directed along the isobar The change in the direction of motion will continue until the particle will move steadily along the isobar at which point the Coriolis force will be in equilibrium with the pressure gradient force as shown in Figure 17 Within the atmospheric boundary layer the direction of the friction force denoted by S coincides with the direction of motion of the particle During the particles steady motion the resultant of the mutually orthogonal Coriolis and friction forces will bal ance the pressure gradient force that is will be normal to the isobars meaning that the friction force and therefore the motion of the particle will cross the isobars Figure 18 Since the friction force which retards the wind flow and vanishes at the gra dient height decreases as the height above the surface increases the velocity increases 1 For straight winds ie winds whose isobars are approximately straight the term geostrophic is substituted in the meteorological literature for gradient k k k k 12 Atmospheric Hydrodynamics 9 Fca Fcb P pressure gradient force High pressure Low pressure Initial direction Direction III Direction II P P Fc Direction of steady wind a b c Figure 17 Frictionless wind balance in geostrophic flow P pressure gradient force Low pressure High pressure A B P Direction of particle motion S friction force Fc Coriolis force S Fc a b Figure 18 Balance of forces in the atmospheric boundary layer Figure 19 Wind velocity spiral in the atmospheric boundary layer α0 Vgr with height Figure 16 The Coriolis force which is proportional to the velocity also increases with height The combined effect of the Coriolis and friction forces causes the angle between the isobars and the direction of motion within the ABL shown as 𝛼0 in Figures 18 and 19 to increase from zero at the gradient height to its largest value at the Earths surface The wind velocity in the boundary layer can therefore be represented by a spiral as in Figure 19 Under certain simplifying assumption regarding the effective flow viscosity the spiral is called the Ekman spiral see Section 231 k k k k 10 1 Atmospheric Circulations If the isobars are curved the horizontal pressure gradient force as well as the centrifugal force associated with the motion on a curved path will act on the elemen tary mass of air in the direction normal to the isobars and the resultant steady wind will again flow along the isobars Its velocity results from the relations Vgrf V 2 gr r dpdn 𝜌 14 where r is the radius of curvature of the air trajectory If the mass of air is in the North ern Hemisphere the positive or the negative sign is used according as the circulation is cyclonic around a center of low pressure or anticyclonic around a center of high pressure 13 Windstorms 131 LargeScale Storms Largescale wind flow fields of interest in structural engineering may be divided into two main types of storm extratropical synoptic storms and tropical cyclones Synop tic storms occur at and above midlatitudes Because their vortex structure is less well defined than in tropical storms their winds are loosely called straight winds Tropical cyclones known as typhoons in the Far East and cyclones in Australia and the Indian Ocean generally originate between 5 and 20 latitudes Hurricanes are defined as tropical cyclones with sustained surface wind speeds of 74 mph or larger Tropical cyclones are translating vortices with diameters of hundreds of miles and counterclockwise clockwise rotation in the Northern Southern hemisphere Their translation speeds vary from about 330 mph As in a stirred coffee cup the column of fluid is lower at the center than at the edges The difference between edge and center atmospheric pressures is called pressure defect Rotational speeds increase as the pressure defect increases and as the radius of maximum wind speeds which varies from 5 to 60 miles decreases The structure and flow pattern of a typical tropical cyclone is shown in Figure 110 The eye of the storm Region I is a roughly circular relatively dry core of calm or light winds surrounded by the eye wall Region II contains the storms most powerful winds Far enough from the eye winds in Region V which decrease in intensity as the distance from the center increases are parallel to the surface Where Regions V and II intersect the wind speed has a strong updraft component that alters the mean wind speed pro file and is currently not accounted for in structural engineering practice The source of energy that drives the storm winds is the warm water at the ocean surface As the storm makes landfall and continues its path over land its energy is depleted and its wind speeds gradually decrease Figure 111 shows a satellite image of Hurricane Irma In the United States hurricanes are classified in accordance with the SaffirSimpson scale Table 112 132 Local Storms Foehn winds called chinook winds in the Rocky Mountains area develop downwind of mountain ridges Cooling of air as it is pushed upwards on the windward side of a 2 See Commentary ASCE 716 Standard 2 k k k k 13 Windstorms 11 25 0 1 10 6 200 500 III II I Eye wall h km V IV R km Figure 110 Structure of a hurricane Figure 111 Satellite view of hurricane Irma Source National Oceanic and Atmospheric Administration photo mountain ridge causes condensation and precipitation The dry air flowing past the crest warms as it is forced to descend and is highly turbulent Figure 112 A similar type of wind is the bora which occurs downwind of a plateau separated by a steep slope from a warm plain Jet effect winds are produced by features such as gorges k k k k 12 1 Atmospheric Circulations Table 11 SaffirSimpson scale and corresponding wind speedsa Category Damage potential 1min speed at 10 m over open water mph 3s gust speed at 10 m over open terrain exposure mph N Atlantic examples 1 Minimal 7495 81105 Agnes 1972 2 Moderate 96110 106121 Cleo 1974 3 Extensive 111129 122142 Betsy 1965 4 Extreme 130156 143172 David 1979 5 Catastrophic 157 173 Andrew 1992 a For the definition of 1minute and 3second wind speeds see Section 21 Official speeds are in mph 20 C 15 C Rain 3000 m 5 C Snow Figure 112 Foehn wind Thunderstorms occur as heavy rain drops due to condensation of water vapor con tained in ascending warm moist air drag down the air through which they fall causing a downdraft that spreads on the earths surface Figure 113 The edge of the spread ing cool air is the gust front If the wind behind the gust front is strong it is called a downburst Notable features of downbursts are the typical difference between the profiles of their peak gusts near the ground and those of largescale storms and the dif ferences among the time histories of various thunderstorms 3 Figure 114 According to 5 the maximum winds ie design level winds rarely occur at the locations where profiles differ markedly from the logarithmic law Microbursts were defined by Fujita 4 as slowrotating smalldiameter columns of descending air which upon reaching the ground burst out violently Figure 115 A number of fatal aircraft accidents have been caused by microbursts According to 5 because of the higher frequency and large individual area of a microburst probabili ties of structural damage by microbursts with 50100 mph wind speeds could be much higher than those of tornadoes k k k k 13 Windstorms 13 Direction of movement 0 1 5 10 km 2 km Figure 113 Section through a thunderstorm in the mature stage Tornadoes are small vortexlike storms and can contain winds in excess of 100 m s1 Figure 116 6 7 For unvented or partially unvented structures the difference between atmospheric pressure at the tornado periphery and the tornado center ie the pressure defect typical of cyclostrophic storms is a significant design factor For such structures the dif ference between the larger atmospheric pressure that persists inside the structure and the lower atmospheric pressure acting on the structure during the tornado passage results in large potentially destructive net pressures that must be accounted for in design see Chapter 27 The National Weather Service and the US Nuclear Regulatory Commission are currently classifying tornado intensities in accordance with the Enhanced Fujita Scale EFscale agreed upon in a forum organized by Texas Tech University in 2001 The EFscale shown in Table 12 replaced the original Fujita scale following a consensus opinion that the latter overestimated tornado wind speeds see eg 8 The EF scale is based on the highest 3second wind speed estimated to have occurred during the tornados life and is shown in Table 12 As noted in 9 no tornado has been assigned an intensity of EF6 or greater and there is some question whether an EF6 or greater tornado would be identified if it did occur For tornadoes that occur in areas containing no objects capable of resisting events with intensity EF0 eg in a corn field no intensity estimate is possible An additional diffi culty is that intensity estimates depend upon quality of construction Since there are no measurements of tornado speeds at heights above ground comparable to typical build ing heights it is necessary to rely on largely subjective estimates based primarily on observations of damage For additional material on tornadoes see Sections 34 and 53 and Chapters 27 and 28 k k k k 0 0 200 400 600 800 Time s Wind Speed ms 10 20 30 40 0 0 500 1000 1500 Time s Wind Speed ms 10 20 30 40 0 0 500 1000 1500 Time s Wind Speed ms 10 20 30 40 0 0 500 1000 1500 Time s a b c d e f g h Wind Speed ms 10 20 30 40 0 0 500 1000 1500 Time s Wind Speed ms 10 20 30 40 0 500 1000 1500 Time s Wind Speed ms 10 20 30 40 0 500 1000 1500 Time s Wind Speed ms 10 0 20 30 40 0 500 1000 1500 Time s Wind Speed ms 10 0 20 30 40 Figure 114 Time histories of eight thunderstorm events Source Reprinted from Ref 3 with permission from Elsevier k k k k 13 Windstorms 15 10 21 6 15 16 22 29 26 12267611254 17 13 15 20 15 17 15 18 14 13 13 13 12 37 62 23 8 20 19 18 17 16 15 14 13 12 11 10 09 08 07 06 05 04 03 02 01 00 59 58 KTS 20 30 40 50 60 70 80 90 100 110 120 130 FRONTSIDE PEAK WIND 130 KTS 112 KTS BACKSIDE PEAK WIND 84 KTS 62 KTS EYE OF MICROBURST AIR FORCE ONE LANDED EDT 1400 Figure 115 Andrews Air Force Base microburst on 1 August 1 1983 Its 1497 mph peak speed was the highest recorded in a microburst in the US 4 Figure 116 Tornado funnel Source National Oceanic and Atmospheric Administration photo k k k k 16 1 Atmospheric Circulations Table 12 Tornado enhanced Fujita Scale Intensity Description Enhanced Fujita Scale 3s peak gust speed mph EF0 Light damage 6585 EF1 Moderate damage 86110 EF2 Considerable damage 111135 EF3 Severe damage 136165 EF4 Devastating damage 166200 EF5 Incredible damage 200 References 1 Humphreys WJ 1940 Physics of the Air New York McGrawHill 2 ASCE Minimum design loads for buildings and other structures ASCESEI 716 in ASCE Standard ASCESEI 716 Reston VA American Society of Civil Engineers 2016 3 Lombardo FT Smith DA Schroeder JL and Mehta KC 2014 Journal of Wind Engineering and Industrial Aerodynamics 125 121132 httpdxdoiorg101016j jweia201312004 4 Fujita TT 1990 Downbursts meteorological features and wind field characteristics Journal of Wind Engineering and Industrial Aerodynamics 36 7586 5 Schroeder J L Personal communication Nov 21 2016 6 Lewellen DC Lewellen WS and Xia J 2000 The influence of a local swirl ratio on tornado intensification near the surface Journal of the Atmospheric Sciences 57 527544 7 Hashemi Tari P Gurka R and Hangan H 2010 Experimental investigation of tornadolike vortex dynamics with swirl ratio the mean and turbulent flow fields Journal of Wind Engineering and Industrial Aerodynamics 98 936944 8 Phan L T and Simiu E Tornado aftermath Questioning the tools Civil Engineering December 1998 08857024980012002A httpswwwnistgovwind 9 Ramsdell J V Jr and Rishel J P Tornado Climatology of the Contiguous United States A J Buslik Project Manager NUREGCR4461 Rev 2 PNNL15112 Rev 1 Pacific Northwest National Laboratory 2007 k k k k 17 2 The Atmospheric Boundary Layer As indicated in Chapter 1 the Earths surface exerts on the moving air a horizontal drag force whose effect is to retard the flow This effect is diffused by turbulent mix ing throughout a region called the atmospheric boundary layer ABL In strong winds the depth of the ABL ranges from a few hundred meters to a few kilometers depending upon wind speed roughness of terrain angle of latitude and the degree to which the stratification of the free flow ie the flow above the ABL is stable Within the ABL the mean wind speed varies as a function of elevation This chapter is devoted to studying aspects of ABL flow of interest from a structural engineering viewpoint Section 21 is concerned with the dependence of the wind speed on averaging time Section 22 presents the equations of mean motion in the ABL Sections 23 and 24 pertain to horizontally homogeneous flows over flat uniform surfaces and contain respectively theoretical as well as empirical results on the dependence of wind speeds on height above the Earths surface and the structure of atmospheric turbulence Section 25 concerns horizontally nonhomogeneous flows ie flows affected by changes of surface roughness or by topographic features and flows in tropical storms and thunderstorms Since the structural engineer is concerned primarily with the effect of strong winds it will be assumed that the ABL flow is neutrally stratified Indeed in strong winds turbulent transport dominates the heat convection by far so that thorough turbulent mixing tends to produce neutral stratification just as in a shallow layer of incompressible fluid mixing tends to produce an isothermal state In flows of interest in structural engineering a layer of stably stratified flow called the capping inversion is present above the ABL and significantly affects the ABLs height 21 Wind Speeds and Averaging Times If the flow were laminar wind speeds would be the same for all averaging times However owing to turbulent fluctuations such as those recorded in Figure 21 the definition of wind speeds depends on averaging time The peak 3second gust speed is the peak of a storms speeds averaged over 3 seconds In 1995 it was adopted in the ASCE Standard as a measure of wind speeds Similarly the peak 5second gust speed is the largest speed averaged over 5 seconds The 5second speed is reported by the National Weather Service ASOS Automated Service Observing Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 18 2 The Atmospheric Boundary Layer 6 AM 5 AM Figure 21 Wind speed record System and is about 2 less than the 3second speed The 28mph peak of Figure 21 is approximately a 3second speed The hourly wind speed is the speed averaged over 1 hour It is commonly used as a reference wind speed in wind tunnel simulations Hence the need to estimate the hourly speed corresponding to a 3second or a 1minute or a 10minute speed specified for design purposes or recorded at weather stations In Figure 21 the statistical features of the record do not vary significantly ie the record may be viewed as statistically stationary see Appendix B over an interval of almost two hours the hourly wind speed is about 185 mph or about 1152 times the peak 3second gust Sustained wind speeds defined as wind speeds averaged over intervals in the order of 1 min are used in both engineering and meteorological practice The fastest 1minute wind speed or for short the 1minute speed is the storms largest 1minute average wind speed The fastestmile wind speed Uf is the storms largest speed in mph averaged over a time interval tf 3600Uf For example a 60 mph fastestmile wind speed is averaged over a 60second time interval Tenminute wind speeds are wind speeds averaged over 10 min and are used in World Meteorological Organization WMO practice as well as in some standards and codes The ratio between the peak gust speed and the mean wind speed is called the gust factor Expressions for the relation between wind speeds with different averaging times are provided in Section 237 as functions of parameters defined subsequently in this chapter k k k k 22 Equations of Mean Motion in the ABL 19 22 Equations of Mean Motion in the ABL The motion of the atmosphere is governed by the fundamental equations of contin uum mechanics which include the equation of continuity a consequence of the principle of mass conservation and the equations of balance of momenta that is the NavierStokes equations see also Chapters 4 and 6 These equations must be supplemented by phenomenological relations that is empirical relations that describe the specific response to external effects of the medium being considered For example in the case of a linearly elastic material the phenomenological relations consist of the socalled Hookes law If the equations of continuity and the equations of balance of momenta are averaged with respect to time and if terms that can be shown to be negligible are dropped the fol lowing equations describing the mean motion in the boundary layer of the atmosphere are obtained U U x V U y W U z 1 𝜌 p x f V 1 𝜌 𝜏u z 0 21 U V x V V y W V z 1 𝜌 p y f U 1 𝜌 𝜏v z 0 22 1 𝜌 p z g 0 23 U x V y W z 0 24 where U V and W are the mean velocity components along the axes x y and z of a Cartesian system of coordinates whose zaxis is vertical p 𝜌 f and g are the mean pressure the air density the Coriolis parameter and the acceleration of gravity respec tively and 𝜏u 𝜏v are shear stresses in the x and y directions respectively The xaxis is selected for convenience to coincide with the direction of the shear stress at the surface denoted by 𝜏0 Figure 22 It can be seen by differentiating Eq 23 with respect to x or y that the vertical vari ation of the horizontal pressure gradient depends upon the horizontal density gradient For the purposes of this text it will be sufficient to consider only flows in which the hor izontal density gradient is negligible The horizontal pressure gradient is then invariant Figure 22 Coordinate axes Isobar αo y x k k k k 20 2 The Atmospheric Boundary Layer with height and thus has throughout the boundary layer the same magnitude as at the boundary layers top p n 𝜌 fV gr V 2 gr r 25 where V gr is the gradient velocity r is the radius of curvature of the isobars and n is the direction of the gradient wind see Eq 14 The geostrophic approximation corresponds to the case where the curvature of the isobars can be neglected The gradient velocity is then called the geostrophic velocity and is denoted by G Eq 25 then becomes 1 𝜌 p x fV g 1 𝜌 p y fUg 26ab where Ug and V g are the components of the geostrophic velocity G along the x and yaxes The boundary conditions for Eqs 2124 may be stated as follows at the ground surface the velocity vanishes while at the top of the ABL the shear stresses vanish and the wind flows with the gradient velocity V gr In addition an interaction between the ABL and the capping inversion occurs see Section 233 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces It may be assumed that in largescale nontropical storms within a flat site of uniform surface roughness with sufficiently long fetch a region exists over which the flow is hor izontally homogeneous The existence of horizontally homogeneous atmospheric flows is supported by observations and distinguishes ABLs from twodimensional boundary layers such as occur along flat plates In the latter case the flow in the boundary layer is decelerated by the horizontal stresses so that the boundarylayer thickness grows as shown in Figure 23 1 In atmospheric boundary layers In atmospheric boundary lay ers however the horizontal pressure gradient which below the free atmosphere is only Figure 23 Growth of a twodimensional boundary layer along a flat plate k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 21 partly balanced by the Coriolis force Figure 18 reenergizes the flow and counteracts the boundarylayer growth Horizontal homogeneity of the flow is thus maintained 2 Under equilibrium conditions in horizontally homogeneous flow Eqs 21 and 22 in which Eq 26ab are used become Vg V 1 𝜌f 𝜏u z Ug U 1 𝜌f 𝜏v z 27ab The Ekman spiral was the first attempt to describe the ABL in mathematical terms and is presented in Section 231 for the sake of its historical interest In the 1960s and 1970s a major advance was achieved in the field of boundarylayer meteorology based on an asymptotic approach As shown in Section 232 the asymptotic approach yields the unphysical result that the mean speed component V vanishes throughout the boundary layers depth except at its top where it has the value V g In addition the 1960s and 1970s work did not consider the important effect of the capping inversion on the ABL height Section 233 introduces the contemporary classification of neutrally stratified ABLs as functions of the BruntVäisäla frequency The latter characterizes the interaction between the ABL and the capping inversion and provides expressions for the height of the ABL that account for that interaction Section 234 presents the logarithmic descrip tion of the mean wind speed within the lower layer of the ABL called the surface layer as well as estimates of the surface layers depth Section 235 presents the power law rep resentation of the wind speed profile which though obsolete is still being used in some codes and standards including the ASCE 716 Standard 3 Section 236 discusses the relation between characteristics of the ABL flows in different surface roughness regimes Section 237 provides details on the relation between wind speeds with different aver aging times 231 The Ekman Spiral The Ekman spiral model is obtained if it is assumed in Eq 27ab that the shear stresses are proportional to a fictitious constant K called eddy viscosity such that 𝜏u 𝜌K U z 𝜏v 𝜌K V z 28ab Equations 27 and 28 then become a system of differential equations with constant coefficients With the boundary conditions U V 0 for height above the surface z 0 and U Ug V V g for z the solution of the system is U 1 2 G1 eazcos az sin az V 1 2 G1 eazcos az sin az 29ab where a f 2K12 Equations 29ab which describe the Ekman spiral are repre sented schematically in Figure 19 Observations are in sharp disagreement with these k k k k 22 2 The Atmospheric Boundary Layer equations For example while according to Eq 29ab the angle 𝛼0 between the surface stress 𝜏0 and the geostrophic wind direction is 45 observations indicate that this angle may range approximately between approximately 5 and 30 see Section 233 The cause of the discrepancies is the assumption mathematically convenient but physically incorrect that the eddy viscosity is independent of height 232 Neutrally Stratified ABL Asymptotic Approach A vast literature is available on the numerical solution of the equations of motion of the fluid A different type of approach based on similarity and asymptotic considerations was developed in 2 The starting point of the asymptotic approach is the division of neutral boundary layers into two regions a surface layer and an outer layer In the surface layer the shear stress 𝜏0 induced by the boundarylayer flow at the Earths surface must depend upon the flow velocity at a distance z from the surface the roughness length z0 that characterizes the surface roughness and the density 𝜌 of the air that is 𝜏0i F Ui Vj z z0 𝜌 210 where U and V are the components of the mean wind speed along the x and y directions and i j are unit vectors Eq 210 can be written in the nondimensional form Ui Vj u 𝜓1x z z0 i 𝜓1y z z0 j 211 where u 𝜏0 𝜌 12 212 is the friction velocity and 𝚿1 𝜓1xi 𝜓1yj is a vector function to be determined Eq 211 known as the law of the wall is applicable in the surface layer and can be written in the form Ui Vj u 𝜓1x z H H z0 i 𝜓1y z H H z0 j 213 where H cuf 214 H denotes the boundarylayer depth ie the height to which the effect of the surface shear stress has diffused in the flow f is the Coriolis parameter and on the basis of data available in the 1960s it was assumed in 2 c 025 As indicated earlier the mean velocity components UH and VH are denoted by Ug and V g respectively and their resultant denoted by G is the geostrophic velocity In the outer layer it can be asserted that at height z the velocity reduction with respect to G must depend upon the surface shear stress 𝜏0 and the air density 𝜌 The nondimensional expression for this dependence is the velocity defect law Ui Vj u Ugi Vgj u 𝜓2x z H i 𝜓2y z z0 j 215 where 𝚿2 is a vector function to be determined k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 23 Consider in Eqs 213 and 215 the x components Ui u 𝜓1x z H H z0 i 216 Ui u Ugi u 𝜓2x z H i 217 From the observation that a multiplying factor inside the function 𝜓1x must be equivalent to an additive function outside the function 𝜓2x the following equations are obtained U u 1 k ln z H ln H z0 218 U u Ug u 1 k ln z H 219 for the surface and the outer layer respectively In Eqs 218 and 219 k 040 is the von Kármán constant and the height z is measured from the elevation z0 above the surface From Eq 218 it follows immediately U u 1 k ln z z0 220 By equating Eqs 218 and 219 in the overlap region there results Ug u 1 k ln H z0 221 The logarithmic law is seen to apply to the U component of the wind velocity through out the depth of the boundary layer Consider now the components Vj u 𝜓1y z H H z0 j 222 Vj u Vgj u 𝜓2y z H j 223 It was assumed in 2 46 that 𝜓1y 0 Then Eqs 222 and 223 yield in the overlap region Vgj u 𝜓2y z H j 0 224 that is 𝜓2y z H Vg u 𝜓2y z H B k 225ab where based on measurements available in the 1960s it was assumed Bk 48 eg 6 It follows from Eqs 223 and 225ab that Vz 0 z H 226 k k k k 24 2 The Atmospheric Boundary Layer Since for z H VH Vg Eq 223 yields Ψ2yHH 0 227 and by virtue of Eq 226 Vz Vg𝛿H 228 where 𝛿 denotes the Dirac delta function This physically unrealistic result is an artifact of the asymptotic approach which transforms the actual profile Vz into the nonphysical profile represented by Eq 228 233 BruntVäisäla Frequency Types of Neutrally Stratified ABLs BruntVäisäla Frequency In much of the theoretical work on ABL flow performed until the 1990s or so ABL flows for which the buoyancy flux at the surface denoted by 𝜇 is 𝜇 0 and 𝜇 0 were defined as neutral and stable respectively This classification did not consider the interaction between the ABL and the free flow ie the flow above the ABL that when stably stratified can have a significant effect on the height of the ABL 79 The interaction between the ABL and the stably stratified free flow above the ABL is characterized by the nondimensional parameter 𝜇N Nf where N is the BruntVäisäla frequency Consider an air particle with density 𝜌z at elevation z in a stably stratified flow If the particle is displaced by a small amount z it will be subjected to an incremental pressure g𝜌z z 𝜌z The motion of the particle will be governed by the equation 𝜌z2z t2 g𝜌z z 𝜌z 229 2z t2 g 𝜌z 𝜌z z z 230 Let g 𝜌z 𝜌z z N2 231 It follows from Eqs 230 and 231 that for positive values of 𝜌zz ie for a sta ble stratification of the free flow z is a harmonic function with frequency N which drives the interaction between the stably stratified free flow and the ABL See also 10 p 136 Truly Neutral and Conventionally Neutral ABL Flows Based on the dependence of the ABL flow upon both 𝜇 and the nondimensional parameter 𝜇N Nf neutrally stratified ABL flows are classified into two categories 79 1 Truly neutral flows 𝜇 0 N 0 observed during comparatively short transition periods after sunset on a background of residual layers of convective origin often treated as irrelevant because of their transitional nature and usually excluded from data analysis 2 Conventionally neutral flows 𝜇 0 N 0 ie neutrally stratified and interacting with the stably stratified layer above the ABL are characterized by negligible buoy ancy and a number 𝜇N 0 typically 50 𝜇N 300 Recall that in strong winds the buoyancy in the ABL may be assumed to be negligible owing to strong mechanical as opposed to thermal turbulent mixing k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 25 Of these two categories it is the conventionally neutral flows that are of interest in structural engineering applications Models of the ABL flow used in structural engineering applications have been based on the assumption that the flow stratification is truly neutral The failure of the asymp totic similarity approach to consider the effect of the capping inversion results in the incorrect prediction of the ABL height as is shown subsequently Integral Measures of the Conventionally Neutral ABL The integral measures of the ABL are the geostrophic drag coefficient the crossisobaric angle and the ABL height For 𝜇N values typical of conventionally neutral flows ie 50 𝜇N 300 the depen dence of the geostrophic drag coefficient Cg u G 232 and of the crossisobaric angle 𝛼0 upon the Rossby number Ro G f z0 233 can be represented by the following expressions based on measurements by Lettau 11 Cg 0205 log10Ro 0556 234 𝛼0 17358 log10Ro 303 235 12 13 p 338 Also for conventionally neutral ABLs 1 H2 f 2 C2 R Nf C2 CN 1 u2 236 where CR 06 and CCN 136 79 Therefore the ABL height is H Ch𝜇Nu f 237 where Ch𝜇N 1C2 R 𝜇NC2 CN12 Note the difference with the expression for H in Eq 214 For any given friction velocity u Coriolis parameter f and surface roughness length z0 the quantities G 𝛼0 and H are obtained by using Eqs 232236 Example 21 ABL integral measures Mean wind speed and veering angle profiles Consider the following parameters f 104 s1 N 0018 s1 so 𝜇N 180 and z0 03 m suburban terrain exposure u 15 m s1 It can be verified by using Eq 236 that Ch 010 so H 010 15104 1500 m According to Eq 214 H 3750 m The trial value G 41 m s1 yields log10Ro 614 uG 0037 to which there corresponds G 41 m s1 and 𝛼0 25 For z 300 m zH 020 for z 800 m zH 053 Figures 24 and 25 show the dependence on height z of the speeds Uz and Vz their resultant and the angle 𝛼0z as obtained in 14 by Computational Fluid Dynamics techniques Note that the component V800 m and a fortiori the component V300 m have negligible contributions to the resultant mean wind speed and that the veering angles 𝛼0300 m and 𝛼0800 m are approximately 2 and 6 respectively Results for Ch 019 based on 15 figure 7 are also included in Figures 24 and 25 k k k k 26 2 The Atmospheric Boundary Layer Uu Vu U2V212u 15 10 5 0 5 10 15 20 25 30 35 zH 0 01 02 03 04 05 06 07 08 09 1 Uu Ch 010 Vu Ch 010 U2V212u Ch 010 Uu Ch 019 Vu Ch 019 U2V212u Ch 019 Figure 24 Dependence of Uu Vu and U2 V2 on zH α0 25 20 15 10 5 0 zH 0 01 02 03 04 05 06 07 08 09 1 Ch 010 Ch 019 Figure 25 Dependence of veering angle on zH k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 27 No mathematical expression that uses the parameters z0 and u is available for the description of the wind profile throughout the depth of the ABL However Section 234 presents the relation between the friction velocity u and the mean wind speed Uz in the lower portion of the ABL and information on surface roughness lengths z0 for various types of surface 234 The Logarithmic Mean Wind Profile The Logarithmic Law Within the lower layer of the ABL whose height is denoted by zs the component Vz of the mean wind velocity is at least one order of magnitude smaller than the component Uz and is therefore negligible in practice see Figure 24 The logarithmic law Eq 220 renumbered here as Eq 238 Uz u 1 k ln z z0 238 is valid for all heights z above the Earths surface within the region z0 z zs By virtue of Eq 238 u Uz 25 lnzz0 239 where z zs According to a belief predating modern ABL research but still persisting among some wind engineers 16 zs 100 m Also according to the ASCE 716 Standard 3 the ABL depth is independent of wind speed In fact the depth H of the ABL is proportional to u see Eq 237 The relation zs 002u f 240 where f is the Coriolis parameter see Section 12 2 46 is a lower bound for the height zs Eq 240 follows from the assumption that in the region z0 z zs the shear stress 𝜏u differs little from the surface stress 𝜏0 and the component V of the velocity is small Integration of Eq 27ab over the height zs yields 𝜏u 𝜏0 𝜌f zs z0 Vg Vdz 𝜏0 𝜌fV gzs 241 or 𝜌fV gzs 𝜂𝜏0 242 where 𝜂 is a small number Since 𝜏0 𝜌u2 Eq 212 and V gu Bk 48 Eq 225ab zs 𝜂u2 fV g 𝜂k f Bu bu f 243 According to 6 the logarithmic law holds for practical purposes even beyond heights at which 𝜂 is in the order of 30 meaning that b 002 Equations 239 and 240 show that the height zs over which the logarithmic law is valid is approximately proportional to the wind speed Uz z0 z zs k k k k 28 2 The Atmospheric Boundary Layer Example 22 Estimation of friction velocity u Assume z 10 m Uz 30 m s1 and z0 003 m open exposure Eq 239 yields u 207 m s1 Example 23 Estimation of surface layer depth zs Assume u 207 m s1 and f 104 s1 According to Eq 240 zs 414 m Surface Roughness Lengths z0 and Surface Drag Coefficients Tables 2123 list surface roughness lengths z0 based respectively on measurements included in the Commentary to the ASCE 716 Standard 3 and specified in the Eurocode 21 Table 21 Values of surface roughness length z0 and surface drag coefficients 𝜅 for various types of terrain Type of Surface z0 cm 103 𝜿 Sanda 00101 12 Snow surface 0106 23 Mown grass 001 m 011 23 Low grass steppe 14 35 Fallow field 23 45 High grass 410 58 Palmetto 1030 813 Pine forest mean height of trees 15 m one tree per 10 m2 zd 12 mb 90100 2830 Sparsely builtup suburbsc 2040 1115 Densely builtup suburbs townsc 80120 2536 Centers of large citiesc 200300 62110 a 17 b 18 c Values of z0 to be used in conjunction with the assumption zd 0 19 Table 22 Surface roughness lengths z0 as listed in ASCE 716 Commentary 3 Type of surface z0 ft m Watera 00160033 0005001 Open terrainb 003305 001015 Urban and suburban terrain wooded areasc 0523 01507 a The larger values apply over shallow waters eg near shore lines Approximate typical value corresponding to ASCE 716 Exposure D 0016 ft 0005 m ASCE Commentary According to 20 for strong hurricanes z0 00010003 m b Approximate typical value corresponding to ASCE 716 Exposure C 0066 ft 002 m ASCE Commentary c Value corresponding approximately to ASCE 716 Exposure B 05 ft 015 m this value is smaller than the typical value for ASCE 716 Exposure B 1 ft 03 m ASCE Commentary k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 29 Table 23 Roughness lengths z0 as specified in Eurocode 21 Type of surface z0 m Sea or coastal areas exposed to the open sea 0003 Lakes or flat and horizontal area with negligible vegetation and no obstacles 001 Areas with low vegetation and isolated obstacles like trees or buildings with separations of maximum 20 obstacle heights eg villages suburban terrain permanent forest 005 Areas with regular cover of vegetation or buildings or with isolated obstacles with separations of maximum 20 obstacle heights villages suburban terrain forests 030 Areas in which at least 15 of the surface is covered with buildings whose average height exceeds 15 m 10 The surface drag coefficient is defined as 𝜅 k ln10z0 244 where k 04 is the von Kármán constant and z0 is expressed in meters Values of 𝜅 corresponding to various values of z0 are given in Table 21 The surface drag coefficient 𝜅 for wind flow over water surfaces depends upon wind speed On the basis of a large number of measurements the following empirical relations were proposed for the range 4 U10 20 m s1 22 𝜅 51 104U10046 𝜅 10475 067U10 245ab where U10 is the mean wind speed in m s1 at 10 m above the mean water level 23 According to 24 for wind speeds U10 40 m s1 𝜅 00015 1 exp U10 125 156 1 000104 246 For additional information on the wind flow over the ocean see 20 2527 The following relation proposed by Lettau 28 may be used to estimate z0 for builtup terrain z0 05Hob Sob Aob 247 where Hob is the average height of the roughness elements in the upwind terrain Sob is the average vertical frontal area presented by the obstacle to the wind and Aob is the average area of ground occupied by each obstruction including the open area surround ing it Example 24 Application of the Lettau formula Check the Eurocode value z0 1 m indicated in Table 23 against Eq 247 assuming the average building height is Hob 15 m the average dimensions in plan of the buildings are 16 16 m and Aob 1600 m2 We have Sob 15 16 240 m2 so the average area occupied by buildings is 16 161600 16 Eq 247 yields z0 1125 m k k k k 30 2 The Atmospheric Boundary Layer The surface roughness length z0 is a conceptual rather than a physical entity and cannot therefore be measured directly It can in principle be determined by measuring the mean wind speeds Uz1 and Uz2 at the elevations z1 and z2 respectively However small errors in the measurement of the speeds can lead to large errors in the estimation of the roughness length Example 25 Errors in roughness length estimates based on mean wind speed measure ments Assume measurements of mean wind speeds Uz1 and Uz2 are available at ele vations z1 and z2 above ground Eq 238 yields Uz2Uz1 r21 lnz2z0lnz1z0 After some algebra it follows that z0 exp r21 ln z1 ln z2 r21 1 248 Let z1 10 z2 25 and z0 0026 m Eq 238 yields Uz2Uz1 1154 It follows then from Eq 248 that indeed z0 0026 m However if measurement errors resulted in a 5 error in r21 that is if in Eq 246 the ratio r21 105 1154 is used the result obtained is z0 013 m rather than 0026 m For a more effective approach to estimating roughness length based on measurements of turbulence intensity see 29 and Example 214 Zeroplane Displacement On account of the finite height of the roughness elements the following empirical modification of Eq 238 is required The quantity z rather than denoting height above ground is defined as z zgr zd 249 where zgr is the height above ground and zd is a length known as the zeroplane displace ment The quantity z is called the effective height It is suggested in 30 that reasonable values of the zero plane displacement in cities may be obtained using the formula zd h z0 k 250 where h is the general rooftop level 235 Power Law Description of ABL Wind Speed Profiles The logarithmic law has long superseded the power law in meteorological practice Unlike the logarithmic law the power law is strictly empirical It was first proposed about a century ago for open terrain in 31 and for builtup terrain in 32 It is still used in the United States 3 Canada 33 and Japan 34 primarily owing to the earlier belief that the logarithmic law is only valid up to 50100 m even in strong winds The variation of wind speed with height can be expressed approximately as Uz Uzref z zref 1𝛼 251 where zref is a reference height for example 10 m above ground in open terrain In Eq 251 the exponent 1𝛼 depends upon surface roughness and upon averaging time the profiles being flatter as the averaging time decreases The power law applied to k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 31 3second peak gust wind profiles has the same form as Eq 251 however in the ASCE 7 Standard its exponent is denoted by â rather than by 𝛼 Fivesecond peak gusts may in practice be assumed to differ negligibly from 3second gusts Eq 251 is assumed in the ASCE 7 Standard and the National Building Code NBC of Canada to be valid up to a height zg purported to represent the geostrophic height and referred to therein as the gradient speed Table 24 lists power law exponents and gradient heights zg speci fied in the ASCE Standard and the NBC specified for four surface exposure categories A centers of large cities B suburban terrain C open terrain and D open water Category A was excluded from later versions of the ASCE 7 Standard on account of the poor agreement of the power law with actual wind speeds over centers of large cities It is shown in 62 that the values of zg assumed in the power law model can result in strongly unconservative estimates of wind effects on supertall buildings designed in accordance with ASCE 716 provisions Example 26 Application of the power law Let zref 328 ft 10 m U3szref 55 mph α 195 open terrain From Eq 251 at 100 ft above ground U3s100 ft 55 100328195 62 mph As noted by Panofsky and Dutton 35 p 131 the power law can be fitted reasonably well to the log law only over small height ranges 236 ABL Flows in Different Surface Roughness Regimes Wind speed maps are developed for structural engineering purposes for open terrain exposure Since most structures are not built in open terrain it is necessary to deter mine wind speeds corresponding to the speeds specified in wind maps for exposures other than open This is done by using the fact that in any given largescale storm the geostrophic speed is independent of surface friction and therefore of terrain roughness Eq 14 We first consider the case in which wind profiles are described by the loga rithmic law Next we consider the power law case Table 24 Power law exponents and gradient heights specified in the 19932016 versions of ASCE 7 Standard and in the National Building Code of Canada NBCC 33 Exposure Aa Bb Cc Dd ASCE 793e 1𝛼 zg ft m 13 1500 457 145 1200 366 17 900 274 110 700 213 NBCf 1𝛼 zg ft m 04 1700 520 028 1300 400 016 900 274 ASCE 7g 19952016 1𝛼 zg ft m 17 1200 366 195 900 274 1115 700 213 a Centers of large cities b Suburban terrain towns c Open terrain eg airports d Water surfaces e Sustained speeds f Mean hourly speeds g Peak 3second gust speeds k k k k 32 2 The Atmospheric Boundary Layer Wind speeds described by the logarithmic law Examples 27 and 28 consider respectively the cases of suburban and ocean versus open exposure Example 27 It can be verified that for f 104 s1 given a surface with open exposure z0 003 m to a storm that produces a friction velocity u 25 m s1 there corre sponds a geostrophic speed G 83 m s1 In accordance with the definition of Ro for suburban terrain exposure z01 03 m to G 83 m s1 there corresponds Ro1 log 83104 03 644 From Eq 234 Cg1 0035 so u1 83 0035 29 m s1 Eq 232 and the crossisobaric angle is 𝛼01 24 Eq 235 From Eqs 236 and 237 there follows for N 001 s1 Ch1 013 and H1 29 013104 3800 m ie about half the asymptotic estimate H1 7250 m Eq 214 Example 28 For ocean surfaces assuming G 83 m s1 and z01 0003 m log10Ro1 log 83104 0003 844 and Cg1 0026 so u1 83 0026 215 m s1 and 𝛼01 18 Eq 236 yields H1 2800 m vs the asymptotic estimate H1 5400 m and Ch1 013 Results close to those obtained by the relatively elaborate procedure used in Examples 27 and 28 can be obtained by Biétrys equation adopted with a minor modification in the Eurocode 21 u1 u z01 z0 00706 252 Example 29 Application of Eq 252 Let z0 003 m If z01 03 m u1u 118 versus 2925 116 as shown in Example 27 if z01 0003 m then u1u 086 versus 21525 086 as shown in Example 28 Wind speeds described by the power law For strong winds given the mean hourly speed Uzopen at the reference height zopen above open terrain with power law exponent 1𝛼open the mean hourly wind speed at height z above builtup terrain with power law exponent 1𝛼 is Uz Uzopen zgopen zopen 1𝛼open z zg 1𝛼 253 where the product of the first two terms in the righthand side is the gradient speed above open terrain Uzgopen Since gradient speeds are not affected by surface rough ness the gradient speed over builtup terrain Uzg is equal to U zgopen The last factor in Eq 253 transforms Uzg into Uz at height z above builtup terrain A relation similar to Eq 253 is also used with the appropriate values of the parameters zg and α from Table 24 for 3second peak gust speeds denoted here by U3s and in the ASCE 7 Standard by V and for sustained wind speeds such as fastestmile speeds or 1minute speeds In the ASCE 7 Standard U3szopen 10 m is the 3second basic wind speed and the product of the last two terms in Eq 253 is denoted in the Standard by Kz Example 210 Relation between wind speeds in different roughness regimes power law description Denote the 3second peak gust speed by U3s Let U3s328 ft 86 mph k k k k 23 Wind Speed Profiles in Horizontally Homogeneous Flow Over Flat Surfaces 33 above open terrain â 95 zg 274 m Table 24 Eq 253 yields U3s45 m 45 m s1 open terrain Using Table 24 and Eq 253 above suburban terrain â 70 and zg 366 m U3s10 m 33 and U3s45 m 40 m s1 237 Relation Between Wind Speeds with Different Averaging Times The mean ratio rt z0 z between the largest average tsecond speed during a storm with a 1hour duration and that storms mean hourly 3600 s speed is a function of the averaging time t the terrain roughness length z0 and the height above ground z Table 25 As noted in Section 21 the ratio U3sU is called the gust factor Terrain with open exposure For the particular case of open terrain exposure z0 003005 m and a height above ground z 10 m the approximate ratio r is listed for selected values of t as follows 36 These values are applicable to largescale nontropical storms over open terrain with open exposure and at the standard 10 m height above ground These values are appli cable only at the standard reference height over terrain with open exposure Example 211 Conversion of fastestmile wind speed to mean hourly speed and to peak 3second gust for open terrain For a fastestmile wind speed at 10 m over open terrain of 90 mph the averaging time is 360090 40 s and the corresponding hourly speed and peak 3second gust are 90129 698 and 698 152 106 mph respectively Example 212 Conversion of peak 3second gust speed to mean hourly speed for open terrain Let the peak 3second gust speed at 10 m above ground in open terrain be 30 m s1 For wind tunnel testing and structural purposes winds characterized by that gust speed are modeled by winds with a 30152 20 m s1 mean hourly speed at 10 m above ground in open terrain Terrain with Exposure Other than Open The following approximate relation may be used Utz Uz ctu2z z012 Utz Uz 1 𝛽z z0ct 25 lnzz0 254ab where Utz is the peak speed averaged over t s within a record of approximately one hour Uz is the mean wind speed for that record over terrain with surface roughness z0 𝛽z0 ct are given in Tables 26 and 27 Following 10 Eq 1825b 𝛽z z0 𝛽z0 exp 15 z H 255 where H is the ABL depth and z z0 and H are in meters Table 25 Ratios r between ts and mean hourly speeds at 10 m above open terrain t s 3 5 40 60 600 3600 152 149 129 125 11 10 k k k k 34 2 The Atmospheric Boundary Layer Table 26 Factor 𝛽 z0 z0 m 0005 003 030 100 𝛽z0 65 60 525 49 Table 27 Factor ct t s 1 10 20 30 50 100 200 300 600 1000 3600 ct 300 232 200 173 135 102 070 054 036 016 0 Note coefficient ct is an approximate empirical peak factor which increases as t decreases Example 213 Conversion of SaffirSimpson scale 1minute speeds at 10 meters over water to peak wind speeds at 10 m above open terrain Category 4 hurricane From Table 11 the 1minute speeds at 10 m above open water that define the weakest and strongest Category 4 hurricanes are 130 and 156 mph respectively The conversion depends on the assumed values of the surface roughness lengths z0 for open water and open terrain Relative large values of z0 are applicable to wind flow over water near shorelines where the water is shallow as opposed to flow over open water Assuming that for hurricane winds over open water z0 0003 m Eq 254ab yields with 𝛽z0 0003 m 65 and c 60 s 129 Tables 26 and 27 Uw 60s10 m Uw10 m 1 255 129 25 ln100003 Uw10 m 086Uw 60s10 m where the superscript w signifies over open water Assuming that over open terrain z0 004 m Eqs 238 and 252 yield Uw10 m U10 m 0003 004 00706 ln100003 ln10004 U10 m 0816Uw10 m where U10 m is the mean hourly wind speed over open terrain It follows that U10 m 086 0816Uw 60s10 m 07Uw 60s10 m Therefore the peak 3second gust over open terrain is Table 27 U3s10 m 152 07 Uw 60s10 m 106Uw 60s10 m To the speed Uw 60s10 m 155 mph there then corresponds a calculated peak 3second gust at 10 m over open terrain U3 s10 m 164 mph In the preceding calculations it was assumed that relations that apply to horizontally homogeneous wind flow ie flow in synoptic storms are also applicable to hurricanes in which the isobars are curved rather than straight and the flow is therefore horizontally inhomogeneous k k k k 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces 35 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces Except for winds with relatively low speeds under special temperature conditions the wind flow is not laminar smooth Rather it is turbulent it fluctuates in time and space that is at any one point in space the wind speed is a random function of time Figure 21 and at any one moment in time the wind speed is a random function of position in space Atmospheric flow turbulence characterization is of interest in structural engineer ing applications for the following reasons First turbulence affects the definition of the wind speed specified in engineering calculations as shown in Sections 21 and 237 Second by transporting particles from flow regions with high momentum into lowspeed regions turbulence can influence significantly the wind flow around a structure and therefore the aerodynamic pressures acting on the structure Chapters 4 and 5 Therefore to simulate correctly fullscale aerodynamic effects in the laboratory it is necessary to achieve laboratory flows that simulate the features of atmospheric turbulence Chapter 5 Third turbulence produces resonant dynamic effects in flexible structures that must be accounted for in structural design Chapter 11 Descriptors of the turbulence used in applications include the turbulence intensity Section 241 integral scales of turbulence Section 242 and the spectra and the crossspectra of the turbulent velocity fluctuations Sections 243 and 244 241 Turbulence Intensities The longitudinal turbulence intensity at a point with elevation z is defined as Iuz u2z z0 12 Uz 256a that is as the ratio of the rms of the longitudinal wind speed fluctuations uz t to the mean speed Uz uz t being parallel to Uz Since u2z z0 12 𝛽z z0 u 256b where approximate values of 𝛽z z0 are given by Eq 255 and Table 26 and by virtue of the log law Iuz 𝛽z z0 25 lnzz0 256c Example 214 Calculation of longitudinal turbulence intensity For z0 003 m z 20 m Eq 256c and Table 26 yield Iuz 015 Equation 256c allows an approximate estimate of the roughness z0 based on the measurement of Iuz Note that if the calculated roughness length z0 were significantly different from 003 m then a corresponding value of 𝛽 60 would be assumed on the basis of Table 26 and z0 would be obtained by successive approximations In the surface layer the decrease of u2z z0 12 with height is relatively slow see eg 35 p 185 and is conservatively typically neglected in structural engineering k k k k 36 2 The Atmospheric Boundary Layer calculations The averaging time in Eq 256 should be equal to the duration of strong winds in a storm Typical durations being considered are 1 hour and 10 minutes The turbulence intensity decreases as the height above the surface increases and vanishes near the top of the ABL Definitions similar to Eq 256 are applicable to the lateral and vertical turbulence intensities Ivz and Iwz In both these definitions the denominator is Uz Measurements suggest that the turbulence intensity is typically higher by roughly 10 in tropical cyclone than in extratropical storms 37 38 see Section 253 242 Integral Turbulence Scales The velocity fluctuations in a flow passing a point are associated with an overall flow disturbance consisting of a superposition of conceptual eddies transported by the mean wind Each eddy is viewed as causing at that point a periodic fluctuation with circular frequency 𝜔 2𝜋n The integral turbulence scales are measures of the spatial extent of the overall flow disturbance In particular the integral turbulence scale Lx u is a measure of the size of the longitudinal velocity components of the turbulent eddies In a structural engineering context Lx u is a measure of the extent to which the overall fluctuating disturbance associated with the longitudinal wind speed fluctuation u will engulf a structure in the alongwind direction and will thus affect at the same time both its windward and leeward sides If Lx u is large in relation to the alongwind dimension of the structure the gust will engulf both sides The scales Ly u and Lz u are measures of the transverse and vertical spatial extent of the fluctuating longitudinal component u of the wind speed The scale Lx w is a measure of the longitudinal spatial extent of the vertical fluctuating component w If the mean wind is normal to a bridge span and Lx w is large in relation to the deck width the vertical wind speed fluctuation w will act at any given time on the whole width of the deck If we now consider a panel normal to the mean wind direction small values of Ly u and Lz u compared with the dimensions of the panel indicate that the effect of the longitudinal velocity fluctuations upon the overall wind loading is small However if Ly u and Lz u are large the eddy will envelop the entire panel and that effect will be significant Mathematically the integral turbulence scale Lx u also called integral turbulence length is defined as follows Lx u 0 1 u2 Ru1u2𝜉d𝜉 257 where the overbar denotes mean value The function Ru1u2𝜉 is defined as the autocor relation function of the longitudinal velocity components u x1 y1 z1 t and u x1 𝜉 y1 z1 t Eq 257 may be interpreted as follows At any given time t the fluctuation ux 𝜉 y z differs from ux y z The difference increases as the distance 𝜉 increases If 𝜉 0 the autocorrelation function is unity if 𝜉 is small the two fluctuations are nearly the same so in Eq 257 the autocorrelation function is close to unity and its product by the elemental length d𝜉 is therefore close to d𝜉 On the other hand if 𝜉 is large the fluctuations ux y z and ux 𝜉 y z differ randomly from each other and their prod ucts are positive for some values of 𝜉 and negative for others so that their mean values tend to be vanishingly small and contribute negligibly to Lx u This interpretation is equiv alent to stating that Lx u is a measure of the size of the largest turbulent eddies of the flow that is of the eddies characterized by large autocorrelation functions k k k k 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces 37 Taylor Hypothesis Frequency Space and Wavenumber Space According to the Taylor hypothesis it may be assumed approximately that the flow disturbance is frozen as it travels with the mean velocity Uz that is ux1 𝜏 t ux1 xU 𝜏 258 where x Ut 𝜏 time and t is a finite time increment This assumption implies that every frequency component of the disturbance also travels essentially unchanged with the mean velocity U During a period T an eddy whose harmonic motion at fixed x has circular frequency 𝜔 2𝜋T 2𝜋n where n 1T denotes the frequency travels with velocity U a distance UT 𝜆 where 𝜆 Un is the wavelength The wavenumber is defined as 𝜅 2𝜋𝜆 2𝜋nU 𝜔U The motion is defined by a cosine function with argument 𝜔t 𝜅x or equivalently 𝜅Ut x meaning that for fixed t it is a harmonic wave in the wavenumber space and for fixed x it is harmonic function in the frequency space By virtue of Taylors hypothesis the integral turbulence length Lx u defined in Eq 257 by following a particles path ie in Lagrangian terms can alternatively be defined at a fixed point ie in Eulerian terms as Lx u U 0 1 u2 Ru𝜏d𝜏 259 where the autocorrelation function is defined by Eq B21 Measurements of Lx u Measurements show that Lx u increases with height above ground and as the terrain roughness decreases The following strictly empirical expression was proposed in 39 for Lx u Lx u Czm 260 where the constants C and m are obtained from Figure 26 Table 28 lists measured values of Lx u and estimates based on Eq 260 The uncertainties in the value of Lx u are seen to be significant On the basis of recent measurements at elevations z of up to about 95 m in open sea exposure at mean speeds Uz 10 to 25 m s1 it was suggested in 40 on a strictly 1000 10 10 01 C m 100 0001 001 01 10 m C zo meters 10 1 10 Figure 26 Values of C and m as functions of z0 Source Reprinted from 39 with permission from Elsevier k k k k 38 2 The Atmospheric Boundary Layer Table 28 Measurements of integral turbulence scales Lx u m Exposure z z0 Range Avg Eq 260 Opena 31 003 60460 200 180 Opena 51 003 130450 200 200 Opena 81 003 60650 300 230 Opena 110 003 110690 350 240 Opena 151 003 120630 400 250 Openb 15 001 82 220 Openb 17 004010 55 120160 Subb 16 100 36 70 a Measurements reported in 61 b Measurements reported in 39 empirical basis that Lx u z 33 Uz 30 where Lx u is in meters and U is in m s1 and it was noted that Lx u increased in the intervals 510 1020 2040 4060 and 6080 m elevation by approximately 7 10 10 8 and 5 respectively The dependence of the inte gral length scale on the velocity at all elevations is not supported by theory however According to 39 it may be assumed Ly u 033 Lx u Lz u 05 Lx u Ly w 04z 261abc Section 243 presents the derivation of the integral turbulence length Lx u from an expression for the spectrum of the longitudinal velocity fluctuations based on theory and validated by measurements reported in 41 see Eq 277 243 Spectra of Turbulent Wind Speed Fluctuations As indicated in Section 242 integral turbulence scales are measures of the average size of the largest turbulent eddies of the flow In some applications a more detailed descrip tion of the turbulent fluctuations is needed For example the resonant response of a flexible structure is induced by velocity fluctuation components with frequencies equal or close to the structures natural frequencies of vibration To calculate that response measures are needed of the size of the turbulent eddies as a function of frequency and of the degree to which the turbulent fluctuations differ from each other as functions of their relative position in space These measures are provided by the spectral density and the crossspectral density functions The Energy Cascade Turbulent velocity fluctuations in a flow with mean velocity U may be viewed as a result of a superposition of eddies each characterized by a periodic motion with circular frequency 𝜔 2𝜋n or of wavenumbers 𝜅 From the equations of balance of momenta for the mean motion the following equation may be derived U x q2 2 V y q2 2 W z q2 2 𝜏u 𝜌 U z 𝜏v 𝜌 V z z w p 𝜌 q2 2 𝜀 0 262 k k k k 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces 39 where the bars indicate averaging with respect to time q u2 v2 w212 263 is the resultant fluctuating velocity u v and w are turbulent velocity fluctuations in the x y and z directions respectively p is the fluctuating pressure and 𝜀 is the rate of energy dissipation per unit mass Eq 262 is the turbulent kinetic energy equation and expresses the balance of turbulent energy advection the terms in the first bracket pro duction the terms in the second bracket diffusion the terms in the third bracket and dissipation It can be shown that the inertial terms in these equations are associated with transfer of energy from larger eddies to smaller ones while the viscous terms account for energy dissipation 42 The latter is effected mostly by the smallest eddies in which the shear deformations and therefore the viscous stresses are large In the absence of sources of energy the kinetic energy of the turbulent motion will decrease that is the turbulence will decay If the viscosity effects are small the decay time is long if compared with the periods of the eddies in the high wavenumber range The energy of these eddies may therefore be considered to be approximately steady This can only be the case if the energy fed into them through inertial transfer from the larger eddies is balanced by the energy dissipated through viscous effects The small eddy motion is then determined by the rate of energy transfer or equivalently by the rate of energy dissipation denoted by 𝜀 and by the viscosity The assumption that this is the case is known as Kolmogorovs first hypothesis It follows from this assumption that since small eddy motion depends only upon the internal parameters of the flow it is independent of external conditions such as boundaries and that therefore local isotropy the absence of preferred directions of small eddy motion obtains It may further be assumed that the energy dissipation is produced almost in its entirety by the smallest eddies of the flow Thus at the lower end of the wavenumber subrange to which Kolmogorovs first hypothesis applies the influence of the viscosity is small In this subrange known as the inertial subrange the eddy motion may be assumed to be independent of viscosity and thus determined solely by the rate of energy trans fer 𝜀 which is equal to the rate of energy dissipation This assumption is known as the Kolmogorov second hypothesis The total kinetic energy of the turbulent motion may correspondingly be regarded as a sum of contributions by each of the eddies of the flow The function E𝜅 representing the dependence upon wavenumber 𝜅 of these energy contributions is defined as the energy spectrum of the turbulent motion It follows that for sufficiently high 𝜅 FE𝜅 𝜅 𝜀 0 264 The dimensions of the quantities within brackets in Eq 264 are L3T2 L1 and L2T3 respectively From dimensional considerations it follows that E𝜅 a1𝜀23𝜅53 265 in which a1 is a universal constant On account of the isotropy the expression for the spectral density of the longitudinal velocity fluctuations1 denoted by Eu𝜅 is to within 1 A mathematical definition of spectra is presented in Appendix B k k k k 40 2 The Atmospheric Boundary Layer a constant similar to the constant in Eq 265 Thus Eu𝜅 a𝜀23𝜅53 266 in which it has been established by measurements that a 05 If expressed in terms of the frequency n the spectral density is denoted by Sun Its expression is determined by noting that 0 Eu𝜅d𝜅 0 Sundn u2 267 see Eq B15 and 𝜅 2𝜋nU Therefore Sundn Eu𝜅d𝜅 268 Mathematically the ordinates of a spectral density function are counterparts of the squares of the amplitudes of a Fourier series In a Fourier series the frequencies are dis crete and the contribution of each harmonic component to the signals variance is finite In a spectral density plot the frequencies are continuous and given a signal gt each component Sgn has an infinitesimal contribution to the variance of gt Spectral den sity plots thus have to plots of squares of Fourier series harmonic components a relation similar to the relation of a probability density function to a discrete probability plot Spectra in the Inertial Subrange Measurements performed in the surface layer of the atmosphere confirm the assumption that in horizontally homogeneous neutrally strat ified flow the energy production is approximately balanced by the energy dissipation It then follows from Eq 262 that the expression for this balance is approximately 𝜀 𝜏0 𝜌 dUz dz 269 where Uz u 1 k ln z z0 238 If Eqs 212 267 and 238 are used 𝜀 u3 kz 270 For the inertial subrange we substitute Eq 270 in Eq 266 Since 𝜅 2𝜋nUz there results nSun u2 026 f 23 271 The lefthand side of Eq 271 and the variable f nz U z 272 are called respectively the reduced spectrum of the longitudinal velocity fluctuations and in honor of Kolmogorovs student who developed Eq 272 the Monin similar ity coordinate Equation 271 was validated by extensive measurements for example 43 Its dependence on height above ground is significant for structural engineering purposes since spectral ordinates within the inertial subrange typically cause the reso nant response of tall structures to wind loads As is the case for the logarithmic law for k k k k 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces 41 mean wind speeds at 10 m above ground greater than say 15 m s1 it is reasonable to apply Eq 271 throughout the height range of interest to the structural engineer Spectra in the LowerFrequency Range The lowerfrequency range also called the energy containing range is defined between n 0 and the lower end of the inertial sub range ns Velocities in this range contribute the bulk of the quasistatic alongwind fluc tuating loading on structures According to theoretical and numerical results reported in 44 and 45 and to measurements reported in 41 for 0 n nl where nl is small ie in the order of 002 Hz or less the spectral density may be assumed to be constant In particular it follows from Eq 259 and B25 that Suz 0 4𝛽u2 Lx uz Uz 273 For frequencies nl n ns Suz n azn where az is determined from the con dition that for n ns Suz n is continuous that is satisfies Eq 271 Expressions for the Spectrum Proposed in the 1960s and 1970s Kaimals spectrum has the form 46 nSuz n u2 105f 1 33f 23 274 where f is the Monin coordinate Eq 272 For open terrain Eq 274 does not satisfy the widely accepted requirement that the area under the spectral curve should be approximately 6u2 To satisfy this requirement the coefficients 105 and 33 are replaced in Eq 274 by the coefficients 200 and 50 respectively nSuz n u2 200f 1 50f 23 275 An expression for the spectrum proposed by Davenport 47 is no longer in use because i it does not account for the dependence of the spectrum on height and ii it implies 0 The ASCE 4912 Standard has adopted the following expression referred to as the von Kármán spectrum 16 48 49 nSuz n u2 4𝛽nLx uzUz 1 708nLx uzUz256 276 Equation 276 was developed for aeronautical applications in conjunction with a value Lx u 760 m 48 at mid to high altitudes It yields the correct expression for the spectrum at n 0 and reflects correctly the decay of the spectrum as a function of n in the iner tial subrange However it is universally accepted in the boundarylayer meteorological community that spectral ordinates in that subrange are well represented by Eq 271 For Eq 276 to be consistent with Eq 271 it would be necessary that Lx u 03𝛽32z 277 According to Eq 277 for open terrain at 10 m above ground 𝛽 60 see Table 26 Lx u 44 m whereas according to ASCE 4912 16 Lx u 110 m Reference 35 p 176 states We recommend that integral scales be avoided in applications to atmospheric data Many investigators have computed integral scales from atmospheric data but the results are badly scattered and cannot be organized k k k k 42 2 The Atmospheric Boundary Layer For this reason it has been proposed to base the estimation of the integral scale Lx u on the frequency nmax for which the curve nSun is a maximum Unfortunately the curves nSun tend to be quite flat and sufficiently variable that nmax is not well defined 35 Reference 50 also warns against the use of this approach and notes that it likely underestimates Lx u by a factor of 2 or 3 Spectral Density Suz n and Integral Scale 63 A model of the spectrum Sun was recently developed on the basis of theoretical studies eg 44 45 and measurements reported in 41 Based on 41 figures 6 and 7 the spectral density of the longitudinal velocity fluctuations can be written as Suz z0 n az z0 nl 0 n nl az z0 n nl n ns 026u2 z Uz z0 23 n53 ns n 278abc Equation 278c was obtained from Eqs 272 and 273 Using the notation nsz Uz fs 279 where according to the measurements of 41 figure 8 f s 0125 For n ns Eq 278c becomes Suz z0 ns 026u2 z Uz z0 23 n53 s 280 The condition that the functions defined by Eqs 278b and 278c be continuous at n ns then yields az 026u2 f 23 s 281 The areas under the spectral curve in the intervals 0 n nl is aznlnl The areas under the spectral curve in the intervals nl n ns and n ns are respectively ns nl 026u2 f 23 s dn n 026u2 f 23 s ln ns nl 282 nd ns 026u2 z Uz 23 n53dn 039u2 f 23 s 283 where nd is the very large frequency corresponding to the onset of dissipation by molec ular friction The total area under the spectral curve is 𝛽z0u2 Therefore 𝛽z0u2 026u2 f 23 s 026u2 f 23 s ln ns nl 039u2 f 23 s 284 Equation 284 yields nl ns exp 𝛽z0 026fs 23 039fs 23 026fs 23 fs Uz z exp 𝛽z0 065fs 23 026fs 23 285 k k k k 24 ABL Turbulence in Horizontally Homogeneous Flow Over Smooth Flat Surfaces 43 0 005 0 50 100 150 200 250 300 350 01 015 02 025 Snn m2 s1 n Hz 03 035 04 045 05 Figure 27 Spectral density plot z 10 m U10 m 30 m s1 z0 003 m f s 0125 Equations 278abc are plotted in Figure 27 for z 10 m U10 m 30 m s1 z0 003 m f s 0125 The integral turbulence scale is Lx uz Su0 zUz 4𝛽z0u2 286 Lx uz 026fs 53 4𝛽z0 exp 𝛽z0 065fs 2 3 026fs 2 3 z 287 The expression for the integral scale is based on values of 𝛽 that are well established and on validated models of the spectrum for both the inertial subrange and the low frequency ranges For sufficiently low values of z 𝛽 is assumed to be constant and Lx uz is independent of wind speed Example 215 Let z 10 m U10 m 539 m s1 z0 004 m Therefore 𝛽 60 and u 039 m s1 According to measurements reported in 41 f s 0125 Then ns 00674 Hz Eq 279 nl 256 103 Hz Eq 285 a10 m 0158 Eq 281 Sunl 10 m 617 m2 s1 Sun 0 10 m Eqs 277ab and 277b The calculated integral length is 911z 911 m Eq 286 The value provided in the ASCE 4912 Standard is 110 m The measurements of 41 have consistently yielded the value f s 0125 at all six eleva tions for which data were obtained Note however that the calculated length is sensitive to the value of the frequency f s In Example 215 assuming f s 01 0125 and as sug gested in 44 f s 016 for z0 004 m and 𝛽 6 Eq 287 yields 10 m 565 91 and 171 m respectively This suggests that the recommendation by Panofsky and Dutton 35 quoted earlier is indeed warranted In addition the finding that the curve nSun is k k k k 44 2 The Atmospheric Boundary Layer flat in the range nl n ns confirms the statement in 35 and 50 that the frequency for which that curve attains a maximum yields no useful information on the integral length Dependence of Lx u on wind speed at higher elevations z It was noted that throughout the sublayer within which the parameter 𝛽 is approximately constant the integral tur bulence length is independent of wind speed However this appears to be no longer the case for higher elevations z Let the height of the ABL be denoted by H Since H is proportional to the friction velocity u Eq 237 for given z the ratio zH is lower for higher winds mean ing that 𝛽z z0 decreases with height Eq 255 Consider for example the case z0 004 m z 55 m and Uz 55 m 678 m s1 as in 41 The logarithmic law yields u 038 m s1 If the order of magnitude of the boundary layer depth is H 01uf where f is the Coriolis parameter see eg Examples 27 and 28 to f 104 s1 there corresponds H 380 m Assuming the validity of Eq 255 𝛽z z0 60 exp 15 55380 48 On the other hand if Uz 55 m 68 m s1 u 38 m s1 H 3800 m and 𝛽z z0 60 exp15 553800 59 It follows from Eq 287 with f s 0125 that the calculated value of the integral scale is 230 m if Uz 55 m 678 m s1 and 463 m if Uz 55 m 68 m s1 This example suggests that estimates of the integral scale at higher elevations depend upon the wind speed at which the measured data were obtained and that the measure ment reports should therefore include that speed Spectra of Vertical and Lateral Velocity Fluctuations According to 51 up to an ele vation of about 50 m the expression for the vertical velocity fluctuations which may be required for the design of some types of bridges is nSwz n u2 336 f 1 10 f 53 288 Equation 288 can be used for suspendedspan bridge design The expression for the spectrum of the lateral turbulent fluctuations proposed in 46 is nSvz n u2 15 f 1 10 f 53 289 In Eqs 288 and 289 the variable f is defined as in Eq 272 244 Crossspectral Density Functions The crossspectral density function of turbulent fluctuations occurring at two different points in space indicates the extent to which harmonic fluctuation components with frequencies n at those points are in tune with each other or evolve at crosspurposes ie are or are not mutually coherent For components with high frequencies the dis tance in space over which wind speed fluctuations are mutually coherent is small For lowfrequency components that distance is relatively large in the order of integral turbulence scales An eddy corresponding to a component with frequency n is said to envelop a structure if the distance over which the fluctuations with frequency n are rel atively coherent is comparable to the relevant dimension of the structure The expression for the crossspectral density of two signals u1 and u2 is Scr u1u2r n SC u1u2r n iSQ u1u2r n 290 k k k k 25 Horizontally NonHomogeneous Flows 45 in which i 1 r is the distance between the points M1 and M2 at which the signals occur and the subscripts C and Q identify the cospectrum and the quadrature spec trum of the two signals respectively The coherence function is defined as Cohr n c2 u1u2r n q2 u1u2r n 291 where c2 u1u2r n SC u1u2r n2 Sz1 nSz2 n q2 u1u2r n SQ u1u2r n2 Sz1 nSz2 n 292ab In Eqs 291 and 292ab Sz1 n and Sz2 n are the spectra of the signals at points M1 and M2 To larger integral turbulence scales there correspond increased values of the coherence For ABL applications it is typically assumed that the quadrature spectrum is negligible The following expression for the cospectrum is used in applications SC u1u2r n S12z1 nS12z2 n expf 293 where f nC2 zz2 1 z2 2 C2 yz2 1 z2 22 1 2Uz1 Uz2 294 yi zi are the coordinates of point Mi i 1 2 and according to wind tunnel mea surements the values of the exponential decay coefficients may be assumed to be very approximately Cz 10 Cy 16 52 Eqs 293 and 294 reflect the intuitively obvious fact that the crossspectrum decreases as i the frequency n increases since for given distance between the points M1 and M2 the mutual coherence is lower for small eddies than for larger eddies andor ii the distance between the points increases For lat eral fluctuations the expression for the cospectrum is similar except that values Cz 7 Cy 11 have been proposed 53 For two points with the same elevation the expres sion for the cospectrum of the vertical fluctuations is also assumed to be similar with Cy 8 53 The exponential decay coefficients are in fact dependent upon surface rough ness and upon wind speed these dependences are typically not accounted for in practice 25 Horizontally NonHomogeneous Flows Horizontal nonhomogeneities of atmospheric flows are due either to conditions at the Earths surface eg changes in surface roughness topographic features or to the meteorological nature of the flow as in the case of tropical cyclones thunderstorms or downbursts While the structure of horizontally homogeneous flows is basically well understood the modeling of horizontally nonhomogeneous flows is to a large extent still incomplete or tentative Computational Fluid Dynamics methods are increasingly being used for a variety of surface roughness and topographic configurations This section contains information of interest for structural engineering purposes 251 Flow Near a Change in Surface Roughness Fetch and Terrain Exposure Sites with uniform surface roughness are limited in size the flows near their bound aries are therefore affected by the surface roughness of adjoining sites Therefore the k k k k 46 2 The Atmospheric Boundary Layer surface roughness is not the sole factor that determines the wind profile at a site The profile also depends upon the distance the fetch over which that surface roughness prevails upwind of the site The terminology used in the ASCE 7 Standard therefore distinguishes between surface roughness and exposure For example a site is defined as having Exposure B if it has surface roughness B and surface roughness B prevails over a sufficiently long fetch for design purposes the wind profile at a site with Exposure B may be described by the power law with parameters corresponding to surface roughness B Sections 23 and 24 consider only the case of long fetch The ASCE 7 Standard provides criteria on the fetch required to assume a given exposure Useful information on the flow in transition zones can be obtained by considering the simple case of an abrupt roughness change along a line perpendicular to the direction of the mean flow Upwind of the discontinuity the flow is horizontally homogeneous and near the ground is governed by the parameter z01 Downwind of the discontinu ity the flow will be affected by the surface roughness z02 over a height hx where x is the downwind distance from the discontinuity This height known as the height of the internal boundary layer increases with x until the entire flow adjusts to the roughness length z02 A wellaccepted model of the internal boundary layer which holds for both smoothtorough and roughto smooth transitions is hx 028 z0r x z0r 08 295 53 where z0r is the largest of the roughness lengths z01 and z02 The validity of Eq 295 is limited to h 02 H where H is the ABL height for very large x Within the internal boundary layer the flow adjusts to the new surface roughness as shown in Figure 28 Example 216 Consider a zone with roughness length z02 030 m downwind of a zone with roughness length z01 003 m The estimated height of the internal boundary layer at a distance x 10 000 m downwind of the line of separation between the two zones is h10 000 m 350 m The same result is valid if the zone with roughness length z01 003 m is downwind of the zone with z02 030 m 252 Wind Profiles over Escarpments Topographic features alter the local wind environment and create wind speed increases speedup effects since more air has to flow through an area decreased with respect to the case of flat land by the presence of the topographical feature The procedure that follows is specified in the ASCE 716 Standard 3 for the calculation of speedup effects on 2 or 3D two or threedimensional isolated hills and 2D ridges and escarpments z0r z02 x hx z z01 Figure 28 Internal boundary layer hx Mean wind speed profile within the internal boundary layer is adjusted to the terrain roughness z02 z01 k k k k 25 Horizontally NonHomogeneous Flows 47 Vz z Vz Lh z x upwind H 2 H 2 H x downwind Speedup Figure 29 Twodimensional escarpment The increase in the wind speeds due to the topography is reflected in the exposuredependent factor Kzt The Standard provides speedup models applica ble to 2D ridges 3D isolated hills and 2D escarpments provided that all the following conditions are satisfied see Figure 29 for notations 1 No topographic features of comparable height exist for a horizontal distance of 100 times the height of the hill H or 32 km whichever is less from the point at which the height H is determined 2 The topographic feature protrudes above the height of upwind terrain features within a 32 km radius by a factor of two or more in any quadrant 3 The structure is located in the upper half of a hill or ridge or near the crest of an escarpment 4 HLh 02 5 The height of the hill H exceeds 525 m for Exposures C and D and 21 m for Exposure B If any of the conditions 15 above is not satisfied Kzt 1 The topographic factor is defined as Kzt Vz xVz2 where Vz 3second peak gust speed at height z above ground in horizontal terrain with no topographic feature The expression for Kzt is Kzt 1 K1K2K32 296 where the factor K1 accounts for the shape of the topographic feature K2 accounts for the variation of the speedup as a function of distance from the crest and K3 accounts for the variation of the speedup as a function of height above the surface of the topographic feature Values of and expressions for K1 K2 K3are given in ASCE 716 For example for HLh 05 K1 aH Lh K2 1 x 𝜇Lh K3 exp 𝛾z Lh 297abc where for 2D escarpments 𝛾 25 𝜇 15 upwind of crest 𝜇 40 downwind of crest a 075 Exposure B a 085 Exposure C and a 095 Exposure D k k k k 48 2 The Atmospheric Boundary Layer Example 217 Topographic factor for a 2D escarpment The escarpment is assumed to have Exposure B and dimensions H 305 m Lh 122 m The topography upwind of the escarpment is assumed to satisfy conditions 1 and 2 The building is located at the top of the escarpment and the downwind distance see Figure 29 between the crest and the buildings windward face is x 122 m In Figure 29 the building would be located to the right of the crest We seek the quantity Kzt for elevation z 76 m above ground at x 122 m Condition 4 is satisfied since HLh 305122 025 02 as is condition 5 since H 305 m 21 m Since HLh 05 Eqs 297abc yield K1 075 305122 01875 K2 1 12240 122 0975 K3 exp25 76122 0855 The topographic factor is Kzt 1 01875 0975 08552 1162 135 This result implies that at x 122 m downwind of the crest and z 76 m above ground the increased peak 3second gust is 116 times larger than the peak 3second gust at 76 m above ground upwind of the escarpment and the corresponding pressures are 1162 135 times larger than upwind of the escarpment 253 Hurricane and Thunderstorm Winds In current structural engineering practice it is assumed that flow models used for syn optic storms are acceptable for hurricanes and thunderstorms as well Although they are not yet sufficient for codification purposes a number of research results on these two types of storm have been obtained in recent years of which the most significant are briefly summarized or cited herein Hurricanes Geophysical Positioning System GPS dropwindsonde or dropsonde measurements of hurricane wind speed profiles yielded the following results i On average in the storms outer vortex wind speeds increase monotonically up to an ele vation of about 1 km where they attain about 14 times their strength at 10 m they then decrease monotonically between 1 and 3 km where they attain about 13 times their strength at 10 m ii On average in the storms eyewall wind speeds increase monoton ically up to an elevation of about 400 m where they attain about 13 times their value at 10 m after which they decrease monotonically between 400 and 3 km where they attain about 11 times their value at 10 m 54 The turbulence intensity in hurricane winds was found to be larger by about 10 in hurricanes than in synoptic storms 37 38 55 Values of the longitudinal integral tur bulence scale Lx u measured at 10 m elevation in hurricane Bonnie varied from 40 to 370 m 37 Table 29 37 lists measured values of Lx u based on 10 and 60min long records at 5 and 10 m above ground as well as values specified in the ASCE 4912 Standard 16 As expected Lx u decreases as the roughness length increases it increases in most cases modestly as the height z increases from 5 to 10 m It is seen in Table 29 that the k k k k 25 Horizontally NonHomogeneous Flows 49 Table 29 Longitudinal integral length scales at 5 and 10 m elevations m Record length 54 Hurricane z0min z0max z 10 min 60 min Eq 260 ASCE 4912 16 Eq 287a Isidore 00011 00060 5 10 98 140 310 450 210400 220420 190 150 Gordon 00002 00014 10 176 365 370450 190 165 Ivan 00080 00551 5 10 126 154 197 240 120180 140190 110 100 Ivan 00116 00497 5 10 105 123 314 366 120130 130140 110 100 Lili 00082 00589 5 10 82 94 189 226 90180 110190 110 100 a Values obtained by using Eq 287 were multiplied by 11 to account for the fact that fluctuations are stronger in hurricanes than in extratropical storms ASCE 4912 Standard 16 values are considerably smaller than the reported 60min measurements It may be assumed that measurements of integral length scales are affected by significant uncertainties as was noted also in Section 242 A hurricane wind speed record that clearly reflects the passage of the eye is shown in Figure 210 The record was obtained at 15 m above ground by an ultrasonic anemome ter unit with a wind speed range of 065 m s1 with a resolution of 001 m s1 capable of measuring instantaneous u v and w wind velocity components with a maximum sam pling rate of 32 Hz The traces shown are 10minute and 3second moving averages of data with a 10 Hz sampling rate Note its nonstationary character which contrasts with the stationarity of Figure 21 Thunderstorms The cold air downdraft that in a thunderstorm spreads horizontally over the ground can be compared to a wall jet Just as in a wall jet the surface friction retards the spreading flow Of particular interest is the first gust or gust front Figures 114 and 211 that is the thunderstorm wind that can exhibit a considerable and relatively rapid change of speed and direction The wind speed increase and the time interval during which it occurs have been called by some authors the gust size ΔV and the gust length Δt respectively 55 Depending upon thunderstorm intensity the gust size may vary approximately from 3 to 30 m s1 while the gust length may range from approximately less than 110 min According to numerical and laboratory simulations 5658 as well as fullscale mea surements 59 near the ground the wind speed profiles along a thunderstorm gust front can be quite different from a loglaw profile However in current design practice it is assumed that thunderstorm characteristics may for practical purposes be assumed to be the same as those of largescale storms This assumption may be warranted given that according to 60 the maximum winds ie design level winds within the thunderstorm are rarely due to storms in which significant deviations from the log law occur Defini tive statements on the micrometeorological and statistical characterization of thunder storms appear to be unwarranted at this time owing to the lack of sufficient fullscale highspeed data k k k k 200 University of Florida Project NSF CMMI 1055744 Tropical Cyclone Harvey Location Aransas Country Airport TX MonthYear 0817 Station ID FCMP UFT2 Height 150 m Lat 280888 Lon 970512 Wind Speed mph Max 1373 mph 3sec 10min 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 251700 251800 251900 252000 252100 252200 252300 260000 260100 260200 260300 260400 260500 260700 260600 260800 260900 261000 10 0 Figure 210 Hurricane wind speed traces Source Courtesy of Professor F J Masters University of Florida k k k k References 51 c177m f 444m b 90m e 355m a 45m d 266m Figure 211 Thunderstorm wind speed records at six elevations above ground near Oklahoma City Source Courtesy of National Severe Storms Laboratory National Oceanic and Atmospheric Laboratory References 1 Schlichting H 1987 Boundary Layer Theory 7th ed New York McGrawHill 2 Csanady GT 1967 On the resistance law of a turbulent Ekman layer Journal of the Atmospheric Sciences 24 467471 3 ASCE Minimum design loads for buildings and other structures ASCESEI 716 in ASCE Standard ASCESEI 716 Reston VA American Society of Civil Engineers 2016 4 Tennekes H and Lumley JL 1972 A First Course in Turbulence Cambridge MIT Press 5 Blackadar AK and Tennekes H 1968 Asymptotic similarity in neutral Barotropic planetary boundary layers Journal of the Atmospheric Sciences 25 10151020 6 Tennekes H 1973 The logarithmic wind profile Journal of the Atmospheric Sciences 30 234238 k k k k 52 2 The Atmospheric Boundary Layer 7 Zilitinkevich SS and Esau IN 2002 On integral measures of the neutral barotropic planetary boundary layer BoundaryLayer Meteorology 104 371379 8 Zilitinkevich S Esau I and Baklanov A 2007 Further comments on the equilibrium height of neutral and stable planetary boundary layers Quarterly Journal of the Royal Meteorological Society 133 265271 9 Zilitinkevich S 2012 The height of the atmospheric planetary boundary layer state of the art and new development In National Security and Human Health Implications of Climate Change ed HJS Fernando Z Klaic and JL McCulley 147161 Netherlands Springer 10 Stull R Practical meteorology an algebrabased survey of atmospheric science University of British Columbia 2015 11 Lettau H 1962 Theoretical wind spirals in the boundary layer of a barotropic atmosphere Beitraege zur Physik der Atmosphaere 35 195212 12 Kung EC 1966 Largescale balance of kinetic energy in the atmosphere Monthly Weather Review 94 627640 13 Hess GD and Garratt JR 2002 Evaluating models of the neutral barotropic planetary boundary layer using integral measures part I overview BoundaryLayer Meteorology 104 333358 14 Hess GD 2004 The neutral barotropic planetary boundary layer capped by a lowlevel inversion BoundaryLayer Meteorology 110 319355 15 Simiu E Shi L and Yeo D 2016 Planetary boundarylayer modelling and tall building design BoundaryLayer Meteorology 159 173181 httpswwwnistgov wind 16 ASCE Wind tunnel testing for buildings and other structures ASCESEI 4912 in ASCE Standard ASCESEI 4912 Reston VA American Society of Civil Engineers 2012 17 Chamberlain AC 1983 Roughness length of sea sand and snow BoundaryLayer Meteorology 25 405409 18 Oliver HR 1971 Wind profiles in and above a forest canopy Quarterly Journal of the Royal Meteorological Society 97 548553 19 Biétry J Sacré C and Simiu E 1968 Mean wind profiles and changes of terrain roughness Journal of the Structural Division ASCE 104 15851593 20 Powell MD Vickery PJ and Reinhold TA 2003 Reduced drag coefficient for high wind speeds in tropical cyclones Nature 422 279283 21 CEN Eurocode 1 Actions on structures Parts 14 General actions Wind actions in EN 199114 European Committee for Standardization CEN 2005 22 Smith SD and Banke EG 1975 Variation of the sea surface drag coeffi cient with wind speed Quarterly Journal of the Royal Meteorological Society 101 665673 23 Wu J 1969 Wind stress and surface roughness at airsea interface Journal of Geophysical Research 74 444455 24 Amorocho J and DeVries JJ 1980 A new evaluation of the wind stress coefficient over water surfaces Journal of Geophysical Research Oceans 85 433442 25 Garratt JR 1977 Review of drag coefficients over oceans and continents Monthly Weather Review 105 915929 k k k k References 53 26 Smith SD 1980 Wind stress and heat flux over the ocean in gale force winds Journal of Physical Oceanography 10 709726 27 Krügermeyer L Grünewald M and Dunckel M 1978 The influence of sea waves on the wind profile BoundaryLayer Meteorology 14 403414 28 Lettau H 1969 Note on aerodynamic roughnessparameter estimation on the basis of roughnesselement description Journal of Applied Meteorology 8 828832 29 Masters FJ Vickery PJ Bacon P and Rappaport EN 2010 Toward objective standardized intensity estimates from surface wind speed observations Bulletin of the American Meteorological Society 91 16651681 30 Helliwell N C Wind over London in Third International Conference on Wind Effects on Buildings and Structures Tokyo 1971 pp 2332 31 Hellmann G Über die Bewegung der Luft in den untersten Schichten der Atmosphäre Königlich Preussischen Akademie der Wissenschaften 1917 32 Pagon WW 1935 Wind velocity in relation to height above ground Engineering NewsRecord 114 742745 33 NRCC National Building Code of Canada Ottawa Ontario Canada Institute for Research in Construction National Research Council of Canada 2010 34 AIJ AIJ Recommendations for loads on buildings Chapter 6 Wind loads Archi tectural Institute of Japan 2004 35 Panofsky HA and Dutton JA 1984 Atmospheric Turbulence Models and Methods for Engineering Applications 1e New York WileyInterscience 36 Durst CS 1960 Wind speed over short periods of time Meteorological Magazine 89 181187 37 Schroeder JL and Smith DA 2003 Hurricane Bonnie wind flow characteristics as determined from WEMITE Journal of Wind Engineering and Industrial Aerodynamics 91 767789 38 Krayer WR and Marshall RD 1992 Gust factors applied to hurricane winds Bulletin of the American Meteorological Society 73 613618 39 Counihan J 1975 Adiabatic atmospheric boundary layers a review and analysis of data from the period 18801972 Atmospheric Environment 9 871905 40 Oh S and Ishihara T A modified von Karman model for the spectra and the spatial correlations of the offshore wind field presented at Offshore 2015 Copenhagen Denmark 2015 41 Drobinski P Carlotti P Newsom RK et al 2004 The structure of the nearneutral atmospheric surface layer Journal of the Atmospheric Sciences 61 699714 42 Hinze JO 1975 Turbulence 2nd ed New York McGrawHill 43 Fichtl GH and McVehil GE 1970 Longitudinal and lateral spectra of turbulence in the atmospheric boundary layer at the Kennedy Space Center Journal of Applied Meteorology 9 5163 44 Hunt JCR and Carlotti P 2001 Statistical structure at the wall of the high Reynolds number turbulent boundary layer Flow Turbulence and Combustion 66 453475 45 Carlotti P 2002 Twopoint properties of atmospheric turbulence very close to the ground comparison of a high resolution les with theoretical models BoundaryLayer Meteorology 104 381410 k k k k 54 2 The Atmospheric Boundary Layer 46 Kaimal JC Wyngaard JC Izumi Y and Coté OR 1972 Spectral characteristics of surfacelayer turbulence Quarterly Journal of the Royal Meteorological Society 98 563589 47 Davenport AG 1961 The spectrum of horizontal gustiness near the ground in high winds Quarterly Journal of the Royal Meteorological Society 87 194211 48 DOD Department of Defense Interface Standard Flying Qualities of Piloted Aircraft MILSTD1797A Department of Defense 2004 49 Lumley JL and Panofsky HA 1964 The Structure of Atmospheric Turbulence New York Interscience Publishers 50 Pasquill F and Butler HE 1964 A note on determining the scale of turbulence Quarterly Journal of the Royal Meteorological Society 90 7984 51 Vickery B J On the reliability of gust loading factors in Technical Meeting Concerning Wind Loads on Buildings and Structures Washington DC 1970 52 Kristensen L and Jensen NO 1979 Lateral coherence in isotropic turbulence and in the natural wind BoundaryLayer Meteorology 17 353373 53 Wood DH 1982 Internal boundary layer growth following a step change in surface roughness BoundaryLayer Meteorology 22 241244 54 Yu B Chowdhury AG and Masters F 2008 Hurricane wind power spectra cospectra and integral length scales BoundaryLayer Meteorology 129 411430 55 Sinclair R W Anthes R A and Panofsky H A Variation of the low level winds during the passage of a thunderstorm gust front NASACR2289 NASA Washington DC 1973 56 Chay MT and Letchford CW 2002 Pressure distributions on a cube in a simulated thunderstorm downburst Part A stationary downburst observations Journal of Wind Engineering and Industrial Aerodynamics 90 711732 57 Letchford CW and Chay MT 2002 Pressure distributions on a cube in a simulated thunderstorm downburstPart B moving downburst observations Journal of Wind Engineering and Industrial Aerodynamics 90 733753 58 Jubayer C Elatar H and Hangan H Pressure distributions on a lowrise building in a laboratory simulated downburst presented at the 8th International Colloquium on Bluff Body Aerodynamics and Applications Boston 2016 59 Lombardo FT Smith DA Schroeder JL and Mehta KC 2014 Journal of Wind Engineering and Industrial Aerodynamics 125 121132 doi 101016jjweia201312004 60 Schroeder J L Personal communication November 2016 61 Shiotani M Structure of Gusts in High Winds Parts 14 Namashino Funabashi Chiba Japan Physical Science Laboratory Nihon University 19671971 62 Simiu E Heckert NA and Yeo D 2017 Planetary Boundary Layer Model ing and Standard Provisions for Supertall Building Design Journal of Structural Engineering 143 06017002 httpswwwnistgovwind 63 Simiu E Potra FA and Nandi TN 2018 Determining longitudinal integral turbulence scales in the nearneutral atmospheric surface layer Boundarylayer Meteorology doi 101007s1054601804004 httpswwwnistgovwind k k k k 55 3 Extreme Wind Speeds Structures are designed to be safe and serviceable meaning that their probabilities of exceeding specified strength and serviceability limit states must be acceptably small These probabilities are functions of the wind speeds to which the structures are exposed The present chapter is concerned with the probabilistic estimation of extreme wind speeds Uncertainties in such estimates are discussed in Chapter 7 Materials that com plement this chapter are provided in Appendices A and C Section 31 provides simple intuitive definitions of exceedance probabilities and mean recurrence intervals MRIs and extends those definitions to wind speeds in mixed wind climates eg climates with both hurricane and nonhurricane winds or with largescale extratropical storm and thunderstorm winds Section 32 defines nondirectional and directional wind speed data in nonhurricane and hurricaneprone regions and reviews main sources of such data for the conterminous United States Section 33 describes and illustrates methods for estimating extreme wind speeds with specified MRIs Section 34 is devoted to tornado climatology 31 Cumulative Distributions Exceedance Probabilities Mean Recurrence Intervals Section 311 introduces these topics intuitively by using the example of a fair die and shows its relevance to the probabilistic characterization of extreme wind speeds Section 312 considers the case of mixed wind climates in regions with for example hurricane winds and significant nonhurricane winds or largescale extratropical storm and thunderstorm winds 311 Probability of Exceedance and Mean Recurrence Intervals 3111 A Case Study The Fair Die We denote the outcome of throwing a fair die once by O The probability denoted by PO n n 1 2 6 that the outcome ie the event O is less than or equal to n is called the cumulative distribution function CDF of the event O The CDF of the outcome O n is PO n n6 The probability of exceedance of the outcome n is PO n 1 PO n 1 n6 The MRI of the event O n is defined as the inverse of the probability of exceedance of that event and is the average number of trials throws Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 56 3 Extreme Wind Speeds required for O n Therefore MRI O n 11 n6 The MRI is also called the mean return period see also Section A51 Example 31 Mean recurrence interval of the outcome of throwing a die For a fair die the probability of exceedance PO 5 1 PO 5 1 56 16 The MRI of the event O 5 is 116 6 trials that is the outcome six occurs on average once in six trials The probability of exceedance of an outcome n increases as the number of trials increases If the probability of nonexceedance of the outcome n in one trial is PO n owing to the independence of the outcomes Section A25 the probability of nonexceedance of the outcome n in m trials is PO nm The probability of exceedance of the outcome n in m trials is 1 PO nm For example the probability of nonexceedance of the outcome five in two throws of a die is 562 2536 and the probability of exceedance of that outcome is 1 2536 1136 3112 Extension to Extreme Wind Speeds Conceptually the difference between the statement the outcome of throwing a die once exceeds n and the statement the largest wind speed V occurring in any one year exceeds v is that the CDF of the largest speed in a year PV v is continuous whereas PO n is discrete For any given n PO n is the same for any one trial throw of a die and is independent of the outcomes of other trials Similarly except for say possi ble global warming effects PV v is the same for any one trial any one year and is independent of speeds occurring in other years The speed v with an Nyear MRI is called the Nyear speed The MRI in years is Nv 1 1 PV v 31 Example 32 Probability of exceedance of the largest wind speed in a given data sam ple Consider the sample of size nine of the largest measured yearly wind speeds 20 18 21 25 17 24 22 20 15 in m s1 the largest speed in the sample is shown in bold type There are n 9 outcomes for which V 25 mph out of n 1 10 possi ble outcomes the 10th outcome being V 25 m s1 Hence the estimated probability PV 25 m s1 910 09 The probability of exceedance of a 25 m s1 largest yearly speed is 1 09 01 The MRI of the event that the 25 m s1 wind speed is exceeded in any one year is 101 10 years The probability of the event V 25 m s1 in 30 years is equal to the probability that V 25 m s1 in the first year and in the second year and in the 30th year that is 0930 004 The probability that V 25 m s1 in 30 years is then 1 004 096 312 Mixed Wind Climates We now consider wind speeds in regions exposed to both nonhurricane and hurricane winds We are interested in the probability that in any one year wind speeds regardless of their meteorological nature are less than or equal to a specified speed v k k k k 32 Wind Speed Data 57 Let the random variables V H and V NH denote respectively the largest hurricane wind speed and the largest nonhurricane wind speed in any one year Further let the probability that V H v and the probability that V NH v be denoted respectively by PV H v and PV NH v The random variable of interest is the maximum yearly speed regardless of whether it is a hurricane or a nonhurricane wind speed and is denoted by maxV H V NH The statement maxV H V NH v and the statement V H v and V NH v are equivalent Therefore PmaxV H V NH v PV H v and V NH v If it assumed that V H and V NH are independent random variables it follows see Section A25 that PmaxVH VNH v PVH vPVNH v 32 The probability distributions PV NH v and PV H v can be obtained as shown in Section 311 With an appropriate change of notation Eq 32 is also applicable to nonthunderstorm and thunderstorm wind speeds The probability of occurrence of the event V H v or V NH v is Section A21 PVH v or VNH v PVH v PVNH v 1 PVH vPVNH v 33ab Example 33 Mean recurrence interval of the event V H v and V NH v Assume that the MRI of the event that nonhurricane wind speeds exceed 45 m s1 is NNH 120 years and that the MRI of the event that hurricane wind speeds exceed 45 m s1 is NH 50 years The respective CDFs are PVNH 45 m s1 1 1NNH 099167 and PVH 45 m s1 1 1NH 098 By Eq 32 the CDF of the 45 m s1 wind speed due to nonhurricane and hurricane winds is PV H 45 and V NH 45 m s1 PV H 45 m s1 PV NH 45 m s1 099167 098 0972 By Eq 31 the MRI of the 45 m s1 wind speed at the site is 11 0972 357 years Example 34 Probability of occurrence of the event V H v or V NH v Assuming again NNH 120 years NH 50 years Eq 33ab yields PV H v or V NH v PV H v PV NH v 1 098 1 099167 0028year Eq A1 32 Wind Speed Data 321 Meteorological and Micrometeorological Homogeneity of the Data Extreme wind speed distributions differ depending upon the meteorological nature of the storms being considered For this reason hurricane synoptic storm and thunder storm data should be analyzed separately In addition wind speed data within a data sample must be micrometeorologically homogeneous meaning that all the data in a set must correspond to the same i height above the surface ii surface exposure eg open terrain and iii averaging time eg 3 s for peak wind gust speeds 1 min 10 min or 1 h Wind speeds at 10 m above terrain with open exposure and with the specified averaging time typically 3 seconds in the United States are referred to as standardized wind speeds If data do not satisfy the micrometeorological homogeneity requirement they have to be transformed so that the requirement is satisfied see Sections 234237 k k k k 58 3 Extreme Wind Speeds Section 241 and Ref 1 which show that as far as the surface exposure is concerned this task can be far from trivial 322 Directional and NonDirectional Wind Speeds Standard provisions for wind loads are based primarily on the use of nondirectional extreme wind speeds that is largest wind speeds in any one year or storm event regard less of their direction Directional extreme wind speeds that is largest wind speeds in any one year or storm event for each of the directional sectors being considered are used to estimate wind effects on special structures at sites for which aerodynamic data are available for a sufficient number of wind directions Denote the directional wind speeds by Uij eg i 1 2 j 1 2 8 where the subscript i indicates the year or the storm event and the subscript j indicates the wind direction For fixed i the corresponding nondirectional wind speed is Ui maxjUij Example 35 Directional and nondirectional wind speeds To illustrate the definitions of directional and nondirectional wind speeds we consider the following largest peak 3second gusts in m s1 recorded in two consecutive 1year periods1 Directional speed Uij Nondirectional speed maxjUij j 1 NE 2 E 3 SE 4 S 5 SW 6 W 7 NW 8 N i 1 45 50 41 48 43 44 47 39 50 i 2 39 47 43 54 40 42 36 38 54 The nondirectional speeds are also shown in bold type in the list of directional speeds 323 Wind Speed Data Sets 3231 Data in the Public Domain Peak Directional Gust Speeds at 10 m Above Open Terrain Standardized Wind Speeds Standardized peak gust speeds averaged over five seconds extracted from Automated Surface Observing Systems ASOS records and transformed to correspond to a 10 m elevation over terrain with open exposure are listed on the site httpswwwnistgov wind The difference between 5second peak gusts and the 3second peak gusts specified in the ASCE 716 Standard 2 is in practice negligibly small The standardized data are separated into thunderstorm and largescale extratropical wind speeds This was accomplished using a procedure described in 3 and software available on httpswww nistgovwind Simulated Synthetic Directional Tropical StormHurricane Wind Speeds Direc tional wind speeds are available for 55 coastline locations milestones along the 1 In the statistical literature a fixed time period is called an epoch k k k k 32 Wind Speed Data 59 Figure 31 Locator map with coastal distance marked in nautical miles Source National Oceanic and Atmospheric Administration Gulf and Atlantic coasts shown in Figure 31 see 4 5 The speeds were obtained by Monte Carlo simulation see Section A8 from approximately 100year records of hurricane climatological data pressure defects radii of maximum wind speeds and translation speeds and directions see Section 131 Probabilistic descriptions of those data were developed and used in conjunction with the physical model described by Eq 14 to obtain probabilistic models of the gradient speeds and directions These models were then transformed via empirical expressions into probabilistic models of surface wind speeds and directions and used for the Monte Carlo simulation of directional speed data at each of the milestones The simulated data based on 4 are listed on httpswwwnistgovwind They consist of i estimated hurricane mean arrival rates and ii sets of 999 1min coastline wind speeds in knots at 10 m above open terrain k k k k 60 3 Extreme Wind Speeds for 16 directions at 225 intervals 1 knot 115 mph 1 mph 0447 m s1 nominal ratios between 3second speeds and 1minute speeds and between 1minute speeds and 1hour speeds are 122 and 125 respectively see Table 25 At any given site as many of 2040 of the total number of simulated hurricane wind speeds are negligibly small Such small or vanishing wind speeds occur for example where the hurricane translation velocity counteracts the rotational velocity For each of the 55 milestones shown in Figure 31 the respective 999 simulated data can be used to obtain by Monte Carlo simulation datasets of any desired size see Section 337 Nondirectional hurricane wind speeds based on more recent simulations than those described in 5 can be obtained both for the coastline and for regions adjacent to the coastline from wind maps in ASCE 716 2 for MRIs of up to 3000 years and from wind maps in 6 for MRIs of up to 107 years 3232 Data Available Commercially Peak Directional Gust Speeds for Each of 36 Directions at 10 Intervals recorded at ASOS stations for periods of about 20 years or less wwwncdcnoaagovoancdchtml Simulated Hurricane Directional Wind Speed Data The methodology for obtaining directional hurricane wind speeds described in 7 is similar to the methodology used in 4 except that the various climatological and probabilistic models used therein have been refined and are based on a larger number of data Unlike the data based on 4 the data based on 7 cover both coastlines and regions adjacent thereto Figure 32 shows approximate estimates of 2000year or 1700year mean hourly hurricane wind speeds at 10 m above open terrain as estimated in 4 the ASCE 710 35 40 45 50 55 60 0 500 1000 1500 2000 2500 3000 ASCE 710 MRI 1700 yrs Georgiou Davenport and B Vickery 1983 MRI 2000 yrs P Vickery and Twisdale 1995 MRI 2000 yrs Batts et al 1980 MRI 2000 yrs milepost nautical miles ms Figure 32 Approximate estimates of mean hourly hurricane wind speeds at 10 m above ground over terrain with open exposure Source After Refs 4 810 k k k k 33 Nyear Speed Estimation from Measured Wind Speeds 61 Standard 8 and Refs 9 10 Note that there are no major differences among the various estimates except for milestones 1100 and 2600 where speeds are likely overestimated in 10 milestones 700 and 1400 where speeds are likely underestimated in 4 and milestones 23002600 where wind speeds are likely underestimated in the ASCE 710 Standard 33 Nyear Speed Estimation from Measured Wind Speeds Estimates of extreme wind speeds based on sets of measured wind speeds can be per formed by using two types of datasets In the traditional epochal approach the dataset being analyzed consists of the largest wind speeds recorded at the site of interest in each of a number of consecutive fixed epochs To avoid seasonality effects the epoch most commonly chosen is 1 year The dataset then consists of the largest yearly wind speed for each year of the period of record In the more modern peaksoverthreshold POT approach the dataset considered in the analysis consists of wind speeds that exceed an optimal threshold Section 331 explains the advantages of the peaksoverthreshold POT over the epochal approach Sections 332 discusses the probability distributions of the largest values and their use in structural engineering Section 333 presents methods for estimating extreme speeds with any specified MRI N based on the epochal approach Section 334 provides information on sampling errors in the estimation of extreme wind speeds modeled by the Type I Extreme Value distribution Section 335 concerns the POT approach Section 336 briefly discusses the spatial smoothing of extreme wind speed estimates performed at multiple stations within meteorologically homo geneous areas Section 337 concerns the development of large extreme wind speed databases from relatively short records Nonparametric estimation methods applicable to extreme wind speeds are presented in Section A9 331 Epochal Versus PeaksOverThreshold Approach to Estimation of Extremes One advantage of the POT approach is that it allows the use of larger data samples than the epochal approach since speeds other than the largest annual speeds can also be included in the data sample This is illustrated in the following example Example 36 Sample sizes in epochal and POT approaches Assume that in Year 1 the largest speed is 36 m s1 and the second largest speed is 34 m s1 and that in Year 2 the largest speed is 43 m s1 and the second and third largest speeds are 35 m s1 and 31 m s1 respectively If a threshold of 32 m s1 is chosen the speeds during Years 1 and 2 included in the sample are 43 m s1 36 m s1 35 m s1 and 34 m s1 four speeds In the epochal approach only two speeds are included in the sample 36 m s1 Year 1 and 43 m s1 Year 2 If the threshold is very high the advantage of a larger sample size is lost For example if the threshold were 40 m s1 only one speed 43 m s1 would be included in the twoyear sample If the threshold were very low the sample would include nonextreme wind speeds this would result in incorrect biased estimates of the extreme wind speeds k k k k 62 3 Extreme Wind Speeds An additional advantage of the POT approach is that it allows an optimal selection of the dataset being analyzed by i excluding from the analysis data lower than an optimal threshold that would result in biased estimates of the extremes and ii ensuring that the size of the dataset is sufficiently large to minimize sampling errors 332 Extreme Value Distributions and Their Use in Wind Climatology As indicated in Section A6 a theoretical and empirical basis exists for the assumption that probability distributions of the largest values are adequate for describing extreme wind speeds probabilistically It has been proven mathematically that three types of such distributions exist characterized by the length of the distribution tail the Gumbel distribution also known as the FisherTippett Extreme Value Type I or EV I distribu tion the Fréchet FisherTippett EV II distribution and the reverse Weibull distribution FisherTippett EV III distribution of the largest values The EV I and the EV II distributions have infinitely long distribution tails This means that their use can lead to estimates of large extremes whose probabilities of being exceeded depend upon the thickness of the distribution upper tails The EV I distributions tails are less thick than the tails of the EV II distributions and entail negligibly small probabilities of exceedance of very large extremes However for EV II distributions the distribution tails are thicker and may result in estimates of unrealistically high extreme wind speeds The EV III distribution has finite tails meaning that for wind speeds larger than the finite value of the distribution tail the probabilities of exceedance are zero Uncertainties inherent in the estimation process can result in extreme wind speed data samples being spuriously best fitted by an EV II distribution when in fact an EV I distribution would be appropriate For this reason the assumption that extreme wind speeds are best fitted by an EV II distribution used in the 1970s for the development of the extreme wind speed maps of the American National Standard A581 was aban doned by consensus of the ASCE 7 Standard Committee on Loads in favor of the EV I distribution Statistical estimates suggested that the EV III distribution may fit extreme wind speed data samples better than the EV I distribution on the basis of such estimates the AustralianNew Zealand Standard 11 Commentary C32 adopted the assumption that the EV III distribution is representative of the behavior of extreme wind speeds However estimates of the tail length of the EV III distribution are in practice prone to large errors and to avoid the underestimation of extreme wind speeds due to spurious best fits the ASCE 7 Standards Committee on Loads also decided against the use of the EV III distribution Unless otherwise indicated it will be assumed in this chapter that the EV I distribution is an appropriate probabilistic model of the extreme wind speeds The CDF of the EV I distribution is FIx exp exp x 𝜇 𝜎 x 𝜇 0 𝜎 34 where 𝜇 and 𝜎 called the location and scale parameter respectively are related to the mean value EX and standard deviation SDX of X by the expressions EX 𝜇 05772𝜎 35a k k k k 33 Nyear Speed Estimation from Measured Wind Speeds 63 SDX 𝜋 6 𝜎 35b Inversion of Eq 34 yields xFI 𝜇 𝜎 ln ln FI 36 or by virtue of Eq 31 xN 𝜇 𝜎 ln ln 1 1 N 𝜇 𝜎 ln N 37ab for large N 333 Wind Speed Estimation by the Epochal Approach This section presents two of the methods for estimating Nyear wind speeds under the assumption that the EV I distribution is appropriate the method of moments and Liebleins BLUE Best Linear Unbiased Estimator method 3331 Method of Moments This method relies on calculated sample means EV and standard deviations SDV of the sample of n wind speeds The wind speed corresponding to an MRI N is obtained from Eqs 37 in which the parameters 𝜇 and 𝜎 are obtained from Eqs 35 Example 37 EV I Extreme Wind Estimation Epochal Approach Method of Moments Assume that in a n 14year record at a site the nondirectional largest yearly peak 3second gust speeds from any direction in m s1 are 36 34 35 37 33 36 40 39 41 43 33 31 28 34 The epochal approach makes use of the mean EV 3571 m s1 and standard deviation SDV 407 m s1 of the n largest annual speeds From Eqs 35 we obtain 𝜎 317 and 𝜇 3390 in m s1 Equations 37ab yield vN 50 years 4627 m s1 and 4630 m s1 vN 3000 years 5928 m s1 and 5928 m s1 respectively BLUE Method In the BLUE method the data are arranged in ascending order that is v1 v2 vn The estimated parameters of the EV I distribution are then given by the expressions 𝜇 n i1 aivi 𝜎 n i1 bivi 38 where the vectors ai bi are listed for n16 in 12 p 20 and for n100 in the MATLAB implementation of the BLUE method which includes a users manual and an example httpswwwnistgovwind Example 38 EV I Extreme Wind Estimation Epochal Approach BLUE Method Consider the dataset of Example 37 The rankordered data are 28 31 33 33 34 34 35 36 36 37 39 40 41 43 k k k k 64 3 Extreme Wind Speeds For the sake of clarity we follow in this example the BLUE method as presented in 12 Using the coefficients ai i 1 2 14 12 p 20 𝜇 28 0163309 31 0125966 33 0108230 33 0095233 34 0084619 34 0075484 35 0067331 36 0059866 36 0052891 37 0046260 39 0039847 40 0033526 41 0027131 43 0020317 3364 𝜎 28 0285316 31 0098775 33 0045120 33 0013039 34 0008690 34 0024282 35 0035768 36 0044262 36 0050418 37 0054624 39 0057083 40 0057829 41 0056652 43 0052642 396 Equation 37a then yields vN 50 years 4909 m s1 vN 3000 years 6533 m s1 The reader can verify that the same result is obtained by using the MATLAB software referenced in this section The method of moments which is less efficient than the BLUE method produces in this case estimates of the 50 and 3000year wind speeds lower than the BLUE estimates by approximately 6 and 9 respectively 334 Sampling Errors in the Estimation of Extreme Speeds The standard deviation of the errors in the estimation of extreme wind speeds with a MRI N may be obtained from the following expression 13 SDvN 078164 146ln N 0577 11ln N 0577212 s n 39 where s is the sample standard deviation of the largest yearly wind speeds for the period of record and n is the sample size Example 39 At Great Falls Montana the largest yearly sustained fastestmile wind speeds in the period 19441977 sample size n 34 were 57 65 62 58 64 65 59 65 59 60 64 65 73 60 67 50 74 60 66 55 51 60 55 60 51 51 62 51 54 52 59 56 52 49 mph The sample mean and the standard deviation of for these data are V 591 mph and SDV 641 mph From Eqs 35 37 and 39 it follows that for N 50 years and N 1000 years v50 758 mph SDv50 371 mph v1000 908 mph SDv1000 636 mph The probabilities that vN is contained in the intervals vN SDvN and vN 2 SDvN are approximately 68 and 95 respectively These intervals are called the 68 and 95 confidence intervals for vN see Section A71 k k k k 33 Nyear Speed Estimation from Measured Wind Speeds 65 335 Wind Speed Estimation by the PeaksOverThreshold Approach Among the methods available for estimating extreme wind speeds by the POT approach we mention the method of moments and the de Haan method both of which are described in Section A72 and the POT Poissonprocesses methods used in 14 which provide information on the uncertainty in the estimates The plots of Figure 33 show estimates by the de Haan method of 100 1000 and 100000year fastestmile wind speeds at 61 m above ground in terrain with open exposure at Green Bay Wisconsin The estimates are functions of threshold speeds in mph The data consisted of the maximum wind speed for each of the successive 8day intervals within a 15year record and included no wind speed separated by less than 5 days For thresholds between about 38 and 32 mph sample sizes of about 35127 the estimated 100year speeds are stable around 60 mph The reliability of the estimate is poorer as the MRI increases this is clearly seen for the 100000year estimates For thresholds higher than 38 mph the estimates are less stable for all three MRIs that is they vary fairly strongly as a function of threshold For thresholds lower than about 32 mph the estimates of the 100yr speed are increasingly biased with respect to the 60 mph estimate owing to the presence in the data sample of low speeds unrepresentative of the extremes Including low speeds in a sample used for inferences on extreme speeds can result in biased estimates as would be the case if the heights of children were included in a sample used to estimate the height of adults For example estimates of extreme wind speeds based on wind speed data recorded every hour the vast majority 41 38 35 32 29 26 23 20 17 MPH 30 40 50 60 70 80 90 100 110 120 130 100000 1000 100 U Figure 33 Estimated wind speeds with 100 1000 and 100000year mean recurrence interval at Green Bay Wisconsin as functions of threshold mph k k k k 66 3 Extreme Wind Speeds of which are low and meteorologically unrelated to the extreme wind speeds would be unrealistic Modern extreme value statistics recognizes that to obtain dependable estimates of extreme values it is necessary to let the tails speak for themselves instead of allowing estimates to be biased by data with small values as is the case in Figure 33 for wind speeds below about 32 mph 336 Spatial Smoothing In developing wind maps results it is appropriate to apply spatial smoothing techniques to reduce discrepancies among results obtained for stations contained in a meteoro logically homogeneous area of appropriate size Such a technique was applied to the development of wind speed maps specified in the ASCE 716 Standard see Section 32 of 14 A technique used for the development of US maps specified in the ASCE 710 maps consisted of considering groups of stations called superstations and including identi cal subgroups of stations in more than one superstation The application of this tech nique led to the demonstrably incorrect result that extreme wind speeds are uniform throughout most of the contiguous United States 337 Development of Large Wind Speed Datasets A number of structural engineering applications require the use of large wind speed datasets for use in nonparametric estimates of wind effects with long MRIs A detailed procedure for generating such data including directional data is presented in 15 For material on Monte Carlo methods used for the development of large wind speed databases see Section A8 34 Tornado Characterization and Climatology Tornado climatology studies and design criteria on tornado action on structures require the characterization of tornadoes from the point of view of their flow modeling and their intensities Section 341 discusses tornado flow modeling based on atmospheric science considerations laboratory testing numerical methods and observations of tornadoes Section 342 is devoted to the use of tornado models observations of tornadoes and their effects and statistical methods for the estimation of wind speeds and associated atmospheric pressure defects Section 343 summarizes simplified conservative models of tornado structure that the US Nuclear Regulatory Commission Office of Nuclear Regulatory Research considers acceptable for the design of nuclear power plants 341 Tornado Flow Modeling Tornadoes are translating cyclostrophic flows that develop within severe thunder storms Because their horizontal dimensions are relatively small typically in the order of 300 m the probability that their maximum speeds at heights above ground in the order of a few tens of meters or less will be measured by a sufficiently strong instrument with fixed location or any other instrument is small For his reason reliable k k k k 34 Tornado Characterization and Climatology 67 measurements of such wind speeds are not available to date Laboratory measurements see Chapters 5 and 27 have shed useful light on tornado flow structure but are only the beginning of efforts to improve current knowledge in this area of research A highly readable generic guide on tornado climatology is available in 16 An anal ysis of information on more than 46 000 tornado segments ie portions of or entire tornadoes reported in the contiguous United States from January 1950 through August 2003 was performed in 17 with a view to determining tornado strike probabilities and maximum wind speeds for use in the development of design criteria for nuclear power plants Section 342 briefly summarizes salient features of 17 Section 343 summarizes US Nuclear Regulatory Commission NRC requirements on atmospheric pressure defects and tornado wind speeds based on the recommendations of 17 342 Summary of NUREGCR4461 Rev 2 Report 17 Of the 46 000 segments more than 39 600 had sufficient information on location inten sity length and width to be used in the analysis Estimates of and confidence intervals for expected values are based in 17 on the assumption first suggested in 1963 18 that lognormal distributions are appropriate As in 16 it is noted in 17 that even though the number of reported tornadoes has been increasing since 1950 owing to improved tornado observation techniques Figure 34 the increase was limited to the least intense tornadoes however the missing information on weaker tornadoes appears not to affect significantly estimates of strike probabilities or maximum wind speeds Comprehensive estimates of tornado characteristics are presented in 17 for the entire contiguous United States for regions thereof and for 1 2 and 4 latitude and 1950 0 200 400 600 800 1000 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Number of Tornadoes F0 F1F5 Figure 34 Number of EF0 tornadoes and total number of EF1 through EF5 tornadoes by year since 1950 19 k k k k 68 3 Extreme Wind Speeds longitude boxes The effect of the variation of the wind speed along and across the tornado footprint was modeled by using results of studies summarized in 19 Methods for estimating i tornado strike probabilities and ii conditional probabil ities that the maximum wind speed will exceed a specified value given that a tornado strike has occurred differ for point and finitesized structures For point structures only estimates of tornado impact areas are necessary For finitesize structures in addition to estimates of tornado impact areas estimates of lengths of tornado paths associated with various wind speeds are needed These were based on 20 For example while for EF0 tornadoes 100 of the length of the tornado path has EF0 speeds for an EF5 tornado it was estimated that on average 0135 0100 0190 0240 0185 and 0150 of the total path length have EF0 EF1 EF2 EF3 EF4 and EF5 wind speeds respectively For point structures the annual probability of exceedance of the speed uo at a point is defined as the probability that a tornado will strike that point times the annual proba bility that the speed u will exceed the speed uo given that a tornado strike has occurred that is Ppu uo Psp Ppu uo s 310 The annual strike probability is Psp At NAr 311 At is the total area in square miles impacted in N years by tornadoes in the region Ar of interest that is the product of the expected area of a tornado in the region Ar by the total number of tornado events that occurred in that region in N years and N is the number of years of record The probability of exceeding a speed uo given that a tornado has occurred is Ppu uo s Auuo At 312 where Auuo is the total area impacted by wind speeds greater than uo see also 18 It is assumed in 17 that Ppu uos is described by a Weibull distribution For the probability of exceedance of a speed uo within a finitesize structure see 17 Uncertainties in the estimation of the tornado strike probabilities and conditional probabilities of tornado wind speeds are due to errors in the tornado footprint mod eling as a rectangle and in the estimation of the length width and area of the tornado footprint the assumption that the structures characteristic dimension is 200 ft and the assignment of an incorrect EF enhanced Fujita scale to tornadoes in the database being used Adjustments for those errors are discussed in 19 Recommendations in 17 of tornado design wind speeds with 105 106 and 107year MRIs for the three regions defined in Figure 35 are based on the spatially averaged estimated speeds for 2 longitudelatitude boxes and are shown in Table 31 The American Nuclear Society ANSIANS232011 R2016 Standards regionaliza tion of tornado wind speeds 22 differs somewhat from the regionalization of Figure 35 343 DesignBasis Tornado for Nuclear Power Plants The NRC Regulatory Guide 176 Revision 1 March 2007 21 provides guidance on designbasis tornado and designbasis tornadogenerated missiles for nuclear power k k k k 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 124 122 120 118 116 114 112 110 108 106 104 102 100 98 96 94 92 90 88 86 84 82 80 78 76 74 72 70 68 66 0 144 130 158 142 91 103 159 168 164 198 180 199 186 186 183 213 195 169 191 117 47 45 43 41 39 37 35 33 31 29 27 25 47 45 43 41 39 37 35 33 31 29 27 25 138 127 148 137 107 124 132 150 154 177 194 200 204 205 212 217 213 212 208 183 210 95 166 84 119 113 71 148 114 147 148 151 162 186 197 201 211 216 221 223 222 220 215 212 215 184 203 176 179 169 172 179 170 133 0 100 0 0 123 146 137 132 158 177 198 207 219 225 225 228 230 223 226 225 220 221 222 204 197 201 200 152 151 130 148 83 146 125 142 123 120 177 191 210 223 229 230 225 227 227 226 225 215 202 203 200 202 172 162 142 81 0 108 130 117 125 151 161 192 215 225 232 229 220 219 224 223 218 205 189 194 201 194 159 134 140 0 138 142 124 112 136 163 194 217 229 235 228 226 227 226 229 219 200 208 206 193 195 76 160 153 140 141 148 72 144 164 198 212 221 226 227 227 226 228 228 219 213 207 214 188 135 140 142 148 145 151 161 187 200 207 214 222 221 223 224 219 215 210 195 193 123 182 183 200 208 211 211 204 211 205 203 198 199 203 186 196 173 212 198 181 192 227 191 Region 1 230 mph Region 2 200 mph Region 3 160 mph Figure 35 Recommended design wind speeds with 107 years mean recurrence intervals 17 k k k k 70 3 Extreme Wind Speeds Table 31 Recommended tornado design wind speeds Wind Speed mph Mean Recurrence Interval years Region I Region II Region III 105 160 140 100 106 200 170 130 107 230 200 160 Table 32 Designbasis tornado wind field characteristics 21 Region Maximum Wind Speed m s1 mph Translational Speed m s1 mph Maximum Rotational Speed m s1 mph Radius of Maximum Rotational Speed m ft Pressure Drop mb psi Pressure Drop Rate mb s1 psi s1 I 103 230 21 46 82 184 457 150 83 12 37 05 II 89 200 18 40 72 160 457 150 63 09 25 04 III 72 160 14 32 57 128 457 150 40 06 13 02 plants in the contiguous United States For the regions shown in Figure 35 Table 32 reproduces the characteristics of the designbasis tornadoes provided in 21 and based on the Rankine model combined with a translational velocity Chapter 27 Designbasis tornadogenerated missiles are considered in Chapter 28 For tornado vertical wind speeds see Chapter 27 Wind field characterization of tor nadoes in the ANSIANS232011 R2016 Standard 22 differs in some respects to that of 21 References 1 Masters FJ Vickery PJ Bacon P and Rappaport EN 2010 Toward objective standardized intensity estimates from surface wind speed observations Bulletin of the American Meteorological Society 91 16651681 2 ASCE Minimum design loads for buildings and other structures ASCESEI 716 in ASCE Standard ASCESEI 716 Reston VA American Society of Civil Engineers 2016 3 Lombardo FT Main JA and Simiu E 2009 Automated extraction and classifi cation of thunderstorm and nonthunderstorm wind data for extremevalue analysis Journal of Wind Engineering and Industrial Aerodynamics 97 120131 4 Batts M E Russell L R Cordes M R Shaver J R and Simiu E Hurricane wind speeds in the United States Building Science Series 124 National Bureau of Standards Washington DC 1980 httpswwwnistgovwind 5 Batts ME Russell LR and Simiu E 1980 Hurricane wind speeds in the United States Journal of the Structural DivisionASCE 106 20012016 httpswwwnist govwind k k k k References 71 6 Vickery PJ Wadhera D and Twisdale LA Technical basis for regulatory guidance on designbasis hurricane wind speeds for nuclear power plants NUREGCR7005 US Nuclear Regulatory Commission Washington DC 2011 7 Vickery PJ Wadhera D Twisdale LA Jr and Lavelle FM 2009 US hurricane wind speed risk and uncertainty Journal of Structural Engineering 135 301320 8 ASCE Minimum design loads for buildings and other structures ASCESEI 710 in ASCE Standard ASCESEI 710 Reston VA American Society of Civil Engineers 2010 9 Vickery P and Twisdale L 1995 Prediction of hurricane wind speeds in the United States Journal of Structural Engineering 121 16911699 10 Georgiou PN Davenport AG and Vickery BJ 1983 Design wind speeds in regions dominated by tropical cyclones Journal of Wind Engineering and Industrial Aerodynamics 13 139152 11 ASNZS Structural design actions wind actions commentary supplement to ASNZS 117022002 Sydney Wellington Standards Australia International Stan dards New Zealand 2002 12 Lieblein J Efficient Methods of ExtremeValue Methodology NBSIR 74602 National Bureau of Standards Washington DC 1974 httpswwwnistgovwind 13 Gumbel EJ 1958 Statistics of Extremes New York Columbia University Press 14 Pintar AL Simiu E Lombardo F T and Levitan M L Maps of NonHurricane NonTornadic Wind Speeds with Specified Mean Recurrence Intervals for the Con tiguous United States Using a TwoDimensional Poisson Process Extreme Value Model and Local Regression NIST Special Publication 500301 National Institute of Standards and Technology Gaithersburg 2015 httpswwwnistgovwind 15 Yeo D 2014 Generation of large directional wind speed data sets for estimation of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 16 US Tornado Climatology National Climatic Data Center Asheville NC 2008 17 Ramsdell JV Jr and Rishel JP Tornado Climatology of the Contiguous United States AJ Buslik Project Manager NUREGCR4461 Rev 2 PNNL15112 Rev 1 Pacific Northwest National Laboratory 2007 18 Thom HCS 1963 Tornado probabilities Monthly Weather Review 91 730736 19 Reinhold TA and Ellingwood BR Tornado Risk Assessment NUREGCR2944 US Nuclear Regulatory Commission Washington DC 1982 20 Twisdale LA and Dunn WL Tornado Missile Simulation and Design Method ology Vols 1 and 2 EPRI NP2005 Electric Power Research Institute Palo Alto California 21 US Nuclear Regulatory Commission Regulatory Guide 176 DesignBasis Tornado and Tornado Missiles for Nuclear Power Plants Revision 1 2007 22 American Nuclear Society ANSIANS232011 Estimating tornado hurricane and extreme straight wind characteristics at nuclear facility sites La Grange Park Illinois reaffirmed Jun 29 2016 k k k k 73 4 Bluff Body Aerodynamics Aerodynamics is the study of air flows that interact with solid bodies Streamlined bodies have shapes that help to reduce drag forces Bodies that are not streamlined are called bluff Bluff body aerodynamics of interest in structural engineering applications is asso ciated with atmospheric flows which are incompressible owing to their relatively low speeds With rare exceptions associated with stably stratified flows see Section 113 atmospheric flows of interest in structural design are turbulent In addition to the turbu lence present in atmospheric flows signature turbulence is generated by the presence of the body in the flow Turbulence significantly complicates the study of bluff body aerodynamics Certain types of engineering structures can be subjected to aerodynamic forces gener ated by structural motions These motions called selfexcited are in turn affected by the aerodynamic forces they generate The structural behavior associated with selfexcited motions is termed aeroelastic and is considered in Part III of the book As pointed out by Roshko the problem of bluffbody flow remains almost entirely in the empirical descriptive realm of knowledge 1 Although much progress is being made in Computational Fluid Dynamics CFD and its application to wind engineering Computational Wind Engineering or CWE its application in structural engineering practice remains limited 2 Indeed the simulation of flows over bluff bod ies in turbulent shear flows is a formidable problem and the approximations required in modeling the flow numerically can produce results that differ significantly and unpre dictably from each other depending upon those approximations To follow Schuster 3 conservative CFD applications are based on the paradigm Develop Validate Apply wherein endusers apply validated software to problems that fall within or at least not too far from its range of validation As pointed out in 3 a modified paradigm Develop Apply Validate may be required under certain circumstances This paradigm entails large uncertainties that must be accounted for how CFD methods may be applied and ultimately developed and validated under those circumstances is discussed in 3 in the context of NASA applications In a civil engineering context an informal Develop Apply Validate approach has been implicit in lowrisk CFD applications wherein the effect of relatively large uncertainties is tolerable for example the prediction of wind flows that cause easily remediable pedestrian discomfort around buildings see Chapter 15 Section 41 reviews fundamental fluid dynamics equations Section 42 considers flows in a curved path and vortex flows Section 43 discusses boundary layers and flow Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 74 4 Bluff Body Aerodynamics separation Section 44 is devoted to wake and vortex formations in twodimensional 2D flow Section 45 concerns pressure lift drag and moment effects on 2D bodies Section 46 presents information on flow effects in three dimensions 41 Governing Equations 411 Equations of Motion and Continuity Consider a fixed elemental volume dV in a fluid The velocity vector is expressed as u u1i1 u2i2 u3i3 41 where i1 i2 i3 are unit vectors along the usual three fixed orthogonal axes The force acting on the fluid contained in the volume dV consists of two parts The first part is the body force caused by gravity and is denoted by F𝜌dV where 𝜌 is the fluid density The second part is due to the net action on the fluid of the internal stresses 𝜎ij i j 1 2 3 For example the contribution to this action of the normal stress 𝜎11 see Figure 41 is 𝜎11dx2dx3 𝜎11 𝜎11 x1 dx1 dx2dx3 𝜎11 x1 dx1dx2dx3 𝜎11 x1 dV 42 It can be similarly shown that the net force component in the i direction due to the action of all stresses 𝜎ij is 3 j1 𝜎ij xj dV 43 Denoting the components of F by Fi i 1 2 3 the force balance equations given by Newtons second law are Dui Dt 𝜌 dV Fi 𝜌 dV 3 j1 𝜎ij xj dV i 1 2 3 44 dx3 x1 dx2 dx2dx3 σ11dx2dx3 dx1 dx1 σ11 σ11 Figure 41 Forces along the i direction on an elementary volume of fluid k k k k 41 Governing Equations 75 where the operator DDt known as the substantial or material derivative is defined as follows D Dt t 3 i1 xi dxi dt t 3 i1 ui xi 45 Since Eq 44 is true for all volume elements it may be divided by the factor dV and the equations of motion of a fluid particle can be written in component form as 𝜌 Dui Dt 𝜌Fi 3 j1 𝜎ij xj 46 We now consider the principle of mass conservation which states that the rate at which mass enters a system is equal to the rate at which mass leaves the system If 𝜌 is constant mass conservation can be shown to imply 3 i1 ui xi 0 47 Equation 47 is called the equation of continuity 412 The NavierStokes Equation Unlike a solid under static conditions a fluid cannot support any stresses other than normal pressures However in dynamic situations it may support shear in a timedependent manner In most fluidmechanical applications it is adequate to assume that the stresses involved are normal pressures or ascribable to viscosity Fluids with internal shear stress proportional to the rate of change of velocity with distance normal to that velocity are termed viscous or Newtonian For example the shear stress 𝜎12 in a simple 2D flow is expressed as 𝜎12 𝜇 u1 x2 48 where the proportionality factor is defined as the fluid viscosity The units of viscosity are 𝜇 force area length velocity force time length2 mass length time Typical values of 𝜇 for air and water at 15C are 𝜇air 1783 105 kg m1 s1 𝜇water 1138 103 kg m1 s1 By distinguishing in the stress tensor 𝜎ij at a fluid point the normal stress p ie pressure and the deviatoric stress defined as dij 2𝜇 eij 1 3𝛿ij 3 k1 ekk i j 1 2 3 49 k k k k 76 4 Bluff Body Aerodynamics where eij 1 2 ui xj uj xi 410 and 𝛿ij 1 i j 0 i j 411 The following expression for the stress 𝜎ij can be obtained 𝜎ij p 𝛿ij 2𝜇 eij 1 3𝛿ij 3 k1 ekk 412 Using the expressions for stress in a Newtonian fluid results in the equations of motion known as NavierStokes equations 𝜌 Dui Dt 𝜌Fi p xi 𝜇 3 j1 2ui x2 j 1 3 3 k1 uk xk xi 413 For an incompressible fluid Eq 47 Eq 413 can be written as ui t uj ui xj 1 𝜌 p xi Fi 𝜈 3 j1 2ui x2 j 414 where 𝜈 𝜇𝜌 is called the kinematic viscosity For air and water at 15C 𝜈air 1455 105 m2 s1 𝜈water 1139 103 m2 s1 415 413 Bernoullis Equation Consider an incompressible inviscid flow experiencing negligible body forces If the flow is steady the fluid element of Figure 42 is subjected in the direction of the stream line ie along the tangent at any instant to the flow velocity to the force p dy dz the force p dp dy dz and the inertial force 𝜌 dx dy dz dU dt 𝜌 dx dt dy dz dU 𝜌 dy dz U dU 416 where dxdt U The equation of equilibrium among those three forces yields dp 𝜌 U dU and upon integration 1 2𝜌U2 p const 417 Equation 417 is known as Bernoullis equation The quantity 12𝜌U2 has the dimen sions of pressure and is called dynamic pressure The quantity dpdx is called the pressure gradient in the x direction k k k k 42 Flow in a Curved Path Vortex Flow 77 p dy dz p dpdy dz dz dx dy ρ dx dy dz dUdt Figure 42 Flowinduced pressures and inertial force on an elemental volume of a fluid in motion Consider the streamline between two points one of which is the stagnation point on the windward face of a body immersed in the flow where U 0 while the other is located in the undisturbed flow far upstream of the body where the static pressure is p0 and the velocity is U0 The pressure at the stagnation point ie the stagnation pressure is pst p0 1 2𝜌U2 0 418 Bernoullis equation is widely used to interpret the relation between pressure and velocity in atmospheric and wind tunnel flows Detailed comments on Bernoullis equation and its applicability including to viscous flows are provided in section 35 of 4 42 Flow in a Curved Path Vortex Flow Consider a 2D flow between two locally concentric streamlines with radii of curvature r and r dr Figure 43 For the flow to maintain its curved path with tangential velocity U at radius r it must experience an acceleration U2r toward the center of curvature Let the pressure acting on the fluid element under consideration be denoted by p The pressure differential between the streamlines at radii r and r dr which is responsible for this acceleration is dp The equation of motion for a fluid element shown in Figure 43 is then dpdA 𝜌 dr dAU2 r 419 where dA is the area of the element in a plan normal to the plan of Figure 43 Therefore dp 𝜌 U2 dr r 420 Bernoullis equation allows the calculation of the pressure along a curved path of the flow In particular one may consider the case wherein the flow is circular and the value of p in Eq 417 is the same on all streamlines This is the case of vortex flow Differentiation of Eq 417 yields 𝜌U dU dr dp dr 0 421 From Eqs 420 and 421 there follows dU U dr r 422 k k k k 78 4 Bluff Body Aerodynamics dA U pdA r pdA STREAMLINES dr dr dpdA dr Figure 43 Flow in a curved path Integration of Eq 422 yields Ur const 423 This law states for an incompressible and inviscid fluid the theoretical hyperbolic rela tion between radius r and tangential velocity U in a free vortex In an actual free vortex however the effects of viscosity are present as well Viscosity locks together a portion of the fluid near the center and causes it to rotate as a rigid body instead of as an inviscid fluid described by Eq 423 Thus at the center of a free vortex the velocity increases with radius whereas according to Eq 423 it decreases with increasing r This decrease actually occurs outward from a transition region in which U attains its maximum value The value of U in this region depends upon the fluid viscosity and the total angular momentum of the vortex Figure 44 illustrates qualitatively the pressure and velocity dependence on radius in a free vortex occurring in a real fluid The free vortex is of interest in many flows that occur in engineering applications For example atmospheric flows along curved isobars are described by generalizations of Eq 420 These have been described in Chapter 1 where additional Coriolis forces have been included 43 Boundary Layers and Separation The viscosity of air at normal atmospheric pressures and temperatures has a relatively small value Nonetheless in some circumstances this small viscosity plays an important role In particular a consequential effect of the viscosity is the formation of boundary layers k k k k 43 Boundary Layers and Separation 79 U U Cr O p r O TRANSITION r Figure 44 Pressure and velocity dependence upon radius in a vortex flow Figure 45 Typical boundarylayer velocity profile Height Velocity Consider an air flow over and along a stationary smooth surface It is an experimental fact that the air in contact with the surface adheres to it This no slip condition causes a retardation of the air motion in a layer near the surface called the boundary layer Within the boundary layer the velocity of the air increases from zero at the surface to its value in the outer flow as opposed to the boundarylayer flow A boundarylayer velocity profile is shown in Figure 45 Since air has mass its motion exhibits inertial effects in accordance with Newtons second law and its application to fluids the NavierStokes equations Viscous flows are therefore subjected to both inertial and viscous effects The relation between these two k k k k 80 4 Bluff Body Aerodynamics effects is an index of the type of flow phenomena that may be expected to occur The nondimensional parameter Re called the Reynolds number is a measure of the ratio of inertial to viscous forces For example consider a volume of fluid with a typical dimen sion L By Bernoullis theorem the net pressure p p0 caused by the fluid velocity U is in the order of 12𝜌U2 and creates inertial forces on the fluid element enclosed by that volume in the order of 𝜌U2L2 The viscous stresses on the element are in the order of 𝜇UL so viscosityrelated forces are in the order of 𝜇ULL2 𝜇UL The ratio of inertial to viscous forces is then in the order of Re 𝜌U2L2 𝜇UL 𝜌UL 𝜇 UL 𝜈 424 A useful approximate value of the Reynolds number in air at about 20 C and 760 mm atmospheric pressure is 67000 UL If Re is large inertial effects are predominant if Re is small viscous effects predominate L is a representative dimension of the body being considered Boundarylayer separation occurs if the kinetic energy of the fluid particles in the lower region of the boundary layer are no longer sufficient to overcome the pressures that increase in the direction of the flow and thus produce adverse pressure gradients The flow in that region then becomes reversed that is separation is taking place Figure 46 Shear layers generate discrete vortices that are shed into the wake flow behind the bluff body Figure 47 Such vortices can cause high suctions near separation points such as corners or eaves A flow around a building with sharp edges is shown schematically in Figure 48 The injection by turbulent fluctuations of highmomentum particles from the outer layer into the zone of separated flow can produce flow reat tachment Figure 49 shows an ageold streamlining measure aimed at reducing flow separation and strong local roof suctions near the ridge under winds normal to the end wall A visualization of flow separation for a bluff shape and of the turbulent flow in the separation zone is shown in Figure 410a in which the separation zone starts close to windward edge If the shape of the deck is streamlined as opposed to being bluff the separation zone is narrower and the turbulent flow about the upper face of the deck almost disappears Figure 410b Figure 411a shows the visualization of flow around a counterclockwise spinning base ball moving from left to right Figure 411b is a schematic of the forces acting on the baseball with velocity U and angular velocity 𝜔 The relative velocity of the flow with respect to the ball is directed from right to left Entrainment of fluid due to friction at the surface of the spinning body increases the relative flow velocities with respect to the Body surface flow boundary Shear layer Reverse flow Outer flow Boundary layer Z U Figure 46 Velocity profile in the boundary layer and in the separation zone of a flow near a curved body surface Source After 5 k k k k 43 Boundary Layers and Separation 81 Figure 47 Flow separation at corner of obstacle Separation points Reattachment point Shear layer Wake Separation zone Figure 48 Flow about a building with sharp edges Source After 5 Figure 49 Three thatched cottages by a road Rembrandt van Rijn 16061669 photo Nationalmuseum Sweden Source Count Kessin collection k k k k 82 4 Bluff Body Aerodynamics a b Figure 410 Visualization of water flow over a a model bridge deck section and b a partially streamlined model bridge deck section Flow velocity is oriented from left to right Source Courtesy of the National Aeronautical Establishment National Research Council of Canada body near its top and decreases them near the bottom By virtue of Bernoullis equation the static pressures are therefore lower near the top and higher near the bottom The flow asymmetry induced by spinning therefore results in a net lift force denoted by FM in Figure 411b called the Magnus force In different aerodynamic contexts flow asym metries due to body motions can under certain conditions be the cause of galloping and other aeroelastic motions 44 Wake and Vortex Formations in TwoDimensional Flow In the following discussion the flow is assumed to be smooth laminar and 2D that is independent of the coordinate normal to the cross section of the body k k k k 44 Wake and Vortex Formations in TwoDimensional Flow 83 a FM b U ω FG FD Figure 411 a Flow around a spinning baseball Source Courtesy of the National Institute of Standards and Technology b Schematic showing forces acting on baseball with velocity U and angular velocity 𝜔 Source Reproduced from 6 with the permission of the American Association of Physics Teachers k k k k 84 4 Bluff Body Aerodynamics a L Re 03 c d Re 250 Re 1000 b Re 10 Figure 412 a Flow past a sharpedged plate Re 03 b Flow past a sharpedged plate Re 10 c Flow past a sharpedged plate Re 250 d Flow past a sharpedged plate Re 1000 Consider a sharpedged flat plate shown in Figure 412a At a very low Reynolds num ber eg Re 03 based on the characteristic length L shown in Figure 412a the flow turns the sharp corner and follows both front and rear contours of the plate At Re 10 obtained by increasing the flow velocity over the same plate the flow separates at the corners and creates two large symmetric vortices that remain attached to the back of the plate Figure 412b At Re 50 the symmetrical vortices are broken and replaced by cyclically alternating vortices that form by turns at the top and at the bottom of the plate and are swept downstream Figure 412c A full cycle of this phenomenon is defined as the activity between the occurrence of some instantaneous flow configuration about the body and the next identical configuration At Re 1000 Figure 412d the inertia forces predominate large distinct vortices have little possibility of forming and instead a gen erally turbulent wake is formed behind the plate its two outer edges forming each a shear layer consisting of a long series of smaller vortices that accommodate the wake region to the adjacent smooth flow regions These results dramatically illustrate the changes in the flow with Reynolds number proceeding from predominantly viscous effects to predominantly inertial effects Next the renowned case of 2D flow about a circular cylinder Figure 413 is briefly examined At extremely low Reynolds number based on the diameter of the cylinder k k k k 44 Wake and Vortex Formations in TwoDimensional Flow 85 a c d e TURBULENT WAKE TURBULENT WAKEnarrower VON KARMAN VORTEX TRAIL Re 1 30 Re 5000 5000 Re 200000 Re 200000 b Re 20 Figure 413 Flow past a circular cylinder a Re 1 b Re 20 c 30 Re 5000 d 5000 Re 200 000 e Re 200 000 Source From 6 by permission of the author and the American Journal of Physics Re 1 the flow assumed laminar as it approaches remains attached to the cylinder throughout its complete periphery as shown in Figure 413a At Re 20 the flow form remains symmetrical but flow separation occurs and large wake eddies are formed that reside near the downstream surface of the cylinder as suggested in Figure 413b For 30 Re 5000 alternating vortices are shed from the cylinder and form a clear vor tex street downstream This phenomenon was first reported by Bénard in 1908 7 in the Englishspeaking world its discovery is attributed to von Kármán who reported k k k k 86 4 Bluff Body Aerodynamics it in 1911 8 the alternating vortices are universally referred to as a von Kármán street although some facetious aerodynamicists use the term boulevard Bénard The finer details of this striking occurrence are still not fully understood and have been the object of both experimental and theoretical studies eg 9 For 30 Re 5000 say there is established behind the cylinder a staggered stable arrangement of vortices that moves off downstream at a velocity somewhat less than that of the surrounding fluid As the Reynolds number increases into the range 5000 Re 200 000 the attached flow upstream of the separation flow is laminar In the separated flow 3D patterns are observed and transition to turbulent flow occurs in the wake farther downstream from the cylinder for the lower Reynolds numbers and nearer the cylinder surface as the Reynolds numbers increase For the larger Reynolds numbers in this range the cylinder wake undergoes transition immediately after separation and a turbulent wake is produced between the separated shear layers Figure 413d Beyond Re 200 000 Figure 413e the wake narrows appreciably resulting in less drag Other bluff bodies notably prisms with triangular square rectangular and other cross sections give rise to analogous vortexshedding phenomena Figure 414 The pronounced regularity of such wake effects was first reported by Strouhal 11 who pointed out that the vortex shedding phenomenon can be described in terms of a nondimensional number the Strouhal number St NsD U 425 where Ns is the frequency of full cycles of vortex shedding D is a characteristic dimension projected on a plane typically normal to the wind velocity and U is the Figure 414 Flow around a rectangular cylinder Re 200 Source Reprinted from 10 with permission from Elsevier k k k k 44 Wake and Vortex Formations in TwoDimensional Flow 87 105 00 REYNOLDS NUMBER Smooth 106 107 01 02 03 STROUHAL NUMBER 04 05 kD 00003 kD 00012 kD 00101 Figure 415 Relation between the Strouhal number and the Reynolds number for circular cylinder Source Reprinted from 12 with permission from Elsevier velocity of the oncoming flow assumed laminar The Strouhal number depends upon the crosssectional shape of the cylindrical body enveloped by the flow Figure 415 shows the relation of St to Re for a circular cylinder in the range 105 Re 107 Coherent vortex shedding was noted to disappear at Reynolds numbers beyond 4105 and contrary to results reported by some observers and summarized in 13 there was no significant increase of the Strouhal number Table 41 lists values of St for different crosssectional shapes for Reynolds numbers in the clear vortexshedding range the approach flow being laminar Figure 416 shows a vortex trail made visible by clouds over Jan Mayen Island Arctic Ocean For additional material on vortex trails over oceans see also 15 As pointed out in 16 the establishment of a vortex trail can be inhibited by a split ter plate as shown in Figure 417 The action of the plate is to prevent flow crossover between the two rows of vortices aft of the cylinder and thus to quiet the entire wake flow Qualitatively the presence of the plate has the same type of effect as lengthening the body in the stream direction and causing it to approach the form of a symmetric air foil Following this type of approach it can be seen that elongated bodies oriented with their long dimension parallel to the main flow tend to elicit relatively narrow wakes If flows about square and rectangular prisms at high Reynolds numbers are compared Figure 418 the square is seen to produce flow separation followed by a wide turbu lent wake whereas the more elongated shapes may exhibit separation at leading corners followed downstream by flow reattachment and finally once more by flow separation at the trailing edge In contrast to the case of Figure 418b if the rectangle is placed with its long dimension normal to the flow the wake exhibits a strong vortexshedding characteristic followed at higher Re by a turbulent wake similar to that produced by the sharpedged plate Figures 412c and d k k k k 88 4 Bluff Body Aerodynamics Table 41 Strouhal number for a variety of shapes Wind Wind Profile dimensions in mm Profile dimensions in mm t 20 t 10 t t 50 Value of St 0120 0147 125 0137 Value of St 0156 Cylinder 11800 Re 19100 0160 0145 50 25 50 50 125 t 10 t 10 0144 0142 0145 0147 0134 0131 0137 50 50 t 05 t 10 25 0120 0150 125 125 50 125 t 10 50 25 25 50 t 15 t 10 50 100 t 15 t 10 0145 50 125 125 25 25 0121 0143 t 10 t 10 0140 0153 50 0114 0145 0200 25 25 25 25 25 25 0135 t 10 t 10 0145 0168 50 25 25 25 125 25 Source From 14 ASCE k k k k 45 Pressure Lift Drag and Moment Effects on TwoDimensional Structural Forms 89 Figure 416 Satellite photo of Jan Mayen Island Arctic Ocean Source Credits NASACSFCLaRCJPL MISR Team Figure 417 Effect of splinter plate on flow behind a circular cylinder Source After 16 45 Pressure Lift Drag and Moment Effects on TwoDimensional Structural Forms Figure 419 shows a section of a bluff body immersed in a flow of velocity U The flow will develop local pressures p over the body in accordance with the Bernoulli equation 1 2𝜌U2 p const 426 where the constant holds along a streamline and U is the velocity on the streamline immediately outside the boundary layer that forms on the bodys surface The integration of the pressures over the body results in a net force and moment The components of the force in the alongwind and acrosswind directions are called drag and lift respectively The drag lift and moment are affected by the shape of the body the Reynolds number and the incoming flow turbulence The body may be designed with the purpose of minimizing drag and maximizing lift resulting in an airfoillike shape In many civil engineering applications the shape of the body is typically fixed by other design objectives than purely aerodynamic ones k k k k 90 4 Bluff Body Aerodynamics WAKE SEPARATION a WAKE SEPARATION REATTACHMENT SEPARATION b Figure 418 Flow separation and wake regions square and rectangular cylinders FL FD Figure 419 Lift and drag on an arbitrary bluff body Nevertheless the lift drag and moment induced by the fluid flows will remain of strong interest because these are effects that must be designed against It is usual to refer to all pressures measured on a structural surface to the mean dynamic pressure 1 2𝜌U2 of the far upstream wind or the free stream wind at a distance from the structure Thus nondimensional pressure coefficients Cp are defined as Cp p p0 12𝜌U2 427 k k k k 45 Pressure Lift Drag and Moment Effects on TwoDimensional Structural Forms 91 where U is the mean value of the reference wind speed and p p0 is the pressure difference between local and far upstream pressure p0 Such nondimensional forms enable the transfer of model experimental results to full scale and the establishment of reference values for cataloging the aerodynamic properties of given geometric forms Similarly the net aerodynamic lift and drag forces per unit span FL and FD in the acrosswind and alongwind direction respectively can be rendered dimensionless and expressed in terms of lift and drag coefficients CL and CD CL FL 12𝜌U2B 428 CD FD 12𝜌U2B 429 where B is some typical reference dimension of the structure For the net flowinduced moment M about the elastic center the corresponding coefficient is CM M 12𝜌U2B2 430 Figure 420 shows the dependence of the mean drag coefficient CD of circular cylin ders immersed in smooth flow CD drops sharply in the range 2 105 Re 5 105 This is called the critical region and corresponds to the transition from laminar to turbulent flow in the boundary layer that forms on the surface of the cylinder The turbulent mix ing that takes place in the boundary layer helps transport fluid with higher momentum toward the surface of the cylinder Separation then occurs much farther back and the wake consequently narrows producing a time averaged CD that is only about one third of its highest value As Re increases into the supercritical and then the transcritical range Re 4106 CD increases once more but remains much lower than its subcritical val ues According to 12 drag coefficients in the transcritical range are about 25 lower than those indicated in Figure 420 Figure 421 depicts a typical distribution of the mean pressure coefficient about the circular cylinder in smooth flow as a function of angular position The pressures corre sponding to 𝜃 0 and 𝜃 180 are referred as the stagnation point and the base pres sure respectively 105 0 Reynolds number Re 106 107 05 10 Subcritical Supercritical Transcritical Critical Drag coefficient CD 15 Figure 420 Evolution of the mean drag coefficient with Reynolds number for a circular cylinder Source After 13 Courtesy of National Physical Laboratory UK k k k k 92 4 Bluff Body Aerodynamics 0 30 θ DEGREES θ 60 180 120 20 Cp 10 0 10 U Re 67 105 Re 84 106 Re 11 105 Figure 421 Influence of Reynolds number on pressure distribution over a circular cylinder Source After 17 Figure 422 illustrates the evolution with Reynolds number of the mean drag coef ficient of a square cylinder in smooth flow during successive modifications of its cor ners Only the sharpcornered square exhibits practically unchanging drag with change of Reynolds number This is accounted for by the early separation of the flow at the upstream corners and the shortness of the afterbody that practically prevents flow reat tachment Squares with rounded corners tend to possess the same kind of critical region as the circular cylinder Note also for the circular cylinder the dependence of the drag upon the roughness of the cylinder surface see 19 Because of such effects certain features of the flow in tests of wind tunnel models can be assumed to be independent of the Reynolds number This will be the case in some situations in which the flow breaks cleanly away at some identifiable flows past a curved body eg a circular cylinder this assumption is not warranted Table 42 shows mean values of CD and CL obtained in smooth flow for sectional shapes used in construction Experiments have shown that for the shapes of Table 42 the effects of turbulence are small The rms value of the fluctuating normal force coefficient CNrms on a square cylinder with side B is shown in Figure 423 as a function of angle of attack 𝛼 with respect to the mean wind direction Here the turbulence with longitudinal integral scale 14B lateral integral scale 04B and 10 turbulence intensity lowers the highest normal force below and raises the lowest normal force slightly above the respective values in laminar flow For the effects of turbulence on the aerodynamics of a square prism see also 21 For a study of unsteady forces acting on rigid circular cylinders see 22 k k k k 46 Representative Flow Effects in Three Dimensions 93 c b Sanded surface Smooth surface 105 104 107 2 4 8 2 4 8 2 4 8 106 12 U 08 04 kh 002 kh 0002 Re kh 0001 rh 05 circular section kh 0007 h 12 U 08 04 rh 0167 h a U 22 18 rh 0021 h Figure 422 Influence of Reynolds number corner radius and surface roughness on drag coefficient square to circular cylinders r is the corner radius k is the grain size of sand Source After 18 For members with rectangular cross section the drag force depends upon i the ratio bh between the sides of the cross section and ii the turbulence in the oncoming flow If the ratio bh is small no flow reattachment occurs Depending upon its intensity the turbulence can enhance the flow entrainment in the wake and therefore cause stronger suctions and larger drag Figure 424a If the ratio bh is sufficiently large the turbulence can cause flow reattachment that could not have occurred in smooth flow and thus results in lower drag Figure 424b The dependence of the drag coefficient upon alongwind turbulence intensity in flow with homogeneous turbulence is shown for two ratios bh in Figure 425 The effect of turbulence in the case of bodies with rounded shapes is essentially to reduce the Reynolds number at which the critical region sets in The roughness of the body surface Figure 422 has a similar effect since it promotes turbulence in the boundary layer that forms on the body surface Fluid particle moments with higher momentum are thus transported into the lower regions of the boundary layer and help to overcome the adverse pressure gradient responsible for flow separation 46 Representative Flow Effects in Three Dimensions Most flows have a 3D character For example if a hypothetical laminar flow consist ing of an air mass displaced uniformly as a single unit encounters an object it will be k k k k 94 4 Bluff Body Aerodynamics Table 42 Twodimensional drag and lift coefficients Profile and wind direction 203 0 CD CD CL CL 196 201 0 204 0 181 0 20 03 183 207 199 009 162 048 201 0 199 119 219 0 Source From 14 k k k k 46 Representative Flow Effects in Three Dimensions 95 0 0 ANGLE OF ATTACK α α 15 5 10 45 20 25 30 35 40 04 10 RMS COEFFICIENT OF FLUCTUATING NORMAL FORCE CNrms 08 06 02 12 14 U SMOOTH STREAM TURBULENT STREAM CNrms12pU2B B Figure 423 Variation of the coefficient of fluctuating normal force CNrms with angle of attack for a square prism Source From 20 reproduced with permission Higher drag Lower drag b h 05 h Higher drag Lower drag a h 01 h Figure 424 Separation layers in smooth flow solid line and in turbulent flow interrupted line Source After 23 k k k k 96 4 Bluff Body Aerodynamics b h 0 0 4 8 uʹ 2 U ½ 12 16 20 10 20 050 10 30 h b CD 40 Figure 425 Dependence of drag coefficient on turbulence intensity Source After 23 diverted in several directions Also the passage of such a flow along a surface sets up boundaryvelocity gradients And threedimensionality is clearly inherent in turbulent flows Although the general equations for fluid flow remain available for application in structural engineering practice most aerodynamic studies rely partially or fully on experiment 461 Cases Retaining TwoDimensional Features The success of the 2D flow models discussed in the previous section has in a few cases been considerable because some actual flows retain certain 2D features at least to a first approximation Consider for example the case of a long rod of square cross section in an air flow with uniform mean velocity normal to one face Except near the ends of the rod the mean flow may in some cases be considered for practical pur poses as 2D However the effects associated with flow fluctuations are not identical in different strips the differences between events that take place at any given time increas ing with separation distance This is shown in Figure 426 for the pressure difference between centerlines of top and bottom faces of the rod under both laminar and turbu lent approaching flow It is observed that the threedimensionality of the flow manifests itself through spanwise loss of correlation rAB between pressure differences measured respectively between point A at section A and point B of section B this correlation loss being accentuated when turbulence is present in the oncoming flow From this example one may infer that fluctuating phenomena including vortex shedding cannot normally be expected to be altogether uniform along the entire length of a cylindrical body even if the flow has uniform mean speed and the body is geometrically uniform The animation of Figure 427 based on wind tunnel measurements in turbulent boundarylayer flow clearly demonstrates the imperfect spatial coherence of pressures on a lowrise struc ture Investigations reported in 24 were among the first to account explicitly for the imperfect spatial coherence of aerodynamic pressures on lowrise structures k k k k 46 Representative Flow Effects in Three Dimensions 97 0 0 2 TURBULENT STREAM SMOOTH STREAM B B D rAB 4 6 rABD 8 12 10 02 CORRELATION COEFFICIENT RAB 04 06 10 08 A A U Figure 426 Spanwise correlation of the fluctuating pressure difference across the center line of a long square cylinder for flow normal to a face Source From 20 reproduced with permission Wind direction Model S32 Cpt Wind angle 0 Scale 1 100 in suburban 100 0 50 100 150 80 60 40 20 0 0 10 20 30 40 50 Figure 427 Fluctuating wind pressure model for 100 ft 200 ft 32 ft building in suburban terrain gable roof with 124 slope Source Based on 1 100 model scale boundarylayer wind tunnel simulation University of Western Ontario animation created by Dr A Grazini Mean wind speed normal to end walls Note asymmetry of pressures with respect to vertical plane containing ridge line Video available at httpswwwnistgovwind k k k k 98 4 Bluff Body Aerodynamics Wind a b IIII IV y v0 I II III IV V VI Wind v v h h y v0 l b c d e IIII IV h l b II V VI 0 10 20 Cp VI II V Figure 428 Summary of model tests in smooth and boundarylayer flow Source From 25 k k k k 46 Representative Flow Effects in Three Dimensions 99 In practice mean flow conditions upwind of tall slender structures are usually not uniform indeed in the atmospheric boundary layer the mean flow velocity increases with height Also certain structures eg stacks are not geometrically uniform These important features in addition to the incident turbulence further decrease the coher ence of vortices shed in the wake of structures 462 Structures in ThreeDimensional Flows Case Studies The complexities of wind flow introduced by the geometries of typical structures and by the characteristics of the terrain and obstacles upstream emphasize the need to carry out detailed studies of wind pressures experimentally using wind tunnel models and simulation Wind flows around buildings are prime examples of 3D flows that cannot be described acceptably by 2D models In order to give some idea of the type of results so obtained and to emphasize the important roles of the boundarylayer velocity profile and of the turbulence in such results a few examples are cited below The existence of significant differences between drag or pressure coefficients mea sured in uniform and boundarylayer flow was first pointed out by Flachsbart in 1932 25 Figure 428b and c show the respective mean wind speed profiles and Figure 428d and e show pressure coefficient measurement results for wind normal to a building face Figure 428a As shown in Chapter 5 a large number of large and fullscale measure ments have been made in the intervening years owing to the need to assess uncertainties in data obtained in conventional wind tunnels Figures 429 and 430 are classic representations by Baines 26 of pressure distribu tions for structures under laminar and shear flows Far more detailed measurement 65 60 08 08 08 05 05 02 020 023 020 023 010 x 020 020 Sym about CL x x x x 06 07 07 05 00 9 5 5 60 55 75 75 80 80 70 70 Wind a Wind b Cp 99 x x 018 x Figure 429 a Pressure distributions on the faces of a cube in a constant velocity field Source From 26 b Pressure distributions on the faces of a cube in a boundarylayer velocity field Source From 26 k k k k 100 4 Bluff Body Aerodynamics 08 10 3 2 Wind Wind 1 2 4 3 09 10 05 05 1 2 3 4 Side Front Back 09 08 09 05 a Figure 430 a Pressure distributions over the sides and top of a tall building in a constant velocity field Source From 26 b Pressure distributions over the sides and top of a tall building in a boundarylayer velocity field Source From 26 results including data on fluctuating pressures are available in modern databases containing results of wind tunnel measurements NISTUWO 27 TPU 28 as well as in reports on large and fullscale measurements eg 2932 Load on secondary structural members eg joists are determined by the algebraic sums of external and internal pressures acting on them Figure 431 depicts the ideal case in which a the building is hermetically sealed so that the internal pressure is k k k k 46 Representative Flow Effects in Three Dimensions 101 06 06 06 06 06 07 Side Front b 04 03 05 09 Wind Wind Back 05 05 05 05 04 to 049 06 056 to 059 06 Figure 430 Continued not affected by the external wind flow b the building has openings on the windward side only in which case wind induces positive internal pressures c the building has openings on the leeward side in which case wind induces internal suctions and d the building has openings on both the windward and leeward sides in which case induces internal pressures that may be either positive or negative Windtunnel data on internal k k k k 102 4 Bluff Body Aerodynamics WIND WIND WIND WIND a HERMETIC BUILDING b WINDWARD OPENING c SUCTION OPENING d OPENINGS ON MORE THAN ONE SIDE pi pi pi 0 pi 0 Figure 431 Mean internal pressures in buildings with various opening distributions Source From 33 with permission from ASCE pressures are reported in 3438 Recent measurements of internal pressures on a largescale model of an industrial building and comparisons with values specified in the ASCE 716 Standard 39 are reported in 40 References 1 Roshko A 1993 Perspectives on bluff body aerodynamics Journal of Wind Engi neering and Industrial Aerodynamics 49 79100 2 Schluenzen K H ed Computational Wind Engineering 2014 CWE 2014 Pro ceedings of the Sixth International Symposium on Computational Wind Engineering Hamburg Germany 2014 3 Schuster D M The Expanding Role of Applications in the Development and Val idation of CFD at NASA in Computational Fluid Dynamics 2010 Proceedings of the Sixth International Conference on Computational Fluid Dynamics ICCFD6 St Petersburg Russia on July 1216 2010 A Kuzmin ed Berlin Heidelberg Springer Berlin Heidelberg 2011 pp 329 4 Batchelor GK 1967 An Introduction to Fluid Dynamics Cambridge Cambridge University Press 5 Centre Scientifique et Technique du Bâtiment 1980 Aérodynamique Nantes France Centre Scientifique et Technique du Bâtiment 6 Nathan AM 2008 The effect of spin on the flight of a baseball American Journal of Physics B76 119124 k k k k References 103 7 Bénard H 1908 Formations de centres de gyration à larrière dun obstacle en movement Comptes rendus de lAcadémie des Sciences Paris 147 839842 8 von Kármán T 1911 Über den Mechanismus des Widerstandes den ein bewegter Körper in einer Flüssigkeit erfährt Nachrichten von der Gesellschaft der Wis senschaften zu Göttingen MathematischPhysikalische Klasse 509517 9 Gerrard JH 2006 The mechanics of the formation region of vortices behind bluff bodies Journal of Fluid Mechanics 25 401413 10 Nakamura Y 1993 Bluff body aerodynamics and turbulence Journal of Wind Engi neering and Industrial Aerodynamics 49 6568 11 Strouhal V 1878 Uber eine besondere Art der Tonerregung Annalen der Physik 241 216251 12 Shih WCL Wang C Coles D and Roshko A 1993 Experiments on flow past rough circular cylinders at large reynolds numbers Journal of Wind Engineering and Industrial Aerodynamics 49 351368 13 Wooton LR and Scruton C 1971 Aerodynamic stability In The Modern Design of WindSensitive Structures ed AR Collins 6581 London Construction Indus try Research and Information Association 14 ASCE Task Committee 1961 Wind Forces on structures Transactions on ASCE 126 11241198 15 Pao HP and Kao TW 1976 On vortex trails over Ocean Islands Atmospheric Science Meteorological Society of the Republic of China Taiwan 3 2838 16 Roshko A 1955 On the wake and drag of bluff bodies Journal of the Aeronautical Sciences 22 124132 17 Roshko A 1961 Experiments on the flow past a circular cylinder at very high Reynolds number Journal of Fluid Mechanics 10 345356 18 Scruton C Rogers EWE Menzies JB and Scorer RS 1971 Steady and unsteady wind loading of buildings and structures and discussion Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences 269 353383 19 Güven O Farell C and Patel VC 1980 Surfaceroughness effects on the mean flow past circular cylinders Journal of Fluid Mechanics 98 673701 20 Vickery BJ 1966 Fluctuating lift and drag on a long cylinder of square cross section in a smooth and turbulent flow Journal of Fluid Mechanics 25 481494 21 Lee BE 1975 The effect of turbulence on the surface pressure field of a square prism Journal of Fluid Mechanics 69 263282 22 So RMC and Savkar SD 1981 Buffeting forces on rigid circular cylinders in cross flows Journal of Fluid Mechanics 105 397425 23 Laneville A Gartshore I S and Parkinson G V An explanation of some effects of turbulence on bluff bodies In Proceedings Fourth International Conference Wind Effects on Buildings and Structures Cambridge University Press Cambridge 1977 24 Stathopoulos T Davenport AG and Surry D 1981 Effective wind loads on flat roofs Journal of the Structural Division 107 281298 25 Flachsbart O 1932 Winddruck auf geschlossene und offene Gebäude In Ergeb nisse der Aerodynamischen Versuchanstalt zu Göttingen ed IV Lieferung L Prandtl and A Betz 128134 Munich and Berlin Verlag von R Oldenbourg k k k k 104 4 Bluff Body Aerodynamics 26 Baines W D Effects of velocity distribution on wind loads and flow patterns on buildings Proceedings Symposium No 1 Wind Effects on Buildings and Structures held at the National Physical Laboratory England UK in 1963 published by HMSO London in 1965 27 NISTUWO NISTUWO aerodynamic database Online Available httpswww nistgovwind 28 TPU TPU aerodynamic database Online Available httpwindarchtkougeiacjp systemengcontentscodetpu 29 Levitan ML Mehta KC Vann WP and Holmes JD 1991 Field measurements of pressures on the Texas Tech building Journal of Wind Engineering and Industrial Aerodynamics 38 227234 30 Richards PJ and Hoxey RP 2008 Wind loads on the roof of a 6m cube Journal of Wind Engineering and Industrial Aerodynamics 96 984993 31 Richards PJ and Hoxey RP 2012 Pressures on a cubic building Part 1 fullscale results Journal of Wind Engineering and Industrial Aerodynamics 102 7286 32 Richards PJ and Hoxey RP 2012 Pressures on a cubic building Part 2 quasisteady and other processes Journal of Wind Engineering and Industrial Aero dynamics 102 8796 33 Liu H and Saathoff PJ 1963 Internal pressure and building safety Journal of the Structural Division ASCE 108 223224 34 Holmes J D Mean and fluctuating internal pressures induced by wind in the Fifth International Conference Fort Collins CO pp 435450 1980 35 Liu H 1982 Internal pressure and building safety Journal of the Structural Divi sion 108 22232234 36 Saathoff PJ and Liu H 1983 Internal pressure of multiroom buildings Journal of Engineering Mechanics 109 908919 37 Harris RI 1990 The propagation of internal pressures in buildings Journal of Wind Engineering and Industrial Aerodynamics 34 169184 38 Vickery BJ 1994 Internal pressures and interactions with the building envelope Journal of Wind Engineering and Industrial Aerodynamics 53 125144 39 ASCE Minimum design loads for buildings and other structures ASCESEI 716 in ASCE Standard ASCESEI 716 Reston VA American Society of Civil Engineers 2016 40 Habte F Chowdhury AG and Zisis I 2017 Effect of windinduced internal pressure on local frame forces of lowrise buildings Engineering Structures 143 455468 k k k k 105 5 Aerodynamic Testing 51 Introduction To date testing remains the predominant means of obtaining aerodynamic data usable for the design of engineering structures It is well established that for most applications the testing has to be performed in flows simulating the main features of atmospheric flows A rigorous simulation of atmospheric flows would require that the nondimensional form of the equations of fluid motion and their attendant boundary conditions be the same in the prototype and at model scale This is not possible in practice owing primarily to the violation of the Reynolds number similarity requirement and the impossibility of rigorously simulating turbulent atmospheric flows Wind tunnel testing is therefore an art that requires consideration of the errors inherent in imperfect simulations see Chapters 7 and 12 Attempts to quantify such errors are made by among other means performing fullscale aerodynamic measurements a difficult endeavor owing to large uncertainties in the prototype wind flow that are often encountered in practice The purpose of this chapter is to discuss similarity requirements Section 52 describe aerodynamic testing facilities used for civil engineering purposes Section 53 consider the dependence of the aerodynamic response of wind tunnel models upon Reynolds number and the turbulence characteristics of simulated atmospheric bound ary layer flows Section 54 discuss blockage effects Section 55 and describe and comment on wind effects based on High Frequency Force Balance HFFB measure ments Section 56 and on pressure measurements Section 57 Aeroelastic testing including testing of suspendedspan bridges is discussed in Part III of the book For a rich source of useful information see 1 52 Basic Similarity Requirements 521 Dimensional Analysis Basic similarity requirements can be determined from dimensional analysis For engi neering structures it may be assumed that the aerodynamic force F on a body is a function of flow density 𝜌 flow velocity U a characteristic dimension D a characteristic Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 106 5 Aerodynamic Testing frequency n and the flow viscosity 𝜇 The following relation governing dimensional consistency then holds F d 𝜌𝛼U𝛽D𝛾n𝛿𝜇𝜀 51 where 𝛼 𝛽 𝛾 𝛿 𝜀 are exponents to be determined Each of the quantities 𝜌 U D n 𝜇 can be expressed dimensionally in terms of the three fundamental quantities mass M length L and time T so Eq 51 can be written as ML T2 d M L3 𝛼 L T 𝛽 L𝛾 1 T 𝛿 M LT 𝜀 52 for the dimensions of the viscosity follow see Section 412 Dimensional consistency requires that M 1 𝛼 𝜀 L 1 3𝛼 𝛽 𝛾 𝜀 T 2 𝛽 𝛿 𝜀 53 from which there follows for example that 𝛼 1 𝜀 𝛽 2 𝜀 𝛿 𝛾 2 𝜀 𝛿 54 Substitution of these relations in Eq 51 yields F d 𝜌1𝜀U2𝜀𝛿D2𝜀𝛿n𝛿𝜇𝜀 55 or F d 𝜌U2D2Dn U 𝛿 𝜇 𝜌UD 𝜀 56 meaning that the dimensionless force coefficient F𝜌U2D2 is a function of the dimen sionless ratios DnU and 𝜇𝜌UD or of their reciprocals Generally an equation involving n physical variables can be written in terms of p n k dimensionless parameters constructed from those original variables where k is the number of physical dimensions involved in the equation This statement is a form of the Buckingham 𝜋 theorem In the preceding example n 5 Eq 51 k 3 ie M L and T and as indicated following Eq 56 p 2 In some wind engineering problems eg the vibrations of suspended bridges the aerodynamic forces are also functions of the acceleration of gravity g By introduc ing g𝜁 into Eq 51 it can easily be shown that the force is also a function of the nondimensional ratio U2Dg called the Froude number The nondimensional ratio 𝜌UD𝜇 UD𝜈 is the wellknown Reynolds number and 𝜈 𝜇𝜌 is the kinematic vis cosity of the fluid Section 412 The parameter nDU is called the reduced frequency and its reciprocal is the reduced velocity If the frequency n being considered is the vortex shedding frequency the reduced frequency is the Strouhal number Section 44 If n is replaced by the Coriolis parameter Section 12 the reduced velocity is called the Rossby number k k k k 52 Basic Similarity Requirements 107 522 Basic Scaling Considerations Similarity requires that the reduced frequencies and the Reynolds numbers be the same in the laboratory and in the prototype This is true regardless of the nature of the fre quencies involved eg vortex shedding frequencies natural frequencies of vibration frequencies of the turbulent components of the flow or of the densities being consid ered eg fluid density density of the structure For example if the reduced frequency is the same in the prototype and in the laboratory ie at model scale applying this requirement to the vortex shedding frequency nv and to the fundamental frequency of vibration of the structure ns we have nvD U p nvD U m 57 and nsD U p nsD U m 58 where the indexes m and p stand for model and prototype respectively It follows from Eqs 57 and 58 that ns nv p ns nv m 59 This is also true of the ratios of all other relevant quantities lengths densities veloc ities Thus for the density of the structure and the density of the fluid it must be the case that 𝜌s 𝜌air p 𝜌s 𝜌f m 510 where 𝜌f is the density of the fluid in the laboratory For the same reason Uz1 Uz2 p Uz1 Uz2 m 511 where z1 and z2 are heights above the surface In particular if in the prototype the veloc ities conform to a power law with exponent 𝛼 it follows from Eq 511 that in the laboratory the velocities must conform to the power law with the same exponent 𝛼 To see this Eq 511 is rewritten as follows z1 z2 𝛼 p Uz1 Uz2 m 512 Since z1z2p z1z2m by virtue of geometric similarity it follows from the preceding equation that similarity is satisfied if z1 z2 𝛼 m Uz1 Uz2 m 513 Since there are three fundamental requirements concerning mass length and time three fixed choices of scale can be made This choice determines all other scales For k k k k 108 5 Aerodynamic Testing example let the length scale the velocity scale and the density scale be denoted by 𝜆L DmDp 𝜆U UmUp and 𝜆𝜌 𝜌m𝜌p The reduced frequency requirement nD U p nD U m 514 controls the frequency scale 𝜆n for all pertinent test frequencies From Eq 514 it fol lows immediately that 𝜆n 𝜆U 𝜆L The time scale 𝜆T is the reciprocal of 𝜆n In principle for similarity between prototype ie fullscale and laboratory flows to be achieved the respective Reynolds numbers Re UD𝜈 must be the same This requirement is referred to as Reynolds number similarity In aerodynamic facilities for testing models of structures the fluid being used is air at atmospheric pressure and Reynolds number similarity is unavoidably violated 53 Aerodynamic Testing Facilities To achieve similarity between the model and the prototype it is in principle necessary to reproduce at the requisite scale the characteristics of atmospheric flows that is i the variation of the mean wind speed with height and ii the turbulence characteristics The purpose of this section is to describe facilities intended to do so including facilities designed to simulate thunderstorm and tornado winds Also described in this section are facilities used for full or largescale tests of special structures such as lamp posts and for providing data on winddriven rain intrusion and on snow deposition In long wind tunnels a boundary layer with a depth of 051 m develops natu rally over a rough floor in test sections with lengths of the order of 20 m in length Figures 5153 In such tunnels as well as in tunnels with considerably shorter test sections eg 510 m the depth of the boundary layer is increased above these values by placing at the test section entrance passive devices such as spires eg Figure 53 grids barriers fences singly or in combination some of which are illustrated subse quently The height of long tunnels may be adjusted to achieve a zeropressure gradient streamwise which owing to energy losses associated with flow friction at the walls and internal friction due to turbulence would otherwise not occur The following procedure for the design of spires with the configuration of Figure 55 was proposed in 41 1 Select the desired boundarylayer depth 𝛿 2 Select the desired shape of the mean velocity profile defined by the power law exponent 𝛼 3 Obtain the height h of the spires from the relation h 139𝛿 1 𝛼2 515 4 Obtain the width b of the spire base from Figure 56 in which H is the height of the tunnel test section 1 The base dimension of the triangular splitter plate in Figure 55 is h4 the lateral dimension is h4 The lateral spacing between the spires is h2 The width of the tunnel need not be an integral multiple of h2 k k k k Third Corner Wind Second Diffuser Axial Blower Second Corner First Corner Pit of Turn Table Pit of Tunnel Balance Working Section Traveling Second Screen Contraction First Screen Honeycomb Setting Chamber Fourth Corner Control Room Working Section Fixed First Diffuser Figure 51 Wind tunnel operated by Kawasaki Heavy Industries Ltd Japan at its Akashi Technical Institute Wind speed range 0225 m s1 test section dimensions 25 3 20 m Source From 2 with permission from ASCE k k k k 110 5 Aerodynamic Testing Approximate depth of boundary layer over a carpet z0 003 cm 10 Distance above tunnel floor m 05 0 20 15 Distance from leading edge of roughness m 10 5 0 Bell mouth Note Ufs is the free stream or undisturbed velocity Ufs Ufs Ufs Ufs Ufs Ufs Approximate depth of boundary layer over rectangular blocks 25 to 10 cm high z0 25 cm Figure 52 Development of boundary layer in a long wind tunnel Source After 3 Figure 53 Wind tunnel Colorado State University Model and turntable are in the foreground and spires are in the background Source Courtesy of Professor B Bienkiewicz The desired mean wind profile occurs at a distance 6h downstream from the spires According to 4 5 the wind tunnel floor downwind of the spires should be covered with roughness elements for example cubes with height k such that k 𝛿 exp 2 3 ln D 𝛿 01161 2 Cf 205 12 516 where D is the spacing of the roughness elements Cf 0136 𝛼 1 𝛼 2 517 and 𝛼 is the exponent of the power law describing the mean wind speed profile According to 4 5 Eqs 516 and 517 are valid in the range 30 𝛿D2k3 2000 Some laboratories have adopted the system proposed in 4 others have used other methods for designing their flow management system see eg Figure 54 k k k k 53 Aerodynamic Testing Facilities 111 Figure 54 Boundarylayer wind tunnel University of Florence Prato Italy Source Courtesy of Professor Claudio Borri Figure 55 A proposed spire configuration Source Reprinted from 4 with permission from Elsevier SPLITTER PLATE h WIND SPIRE FRONT FACE b k k k k 112 5 Aerodynamic Testing 024 020 016 012 008 004 00 01 02 03 05 03 01 0 α 04 05 b h δ 08 H Figure 56 Proposed graph for obtaining spire base width Source Reprinted from 4 with permission from Elsevier Various aerodynamic testing facilities are described in the following National Aeronautical Establishment National Research Council of Canada A short wind tunnel with 9 m 9 m cross section designed for aeronautical applications has occasionally been used for civil engineering purposes and is shown in Figure 57 The drawback of this facility from a civil engineering point of view is that the test section is too short to allow the flow to develop features with an acceptable resemblance to those of the atmospheric boundary layer Figure 57 Spire and roughness arrays in a short wind tunnel Source Courtesy of the National Aeronautical Establishment National Research Council of Canada k k k k 53 Aerodynamic Testing Facilities 113 Figure 58 Interior view of IBHS Research Center with fullscale specimens placed on the 168 m diameter turntable with a surface area of 220 m2 The 105fan array with 300 hp motors is located on the left side of the picture Source Courtesy of the Institute for Business Home Safety IBHS Research Center Figure 58 shows an outside and inside view of the Institute for Business Home Safety IBHS Research Center in South Carolina a multiperil facility capable of testing structures subjected to realistic Category 1 2 and 3 hurricanes extratropical windstorms thunderstorm frontal winds wildfires and hailstorms One purpose of the test performed on the two buildings shown in Figure 58 was to offer a vivid illustration of the benefits of robust construction by contrasting in a video the good performance of the stronger of the two buildings and the collapse of the weaker building Florida International University Wall of Wind Experimental Facility The Wall of Wind WoW is powered by twelve 49 m diameter fans and is capable of testing in up to 70 m s1 157 mph wind speeds Figures 59 and 510 The test section is 61 m 43 m and the turntable diameter is 49 m Testing can be performed at scales approximately twice as large and Reynolds numbers approximately five times as large as in facilities such as for example the wind tunnel in Figure 51 As can be seen in Figure 510 the spires and floor roughness elements for the simulation are similar to those used in typical wind tunnels The facility can be used for destructive testing and for the simulation of water intrusion due to winddriven rain University of Florida UF Boundary Layer Wind Tunnel The University of Floridas major aerodynamic testing facility is its boundary layer wind tunnel with a 6 m wide 3 m high and 40 m long test section and a 16 m s1 maximum flow speed Figure 511 The floor roughness elements which help to simulate various surface exposures are auto mated and individually controlled This feature allows fine tuning of the boundary layer at the test section and rapid reconfiguring for efficient testing using multiple exposures Tornado Simulator Iowa State University ISU Basic ideas on facilities for tornado simulation were developed in 6 and 7 among others The ISU tornado simulator k k k k 114 5 Aerodynamic Testing Figure 59 Twelvefan wall of wind Florida International University Source Courtesy of Professor A Gan Chowdhury Figure 510 Twelvefan wall of wind Florida International University view of test section Source Courtesy of Professor A Gan Chowdhury k k k k 53 Aerodynamic Testing Facilities 115 Figure 511 University of Florida boundarylayer wind tunnel Source Courtesy of Professor K R Gurley is a modern version of the facility described in 6 and is shown schematically in Figure 512 8 WindEEE Dome The Wind Engineering Energy and Environment WindEEE Dome 9 10 is an innovative hexagonal wind tunnel that allows for atmospheric boundary layer simulations over extended areas and complex terrain and of tornadoes down bursts and microbursts Figures 513 and 514 For the atmospheric boundary layer simulation mode Figures 513a and 514 the test section is 14 m wide 38 m high and 25 m long and the maximum flow velocity is 35 m s1 The tornado simulation mode Figures 513b allows the modeling of cate gory F0F3 tornado flows with vortex diameters of up to 45 m translation speeds of up to 2 m s1 and flow velocities of up to 25 m s1 The downburstmicroburst simulation mode Figure 513c can achieve flows with up to 2 m s1 translation speeds and 30 m s1 velocities One of the six walls shown in Figure 513 has four rows of 15 independently adjustable fans each used to simulate the atmospheric boundary layer flow The other five walls have each eight fans at their base For the tornado simulation mode directional vanes are placed in front of each of those fans The angle of orientation of the vanes can be adjusted to impart the desired swirl ratio to the flow ie the ratio between the tangential velocity and the radial velocity in the vortex Six large fans placed in the upper chamber Figure 513 produce an updraft shown schematically in Figure 513b For details on various capabilities of the WindEEE facility including measurement capabilities see 9 10 k k k k 116 5 Aerodynamic Testing Turning Vane 03 m 1 ft Honeycomb Screen 55 m 18 ft 122 m to 244 m Floor height H 023 m to 152 m Rotating downdraft Adjustable ground plane 152 m 5 ft Fan Motor 183 m 6 ft Figure 512 Iowa State University tornado simulator Source Courtesy of Professor P Sarkar a b c Figure 513 Schematic cross section a Atmospheric boundarylayer simulation mode b tornado flow simulation mode c downburstmicroburst simulation mode Source Courtesy Professor H Hangan k k k k 53 Aerodynamic Testing Facilities 117 Figure 514 View of test section Source Courtesy Professor H M Hangan Politecnico di Milano Milan Italy The test section of its largescale aerodynamic test ing facility is 1385 m wide 385 m high and 35 m long and the maximum wind speed is 16 m s12 Centre Scientifique et Technique du Bâtiment CSTB Nantes France The test section of its largescale boundarylayer wind tunnel Figures 515 and 516 is 4 m wide 1735 m high and 15 m long and the maximum wind speed is 30 m s1 Note in Figures 515 and 516 that the passive flow management devices being used are different depending upon type of application Like other prominent laboratories CSTB Figure 515 Test section of boundarylayer wind tunnel Source Photo Florence Joubert courtesy of CSTB 2 No picture available at the time of printing k k k k 118 5 Aerodynamic Testing Figure 516 Test section of boundarylayer wind tunnel Source Courtesy of CSTB Note that for this application the flow management devices placed at the entrance to the test section are radically different from the typical spires operates large facilities for testing winddriven rain intrusion Figure 517 roofing Figure 518 snow deposition Figure 519 and other applications Technical University Eindhoven TUE The TUE boundarylayer wind tunnel test section is 27 m long 3 m wide and 2 m high Wind speeds can be as high as 30 m s1 The wind tunnel is designed for build environment maritime sports vehicle aerodynamics air quality and wind energy applications Both open and closed circuit modes are feasi ble Figure 520 Measurement equipment includes 3D Laser Doppler Anemometry Figure 517 Winddriven rain intrusion test Source Courtesy of CSTB k k k k 53 Aerodynamic Testing Facilities 119 Figure 518 Roofing test Source Courtesy of CSTB Figure 519 Snow deposition test Source Courtesy of CSTB k k k k 120 5 Aerodynamic Testing Corner vanes Diffusor 4 fans with individual control Lowangle diffusor Corners with corner vanes Contraction Translation stage below besides and on top of modules for 3D LDA over 27 m length 27 m long test section with 9 independently movable and instrumented modules cross section 3 2 m2 Screens and honeycomb Figure 520 TUE boundary layer wind tunnel Source Courtesy of Professor B Blocken 54 Wind Tunnel Simulation of Atmospheric Boundary Layers 541 Effect of Type of Spires and Floor Roughness Elements Figure 521 11 shows the mean velocity and the longitudinal and vertical turbulence intensity profiles at i 61 m and ii 183 m downwind of the test section entrance for flows obtained by using three different types of spires the wind floor being covered by staggered 127 mm cubes spaced 508 mm apart In Figure 521 the boundarylayer thick ness 𝛿 the mean wind speed at elevation 𝛿 and the power law exponent 𝛼 are denoted by delta Uinf and EXP respectively It was assumed in the study that the mean flow with power law exponent 𝛼 016 at station x 61 m and 𝛼 029 at station x 183 m are approximately representative of open terrain and suburban terrain respectively Some modelers adopt a geometric scale equal to the ratio between the boundarylayer thickness measured in the laboratory and values zg of Table 24 even though the latter are nominal rather than physically significant The use of this geometric scaling criterion for the simulations of Figure 521 yielded the geometric scales 𝛿 zg 075274 1365 for the flow with open exposure 𝛼 016 and 1400 for the flow with builtup terrain exposure 𝛼 029 The respective measured longitudinal turbulence intensities at 50 m above ground are 007 and 015 versus about 015 and 0225 estimated using Eq 256 for atmospheric boundarylayer flows As expected the discrepancy between the longitudinal turbulence intensity in the wind tunnel and the target value in the atmosphere is more severe at the station x 61 m which would correspond to the fetch available in a typical short wind tunnel Figure 522 11 shows spectra of the longitudinal velocity fluctuations measured at station x 183 m and elevation z𝛿 005 in the three flows described in Figure 521 For nzUz 10 the spectra corresponding to two of the three types of spires differ from each other by a factor greater than two k k k k 54 Wind Tunnel Simulation of Atmospheric Boundary Layers 121 120 MEAN VELOCITY PROFILES MEAN VELOCITY PROFILES LONGITUDINAL TURBULENCE INTENSITY Uinf ms 934 921 883 744 6000 26 20 29 10000 13000 4000 24 16 18 7500 10000 886 799 delta cm Uinf ms delta cm Uinf ms delta cm EXP Uinf ms delta cm EXP 100 80 60 Zdelta 40 20 00 20 40 60 UUINF a b 80 100 120 100 80 60 Zdelta 40 20 00 20 40 60 UUINF 80 100 120 934 886 799 4000 7500 10000 921 883 744 6000 10000 10000 100 80 60 Zdelta Zdelta 40 20 00 00 800 1600 UrmsU 100 a 2400 3200 LONGITUDINAL TURBULENCE INTENSITY 120 100 80 60 40 20 00 00 800 1600 UrmsU 100 b 2400 3200 Figure 521 Wind tunnel flow features at a 61 m and b 183 m downwind of spires obtained by using three types of spire configurations Source Reprinted with permission from 11 k k k k 122 5 Aerodynamic Testing 1000 100 010 nSnu2 EXP 26 20 29 001 0001 0010 0100 nzU 1000 10000 100000 Figure 522 Spectra of longitudinal velocity fluctuations measured at 183 m downwind of spires Source Reprinted with permission from 11 That wind tunnels with different flow management devices can result in flows with different properties and hence in different aerodynamic pressures on bodies immersed in those flows was confirmed by a round robin set of tests reported in 12 https wwwnistgovwind The tests were conducted by six reputable wind tunnels in the US Canada France and Japan on a model of an industrial building with both open and suburban terrain exposure Coefficients of variation CoV of wind effects determined on the basis of the test results differed significantly from laboratory to laboratory and were found to be as high as 40 542 Effect of Integral Scale and Turbulence Intensity It is assumed in current practice see eg ASCE 4912 Standard 13 and ASCE 716 Standard 14 that wind tunnel flows are satisfactory if in addition to the mean wind profiles they reproduce the longitudinal turbulence intensity and to some degree at least the longitudinal integral scale of turbulence typical of atmospheric boundarylayer flows This section discusses the extent to which this assumption is warranted Integral Scale and Turbulence Intensity Some laboratories assume that the integral length is a valid characterization of turbulence for wind tunnel testing purposes In prin ciple the geometric scale of the simulation should be consistent with the relation Dm Dp Lx um Lx up 518 where the indexes m and p stand for model and prototype respectively However the usefulness of Eq 518 is questionable for three reasons First estimates of integral k k k k 54 Wind Tunnel Simulation of Atmospheric Boundary Layers 123 lengths are typically highly uncertain Second the smallscale turbulence transports into the separation bubble free flow particles with large momentum thus promoting flow reattachment and strongly affecting pressure distributions near separation points 15 the integral scale is not a significant factor in this phenomenon Third integral tur bulent scales similar to those occurring in the atmosphere are not achievable in typical conventional wind tunnels at geometric scales used for the simulation of wind effects on lowrise buildings eg 1 501 100 This is the case because the size of the eddies associated in the atmosphere with lowfrequency flow fluctuations is too large to be accommodated at such scales in wind tunnels with testsection widths of the order of 153 m These fluctuations contribute most of the turbulence intensity in atmo spheric boundarylayer flows For these reasons the ASCE 716 considerably relaxes the requirement inherent in Eq 518 In addition it follows from the lowfrequency fluctuation deficit in conventional wind tunnels that equal simulated and prototype tur bulence intensities may not produce similar aerodynamic effects because the respective flows have different frequency content Compensating for Missing LowFrequency Fluctuations The effect of the low frequency fluctuation deficit in conventional boundarylayer wind tunnel tests at geometric scales of the order of 1 100 can be compensated for by assuming that the energy of those fluctuations is concentrated at frequencies close to or equal to zero Since zerofrequency infiniteperiod velocity fluctuations are in effect constant velocities this assumption entails adding to the aerodynamic pressures measured in the wind tunnel via postprocessing a constant pressure pd 1 2𝜌CpU2 def 519 In Eq 519 𝜌 is the air density Cp is the mean pressure coefficient measured in the wind tunnel and U2 def is the estimated area under the spectral density function of the lowfrequency contributions not reproduced in the wind tunnel This approach is conservative because it implies perfect spatial coherence of the pressures that would be induced by the missing fluctuation components when in reality that coherence is imperfect For an alternative approach see 16 543 Effects of Reynolds Number Similarity Violation In principle for similarity between prototype and wind tunnel flows to be achieved the respective Reynolds numbers must be the same This requirement is referred to as Reynolds number similarity In aerodynamic facilities for testing models of structures the fluid being used is air at atmospheric pressure and Reynolds number similarity is unavoidably violated The aerodynamic behavior of the bodies depends upon whether the boundary layers that form on the curved surfaces are laminar or partially or fully turbulent Since boundary layers occurring at high Reynolds numbers are turbulent it is logical to attempt the reproduction of fullscale flows around smooth cylinders by changing laminar boundary layers into turbulent ones This can be done by providing the surface with roughness elements 17 k k k k 124 5 Aerodynamic Testing According to 18 the thickness e of the roughness element should satisfy the relations Ue𝜈 400 and eD 001 where U is the mean speed 𝜈 is the kinematic viscosity and D is the characteristic transverse dimension of the object For the tower shown in plan in Figure 523 the roughness was achieved by fixing onto the surface of the 1200 model 32 equidistant vertical wires Three sets of experiments are reported in 18 in which the surface of the cylinder was i smooth ii provided with 06 mm wires eD 7103 and iii provided with 1 mm wires respectively It was found that the highest mean and peak pressures were more than twice as high on the smooth model than on the models provided with wires The differences between pressures on the model and with 06 mm and the model with 1 mm wires were small The influence of the roughness on the magnitude of the mean pressures at 20 m full scale below the top of the building is shown in Figure 523 in which Cp p pr 1 2𝜌U2 r 520 where p is the measured mean pressure pr is the static reference pressure Ur is the mean speed at the top of the building and 𝜌 is the air density Unlike bodies with rounded shapes bodies with sharp edges have fixed separation points Figure 418 whose separation at the edges is independent of Reynolds num ber It has therefore been hypothesized that flows around such bodies are similar at full scale and in the wind tunnel even if Reynolds number similarity is violated However in the wind tunnel friction forces are larger in relation to inertial forces than at full scale Smooth model 2 1 1 SW Wind 0 Models with 06 mm and with 1 mm wires Cp Figure 523 Influence of model surface roughness on pressure distribution Source Courtesy of Cebtre Scentifique et Technique du B atiment Nantes France k k k k 54 Wind Tunnel Simulation of Atmospheric Boundary Layers 125 This affects the local vorticity at edges and corners in the wind tunnel resulting in local pressures typically weaker than at full scale Examples are shown in Section 544 544 Comparisons of Wind Tunnel and FullScale Pressure Measurements Figure 524 shows that the negative peak pressures measured at a corner of a lowrise building can be significantly stronger at full scale than in the wind tunnel Additional comparisons of pressures on the Texas Tech building and its wind tunnel models tested at Colorado State University and the University of Western Ontario were published in 20 Figure 525 shows that wind tunnel measurements are acceptable for the wall pressures but inadequate for the roof corner Figure 526a and b show comparisons between wind tunnel and fullscale measure ments of pressures at the Commerce Court tower Toronto The wind tunnel values were provided at the design stage and are represented by open circles The solid lines join aver age values of estimates derived from fullscale measurements the shaded areas indicate the standard deviation of the fullscale estimates in Figure 526 the notation RMSM denotes the root mean square value about the mean Note that fluctuating pressures attributable to fluctuating lift differ at some points significantly in the wind tunnel from their fullscale counterparts For some tall buildings the loss of highfrequency velocity fluctuations content in the laboratory can also reduce the strength of the resonant fluctuations induced on the model by the oncoming flow 0 0 50 100 150 200 250 300 350 400 1 2 3 4 Min Pressure Coefficients 5 6 7 8 Angle of Attack Full Scale Cp Wind Tunnel Figure 524 Minimum pressure coefficients at building corner eave level Texas Tech University experimental building fullscale and wind tunnel measurements Source From 19 k k k k 126 5 Aerodynamic Testing 30 120 90 60 30 00 20 10 00 Cp max Cp rms Cp rms Cp min Cp mean Cp mean 10 20 30 04 03 02 01 00 05 00 05 10 0 100 200 Azimuth degrees a b 300 400 0 00 10 20 00 04 08 12 16 100 200 300 400 UWO smooth exp UWO rough exp Full Scale CSU Azimuth degrees UWO smooth exp UWO rough exp Full Scale CSU Figure 525 Wind pressure coefficients on the Texas Tech Experimental Building full scale and wind tunnel measurements a wall pressures b corner roof pressures Source Reprinted from 20 with permission from Elsevier 05 N N SENSOR HEIGHT 409 069 H 310 084 H SENSOR HEIGHT 04 03 RMSM PRESSURE COEFF MEAN PRESSURE COEFF RMSM PRESSURE COEFF MEAN PRESSURE COEFF 02 01 10 05 05 WEST WEST NORTH EAST WIND DIRECTION SOUTH WEST WEST NORTH EAST WIND DIRECTION a b SOUTH 10 0 0 05 04 03 02 01 10 05 05 10 0 0 Figure 526 Pressures measured on a west wall at 206 m from NW corner at 46th floor and b east wall at 206 m from NE corner at 50th floor Commerce Court Tower Source Reprinted from 21 with permission from Elsevier k k k k 55 Blockage Effects 127 55 Blockage Effects A body placed in a wind tunnel will partially obstruct the passage of air causing the flow to accelerate This effect is called blockage If the blockage is substantial the flow around the model and the models aerodynamic behavior are no longer representative of prototype conditions Corrections for blockage depend upon the body shape the nature of the aerodynamic effect of concern ie whether drag lift Strouhal number and so forth the characteris tics of the wind tunnel flow and the relative bodywind tunnel dimensions Basic studies of blockage are summarized in 22 which contains a bibliography on this topic For drag measured in closed wind tunnels it is concluded in 22 that the following approximate relation may be used for the great majority of model configurations in all flows including boundarylayer flows CDc CD 1 K SC 521 where CDc is the corrected drag coefficient CD is the drag coefficient measured in the wind tunnel S is the reference area for the drag coefficients CDc and CD and C is the wind tunnel crosssectional area The ratio SC is called the blockage ratio The coefficient K has been determined only for a limited number of situations For example for a bar with rectangular cross section spanning the entire height of a wind tunnel with nominally smooth flow K was determined to depend upon the ratio ab as shown in Figure 527 where a and b are the dimensions of the alongwind and acrosswind sides of the rect angular cross section respectively In practice it may be assumed that for 2 blockage ratios the blockage corrections are about 5 and that to a first approximation the blockage correction is proportional to the blockage ratio 22 For a basic study of blockage effects on bluffbody aerodynamics see 24 26 24 20 18 K 16 14 12 10 08 0 0 05 10 15 ab 20 25 30 Figure 527 Blockage correction factor K for twodimensional prism ratio ab in nominally smooth flow 23 k k k k 128 5 Aerodynamic Testing 56 The HighFrequency Force Balance The HFFB approach uses rigid test models supported at the base by a highfrequency force ie a rigid balance The balance allows measurements of strains proportional to the base bending moments shears and torsional moments and experiences very small deformations that render the model motions negligibly small Figure 528 The HFFB approach is applicable primarily to buildings with approximately straightline fundamental modal shapes in sway along the principal axes of the building The expression for the base moment generated by the wind load in the xdirection is Mbxt H 0 wxz tzdz 522 where H building height wxz t wind loading parallel to the xdirection per unit height and z elevation above ground Assuming that the fundamental modal shape is a straight line the generalized force in the xdirection is also given by righthand side of Eq 522 Owing to this coincidence measurement of the base moment yields the generalized force Qx1t Qx1t H 0 wxz tzHdz 523 where zH is the fundamental modal shape The estimation of the fundamental frequency of vibration from the analysis of the structure and the specification of the damping ratio then allow the approximate estimation of the dynamic response see Chapter 11 Similar statements apply to the generalized force in the ydirection 25 While the generalized aerodynamic torsional moment has the expression Q𝜙1t H 0 Tz t𝜑T1zdz 524 BUILDING MODEL WIND TUNNEL FLOOR Figure 528 Schematic of forcebalance model k k k k 57 Simultaneous Pressure Measurements at Multiple Taps 129 where Tz t is the aerodynamic torsional moment per unit height and 𝜑T1z is the fundamental mode of vibration in torsion the base aerodynamic torsional moment mea sured in the wind tunnel is Q𝜙1HFFBt H 0 Tz tdz 525 Since 𝜑T1z 1 the measured base torsional moment cannot be a substitute for the fundamental generalized torsional moment Q𝜑1t In addition the HFFB approach provides no information on the contribution of higher modes of vibration to the response If the fundamental modes of vibration in the x and y directions do not vary linearly with height the measured base bending moments are inadequate substitutes for the expressions of the respective modal generalized forces Corrections accounting for the actual modal shapes can be applied but they depend upon the distribution of the wind pressures which until the 1990s could not be obtained by measurements and was therefore generally unknown especially for buildings affected by aerodynamic interfer ence effects The corrections and the corresponding approximations of the generalized torques and moments therefore depended upon educated guesses concerning the wind pressure distribution In the 1980s 1990s and even the first years of the 2000s the design of tall buildings was based on the HFFB approach that in spite of its limitations was a step forward with respect to earlier practices The HFFB procedure has two advantages it is relatively fast and inexpensive and it is compatible with the presence of architectural details that may render difficult the use of pressure taps in some cases The procedure is convenient for use in preliminary studies of aerodynamic alternatives for which only qualitative results are required 57 Simultaneous Pressure Measurements at Multiple Taps Figure 529 shows a model with the large number of pressure taps for which simulta neous pressure measurements are enabled by modern electronic scanning systems In contrast Figure 530 shows typical tap locations for models subjected to tests compatible with the capabilities available in the late 1970s on the basis of which ASCE 7 Standard provisions were developed in the 1980s In addition of the fact that the spatial resolu tion of the pressure taps is two orders of magnitude higher in modern practice than in the 1970s the quality of the inferences based on the models with large numbers of taps is due to the fact that unlike their 1970s predecessors all data obtained by electronic scanning systems can be recorded and therefore allow transparent postprocessing A widely used simultaneous pressure measuring system is the Electronic Pres sure Scanning System developed by Scanivalve Corporation wwwscanivalvecom Figure 531 A pressure measuring system includes an Electronic Pressure Scan ning Module eg ZOC33 with 64 pressure sensors a Digital Service Module eg DSM4000 which can service up to eight Electronic Pressure Scanning Modules ie up to 512 sensors and contains an embedded computer RAM memory and a hard disk drive a pressure calibration system auxiliary instrumentation to regulate supply of clean dry air and data acquisition software k k k k 130 5 Aerodynamic Testing Figure 529 Building model in wind tunnel Source From 26 111 68 305 m 6 17 47 244 m 48 m 49 m 10 m 29 Figure 530 Pressure tap arrangement in typical 1970s tests Source After 27 The connection between the Electronic Pressure Scanning Module and the pressure taps is made through plastic tubes A test model with tubes connecting the pressure taps to the scanning module is shown in Figure 532 Tube characteristics must conform to requirements assuring that no significant distortion of pressures acting at the taps occurs 28 29 k k k k 57 Simultaneous Pressure Measurements at Multiple Taps 131 Figure 531 View of electronic multichannel pressure scanning system wwwscanivalvecom Figure 532 Tubes installed on a smallscale test building k k k k 132 5 Aerodynamic Testing References 1 Kopp G A ed LargeScale and FullScale Laboratory Test Methods for Examin ing Wind Effects on Buildings Frontiers in the Built Environment series frontiersin org Online Available wwwfrontiersinorgresearchtopics4739largescaleand fullscalelaboratorytestmethodsforexaminingwindeffectsonbuildings 2018 2 Marshall RD 1984 Wind tunnels applied to wind engineering in Japan Journal of Structural Engineering 110 12031221 3 Davenport A G and Isyumov N The application of the boundarylayer wind tunnel to the prediction of wind loading in Proceedings of the International Research Seminar on Wind Effects on Buildings and Structures Vol 1 p 221 Copyright Canada University of Toronto Press 1968 4 Irwin PA 1981 The design of spires for wind simulation Journal of Wind Engineering and Industrial Aerodynamics 7 361366 5 Wooding RA Bradley EF and Marshall JK 1973 Drag due to regular arrays of roughness elements of varying geometry BoundaryLayer Meteorology 5 285308 6 Ward NB 1972 The exploration of certain features of tornado dynamics using a laboratory model Journal of the Atmospheric Sciences 29 11941204 7 DaviesJones RP 1973 The dependence of core radius on swirl ratio in a tornado simulator Journal of the Atmospheric Sciences 30 14271430 8 Haan FL Sarkar PP and Gallus WA 2008 Design construction and perfor mance of a large tornado simulator for wind engineering applications Engineering Structures 30 11461159 9 Refan M and Hangan H 2016 Characterization of tornadolike flow fields in a new model scale wind testing chamber Journal of Wind Engineering and Industrial Aerodynamics 151 107121 10 Refan M Hangan H and Wurman J 2014 Reproducing tornadoes in laboratory using proper scaling Journal of Wind Engineering and Industrial Aerodynamics 135 136148 11 Cermak JE 1982 Physical modeling of the atmospheric boundary layer in long boundarylayer tunnels In Wind Tunnel Modeling for Civil Engineering Applica tions Proceedings of the international workshop on wind tunnel modeling criteria and techniques in civil engineering applications Gaithersburg MD USA April 1982 1st ed ed TA Reinhold 97125 Cambridge UK Cambridge University Press 12 Fritz WP Bienkiewicz B Cui B et al 2008 International comparison of wind tunnel estimates of wind effects on lowrise buildings testrelated uncertainties Journal of Structural Engineering 134 18871890 13 ASCE Wind tunnel testing for buildings and other structures ASCESEI 4912 in ASCE Standard ASCESEI 4912 Reston VA American Society of Civil Engi neers 2012 14 ASCE Minimum design loads for buildings and other structures ASCESEI 716 in ASCE Standard ASCESEI 716 Reston VA American Society of Civil Engineers 2016 15 Li QS and Melbourne WH 1995 An experimental investigation of the effects of freestream turbulence on streamwise surface pressures in separated and reattaching flows Journal of Wind Engineering and Industrial Aerodynamics 5455 313323 k k k k References 133 16 Mooneghi MA Irwin PA and Chowdhury AG 2016 Partial turbulence simulation method for predicting peak wind loads on small structures and building appurtenances Journal of Wind Engineering and Industrial Aerodynamics 157 47 17 Szechenyi E 1975 Supercritical Reynolds number simulation for twodimensional flow over circular cylinders Journal of Fluid Mechanics 70 529542 18 Gandemer J Barnaud G and Biétry J Études de la tour DMA Partie I Étude des efforts dûs au vent sur les façades Centre Scientifique et Technique du Bâtiment Nantes France 1975 19 Long F Uncertainties in pressure coefficients derived from full and model scale data report to the National Institute of Standards and Technology Wind Science and Engineering Research Center Texas Tech University 20 Tieleman HW 1992 Problems associated with flow modelling procedures for lowrise structures Journal of Wind Engineering and Industrial Aerodynamics 42 923934 21 Dalgliesh A 1975 Comparisons of model fullscale wind pressures on a highrise building Journal of Wind Engineering and Industrial Aerodynamics 1 5566 22 Melbourne WH 1982 Wind tunnel blockage effects and correlations In Wind Tunnel Modeling for Civil Engineering Applications 1st ed ed TA Reinhold 197216 Cambridge UK Cambridge University Press 23 Courchesne J and Laneville A 1979 A comparison of correction methods used in the evaluation of drag coefficient measurements for twodimensional rectangular cylinders Journal of Fluids Engineering 101 506510 24 Utsunomiya H Nagao F Ueno Y and Noda M 1993 Basic study of blockage effects on bluff bodies Journal of Wind Engineering and Industrial Aerodynamics 49 247256 25 Tschanz T and Davenport AG 1983 The base balance technique for the deter mination of dynamic wind loads Journal of Wind Engineering and Industrial Aerodynamics 13 429439 26 Ho C E Surry D and Moorish D NISTTTU Cooperative Agreement Windstorm Mitigation Initiative Wind Tunnel Experiments on Generic Low Buildings Alan G Davenport Wind Engineering Group The University of Western Ontario 2003 27 Davenport A G Surry D and Stathopoulos T Wind loads on lowrise build ings Final report on phase I and II BLWTSS81977 University of Western Ontario London Ontario Canada 1977 28 Irwin PA Cooper KR and Girard R 1979 Correction of distortion effects caused by tubing systems in measurements of fluctuating pressures Journal of Wind Engineering and Industrial Aerodynamics 5 93107 29 Kovarek M Amatucci L Gillis K A Potra F A Ratino J Levitan M L and Yeo D Calibration of dynamic pressure in a tubing system and optimized design of tube configuration a numerical and experimental study NIST TN 1994 National Institute of Standards and Technology Gaithersburg MD 2018 https wwwnistgovwind k k k k 135 6 Computational Wind Engineering 61 Introduction Computational Fluid Dynamics CFD is a vast field aimed at describing fluid flows using numerical methods Computational Wind Engineering CWE is a CFD subfield whose main objective is to produce descriptions of aerodynamic wind effects on the built environment In particular descriptions are sought for use in the structural design of buildings and other structures It is symptomatic that while addressing recent CWE accomplishments stateoftheart surveys 13 mention few if any applications to structural design practice This is because to date with rare exceptions 4 structural designers cannot rely on CWE with the degree of confidence required to ensure the safety of structures whose failure may result in loss of life However CWE is increasingly being used in such applications as the evaluation of pedestrian comfort in zones of intensified wind speeds see Chapter 15 and the estimation of wind effects on solar collectors in solar power plants 5 In a number of cases CWE can provide solutions that may be used for preliminary design purposes if backed by proper validation see the UK Design Manual for Roads and Bridges BD 4901 6 the Eurocode prEN 199114 7 and the Architectural Institute of Japan Guidebook 8 Currently CWE research is aimed at creating tools allowing the development of aerodynamic data usable for structural design even in the absence of closely related adhoc experimental validation The purpose of this chapter is to present a brief compendium of selected informa tion on CWE modeling numerical issues and verification and validation procedures with a view to acquainting wind and structural engineers with the CWE vocabulary and facilitating dialogue between wind and structural engineers on the one hand and CWE professionals on the other It is shown in Chapter 12 that uncertainties in the aerodynamic pressures have con siderably less weight in the global uncertainty budget than do uncertainties in the wind speeds for this reason their effect on the estimates of overall effects of the flow on the structure are less severe than is the case in automotive or aeronautics applications The mathematical model used in CWE simulations consists of the governing equations of the flow Section 62 The governing equations need to be discretized and grids within a computational domain are generated for implementing the dis cretization Section 63 The requisite initial and boundary conditions are considered in Section 64 Numerical solutions for the flow as represented by the discretized com putational model are briefly discussed in Section 65 Section 66 concerns numerical Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 136 6 Computational Wind Engineering stability issues Section 67 summaries turbulence models Section 68 is a concise introduction to verification and validation VV and uncertainty quantification UQ Section 69 considers the role of wind tunnel testing and CWE prospects in the simulation of aerodynamic effects Section 610 briefly discusses best practice guidelines PBG 62 Governing Equations Fluid flows are described by the equation of continuity conservation of mass and the NavierStokes equations conservation of momentum CWE only considers incom pressible air flow see Chapter 4 for which the equation of continuity is Ui xi 0 61 where Ui are the velocity components in the xi directions in a Cartesian coordinate system i 1 2 3 Einstein notation is used in Eq 61 and subsequent equations The NavierStokes equations are Ui t Uj Ui xj 1 𝜌 p xi 𝜈 2Ui xjxj fi 62 where p is the pressure ν is the fluid kinematic viscosity and f i is the vector representing body forces eg the gravity or the pressuregradient force The nondimensional form of Eqs 61 and 62 highlights the dependence of the flow on the Reynolds number Re U i x i 0 63 U i t U j U i x j p x i 1 Re 2U i x j x j f i 64 where the nondimensional variables based on reference length Lref and velocity Uref are defined as x i xi Lref U i Ui Uref t t LrefUref tUref Lref p p 𝜌U2 ref f fi U2 refL1 ref 65 63 Discretization Methods and Grid Types Discretization of the governing equations Eqs 61 and 62 in CWE is commonly performed using finite difference finite volume or finite element methods FDM FVM or FEM respectively All methods discretize the computational domain using grids and approximate the governing partial differential equations by systems of algebraic equations FDM typically restricted to simple geometries uses Taylor series or polyno mial fitting to approximate at each grid point the derivatives that appear in the governing equations FVM the most commonly used discretization technique solves the integral form of the governing equations in a domain subdivided into small contiguous control k k k k 64 Initial and Boundary Conditions 137 a Structured regular grids c Blockstructured grids b Unstructured grids Figure 61 Types of grid volumes The method defines the control volume boundaries rather than the computa tional nodes and the values of the variables are approximated at the cell faces from the values at the control volume centers FEM is similar to FVM but uses weight functions aimed at minimizing approximation errors 9 Its main advantage is that it is readily applicable to flows with complex geometries Through the grid generation process the computational domain is composed of by a large number of cells consisting of nodes vertexes and lines joining adjacent nodes thus defining a grid also called mesh Grids can be structured or unstructured Struc tured grids Figure 61a are defined as families of grid lines such that lines of a single family do not cross each other and lines of a family cross lines of other families only once 10 Unstructured grids Figure 61b usually consist of triangles or quadrilaterals in two dimensions and tetrahedra and hexahedra in three dimensions typically in irreg ular patterns The generation of unstructured grids can be automated in computational domains with any level of geometric complexity However unstructured grids require more computational memory and entail higher costs than structured grids For parallel computing structured grids can be based on a multiblock approach in which a domain with complex geometries is decomposed into multiple blocks zones with simple geometries Figure 61c Interfaces between blocks should be located in regions in which the flow characteristics eg pressure and velocity gradient are not rapidly changing Unstructured grids are typically decomposed into zones by an algo rithm embedded in a mesh generation program 64 Initial and Boundary Conditions Simulations are of two generic types i steadystate simulations applied to equilibrium problems and ii marching simulations applied to transient problems In equilibrium problems the governing equations are solved once to determine the timeindependent solution In marching problems the equations are solved at each time step starting from the initial conditions to determine the timedependent solution as it advances in time Appropriate initial and boundary conditions in conjunction with the governing equations are required for constructing a wellposed mathematical model of the flow 641 Initial Conditions For timedependent simulations initial values are generally imposed in the computa tional domain The most effective initial conditions are solutions of the fully developed k k k k 138 6 Computational Wind Engineering flow obtained from previous simulations Results from steadystate simulations can be employed to expedite the turbulence development in transient flow simulations 642 Boundary Conditions Boundary conditions BC are typically defined in terms of boundary values of the unknown field and their derivatives BC commonly used in CWE applications are listed next Dirichlet boundary conditions assign at the boundary a constant 𝜙0 value for the vari able 𝜙 𝜙 𝜙0 66 If 𝜙 is a pressure or a velocity Eq 66 describes a constant pressure or a constant veloc ity field at the boundary condition respectively Von Neumann boundary conditions assign at the boundary a constant gradient of the variable 𝜙 𝜙 n 𝜙0 67 where n is normal to the boundary Convective boundary conditions also called nonreflective BC approximate the variable 𝜙 at a boundary near which the flow is convective but exhibits no diffusive effects that is for an upstream reference velocity Uref 𝜙 t Uref 𝜙 0 68 Periodic boundary conditions also called cyclic BC approximate cyclically repeating behavior as follows 𝜙tB 𝜙tBL 69 where B represents a boundary and L is the characteristic length of periodicity Noslip wall boundary conditions are applied to viscous flow bounded by a solid wall where the flow velocity relative to the wall vanishes that is for a stationary wall UP U 0 610 where UP and U are the tangential and normal components of the velocity vector respectively This boundary condition is typically used near the wall when the grids in that region are fine enough to resolve the flow throughout the viscous sublayer ie for z 1 where z uz𝜈 z is the direction normal to wall and u is the friction velocity Slip or inviscid wall boundary conditions model a zeroshear solid wall ie no friction at the interface of fluid and structure Thus the velocity component normal to the wall is zero U 0 611 and the gradients normal to the wall of the velocity components are assumed to be zero UP n U n 0 612 This can be used for a wall above which viscous effects are negligible or for a far boundary field that influences negligibly the flow physics of interest k k k k 66 Stability 139 Symmetry Boundary Conditions are employed on a plane when the flow is assumed to be symmetric with respect at that plane Thus there is no fluxes across the plane meaning that the velocity normal to the boundary is zero U 0 613 In addition the gradient of the velocity tangent to the symmetry plane in the direction normal to that plane is zero UP n 0 614 which means that the shear stress is zero but the normal stress is not zero U n 0 on the symmetry plane which is not the cases in the noslip and slip wall boundary condition Another requirement is zero gradient of all scalar quantities 𝜙s normal to the symmetry plane 𝜙s n 0 615 65 Solving Equations For CWE applications modeled by nonlinear partial differential equations matrix equations are solved by iterative methods in which the initial solution is assumed the equation is linearized and the solution is improved by repeating the process until an acceptable solution is obtained More details of the solutions of the systems of equations are provided in 10 Chapter 5 In incompressible flow a difficulty arises in the solution of the governing equations since no independent equation for the pressure is available The conservation of momen tum equations contain pressure gradient terms and in combination with the continuity equation can be used to determine the pressure field as a function of time and space using methods discussed in 10 Chapter 7 11 and 12 66 Stability Numerical approximations to the governing equations may exhibit unstable behavior that is they may magnify errors that occur as a result of discretization Stability is assured by satisfying the CourantFriedrichsLewy CFL condition which requires that the distance traveled by a fluid element per time step not be larger than the distance between adjacent grid points 13 14 In 1D simulations the CFL condition is CFL U Δt Δx Cmax 616 where U and Δx are the flow velocity and the grid size in the x streamwise direction respectively Δt is the chosen time step and Cmax is the upper bound of the CFL number which is less than unity and can vary depending on numerical schemes employed for solving the equations If Cmax 08 is chosen the largest time step used in the simulation is estimated as Δtmax 08Δx U 617 k k k k 140 6 Computational Wind Engineering The CFL condition can be extended to 3D simulations as follows CFL max U Δx V Δy W Δz Δt Cmax 618a or CFL U Δx V Δy W Δz Δt Cmax 618b The corresponding largest time step can be estimated by Eq 618a or more conserva tively by Eq 618b 67 Turbulent Flow Simulations 671 Resolved and Modeled Turbulence A turbulent flow consists of turbulent motions over broad range of length and time scales as illustrated by the energy spectrum E𝜅 per unit of wave number 𝜅 in Figure 62 from energycontaining eddies to energydissipation eddies The smallest scales of turbulent flow associated with energydissipation eddies 15 are l𝜂 𝜈3 𝜀 14 length 𝜏𝜂 𝜈 𝜀 12 time u𝜂 𝜈𝜀14 velocity 619abc where 𝜈 is the kinematic viscosity and 𝜀 is the rate of energy dissipation of the turbu lent kinetic energy k defined as k 12uiui For details on the energy spectrum see Section 243 Strategies for the simulation of turbulence motions depend on the extent to which eddy motions are resolved on the one hand and modeled empirically on the other Figure 62 Direct Numerical Simulation DNS resolves all turbulent scales and uses no turbulence modeling Section 672 Large Eddy Simulation LES resolves the largescale turbulent eddies and models the smallscale eddies Section 673 In steady ReynoldsAveraged NavierStokes Simulation RANS1 all turbulent eddies are modeled Unsteady ReynoldsAveraged NavierStokes simulation URANS models all turbulent eddies but resolves lowfrequency motions associated with unsteadiness in the mean flow such as vortexshedding Section 674 Hybrid RANSLES employs the RANS approach near walls and LES in regions far from the walls Section 675 Simulation costs increase as the resolved part of the simulation increases The resolved and modeled parts in each turbulence model are illustrated in Figure 62 672 Direct Numerical Simulation DNS DNS is the most reliable approach to the simulation of turbulent flows It consists of solving the discretized governing equations of the fluid motion by explicitly resolving all scales of turbulence down to the dissipation scale without resorting to empirical 1 ReynoldsAveraged NavierStokes Simulation is also referred to as ReynoldsAveraged Numerical Simulation k k k k 67 Turbulent Flow Simulations 141 Energy input from mean flow Energy dissipation Energy containing eddies Wave number k length scale1 Modeled Resolved Resolved Modeled Inertial subrange Energy dissipation eddies Normalized energy spectrum Ek Resolved 0l1 SGS l1 Ek ε 23 κ 53 Modeled DNS LES Hybrid RANSLES URANS Steady RANS Energy cascade ηl1 Figure 62 Turbulence spectrum turbulence modeling DNS must satisfy the following conditions First the smallest resolved scales must be in the order of the dissipation scales that is in the order of 1 mm for atmospheric boundary layer ABL flow Second the dimensions of the computational domain Lx Ly Lz in the x y z directions must be significantly larger than i the largest scales of the turbulent flow the scales can be in the order of hundreds of meters for ABL flow and ii the characteristic length of the structure for signature turbulence 11 16 In addition the domain must be sufficiently large to reduce the blockage effect to an acceptable level eg 25 blockage ratio Under the assumption that in the of energydissipating range the eddies can be resolved by fourpoint grids in each direction Δx Δy Δz l𝜂4 the number of cells can be estimated as the ratio of the volume of the computational domain to the volume of a cell that is Nxyz LxLyLz l𝜂43 620 Assuming for a fullscale simulation that Lx 1000 m Ly Lz 100 m and l𝜂 0001 m Nxyz is in the order of 1018 It can be shown that the corresponding minimum number of timesteps in a simulation with turnover time T0 over which the largest eddies with scale l0 break down into eddies with dissipation scales l𝜂 is in the order of Nt T0 Δt l0k12 l𝜂4k12 4l0 l𝜂 621 k k k k 142 6 Computational Wind Engineering where the square root of the turbulent kinetic energy has the dimension of a velocity For Lx 1000 m Ly Lz 100 m l0 150 m and l𝜂 0001 m Nt 6 105 and the computational cost of the DNS simulation for this example is commensurate with Nxyz Nt 1023 For boundary layer flows near a wall the first in a direction normal to the wall should be located at a distance z 1 from the wall there should be 35 cells in the direction normal to the wall up to z 10 The grid sizes should be Δx 1015 in the direc tion of the tangent to the wall and Δy 5 in the crossstream direction Δx uΔx𝜈 and Δy uΔy𝜈 17 Therefore the grid sizes are inversely proportional to friction velocity u and therefore to the Reynolds number of the flow Using current computer technology DNS can only be applied to practical problems for which the Reynolds numbers are low For CWE applications time and memory requirements for DNS simulations are prohibitive to date It has been estimated that DNS simulations may become feasible for the analysis of common engineering problems by 20502080 18 19 673 Large Eddy Simulations LES LES resolves the timeaveraged and unsteady motions of largescale turbulent eddies and models small subgridscale SGS eddies The largescale eddies contain most of the energy of the flow and have the largest contribution to the Reynolds stress tensor 𝜏ij 𝜏ij uiuj i j 1 2 3 622 where ui is the fluctuating velocity component in the ith direction the subscripts 1 2 and 3 represent the x y and z directions respectively and the overbar denotes timeaveraging The size of the small eddies to be modeled is determined by the filter width ΔSGS The small eddies are approximately isotropic and do not depend upon the characteristics of largescale flow The velocity field for the unfiltered motion can be written as Ux t ux t u SGSx t 623 where ux t is the velocity in the filtered motion and u SGSx t is the subfiltered tur bulent velocity The filtered velocity can be obtained using explicit filter functions eg tophat or Gaussian filter function 20 or through an implicit filtering process by grid scales While the former approach is used for fundamental turbulence studies the lat ter is commonly used in applications The filtering approach attenuates small eddies whose sizes are smaller than ΔSGS and leaves the large and intermediatescale eddies unchanged Figure 63 illustrates the spatially filtered velocity as affected by the filter width 21 To resolve the motion of large and intermediatescale eddies LES uses the governing equations based on filtered variables ui xi 0 624 ui t uiuj xj 1 𝜌 p xi 𝜈 2ui xixj 𝜏R ij xj fi 625 k k k k 67 Turbulent Flow Simulations 143 U Wind velocity Space Δ1 Δ2 x u u1 u2 SGS1 uʹ SGS2 uʹ Figure 63 Illustration of unfiltered velocity field Ux and resolved velocity field uix based on filter width Δi i 1 2 21 Reprinted from Figure 24 of LargeEddy Simulation in Hydraulics W Rodi G Constantinescu and T Stoesser 2013 CRC Press 20 with permission from Taylor Francis where f i is the filtered external force vector per unit mass and the SGS stress tensor called residual stress tensor is 𝜏R ij UiUj uiuj 626 The SGS stress can be decomposed into an isotropic and a deviatoric part 𝜏R ij 1 3𝛿ij𝜏R kk 𝜏r ij 627 where 𝛿ij is the Kronecker delta and k 1 2 3 Substituting Eq 627 into Eq 625 the LES governing equations become ui t uiuj xj 1 𝜌 p xi 𝜈 2ui xjxj 𝜏r ij xj fi 628 where p p 1 3𝜌𝛿ij𝜏R kk 629 Closure of Eq 628 requires the development of SGS models that is models of the deviatoric SGS stress 𝜏r ij The models predict effects of the SGS stresses on the resolved motion whose length scales depend upon the filter width ΔSGS For uniform grids with mesh size Δ ΔSGS Δ For nonuniform grids ΔSGS ΔxΔyΔz13 for example The widely used Smagorinsky SGS model 22 approximates the deviatoric SGS stress 𝜏r ij by assuming the validity of Boussinesqs eddy viscosity hypothesis 23 according to which the deviatoric part of the Reynolds stress is proportional to the strain rate tensor of the filtered resolved velocities Sij 12uixj ujxi that is 𝜏r ij 2𝜈tSGSSij 630 where 𝜈t SGS is the kinematic eddy viscosity to be modeled under the assumption that the eddy viscosity is proportional to a typical length scale lSGS and a velocity scale k k k k 144 6 Computational Wind Engineering qSGS of the flow The SGS turbulent eddy viscosity in the Smagorinsky model can be expressed as 𝜈tSGS lSGSqSGS CsΔSGS2S 631 where the characteristic length and velocity scales are Cs ΔSGS and CsΔSGS S respectively the Smagorinsky constant Cs varies depending upon the flow between 01 and 02 24 and S 2SijSij12 This model has been widely used on account of its simplicity and computational efficiency However the use of a constant value for Cs makes it difficult to predict accurately complex flows For example Cs 017 as determined for isotropic homogeneous turbulence 25 should be decreased for flow with strong mean shear especially near a wall 26 in order to reduce the amount of dissipation introduced by the SGS model and the resulting spurious SGS stresses 11 For this reason in the Smagorinsky model a nearwall correction is required to capture the nearwall effects To address the shortcomings of the Smagorinsky model dynamic SGS models have been proposed for nonisotropic flows 27 28 in which the model parameter is automatically reduced near the wall from its value for isotropic flow Improved SGS models still need to be developed for complex geometry and highly anisotropic flow applications Reliable LES simulations require sufficiently fine spatial and temporal scales The grid sizes should be l𝜂 ΔSGS l0 see Figure 62 The computational domain size required for LES simulations is the same as for DNS To resolve flows in the wall region the typical requisite grid sizes close to the wall are Δx 50 in the alongwall streamwise direction and Δy 15 in the crossstream direction in the normaltowall direction the first grid point from the wall should be at z 1 while at least three grid points in the viscous region 1 z 10 and 3050 grid points within the boundary layer are required 20 The total number of grid points for wallresolving LES is smaller than for DNS 29 but it is still prohibitively expensive particularly for high Reynolds number flows over wallmounted structures Approaches to reducing the computational cost include using walllayer models called WallModeled LES or WMLES 30 or using hybrid RANSLES methods 31 are discussed in Section 675 674 ReynoldsAveraged NavierStokes Simulation RANS RANS are a primary approach for practical turbulent flow simulations owing to their simplicity and relatively low computational cost RANS simulates the averaged fields of turbulent flows by solving the Reynoldsaveraged NavierStokes equations In RANS the flow field is divided by Reynolds decomposition into a mean flow field and a fluctuating field For example the flow velocity can be expressed as Ux t ux t u RANSx t 632 where u is the timeaveraged velocity and u RANS is the fluctuating component Steady RANS based on timeaveraging is used to simulate timeindependent flow URANS based on ensembleaveraging simulates time and spacedependent flow It has been noted that while all turbulent flows are unsteady not every unsteadiness is turbulence k k k k 67 Turbulent Flow Simulations 145 U t ut uʹRANSt Wind velocity Time u t RANSt uʹ t u t Figure 64 Illustration of Reynolds decomposition Ut ut u RANSt For example in flows with a largescale periodicity due to vorticity shed in the wake of a structure that periodicity would be suppressed by timeaveraging but is preserved under ensembleaveraging URANS is applied to such flows Figure 64 The equations of Reynoldsaveraged flow field are derived by applying the ensemble averaging operation Eq 632 to the governing equations Eqs 61 and 62 Using the decomposition and noting that Ui u the governing equations for URANS are derived as ui xi 0 633 ui t xj uiuj 1 𝜌 p xi 𝜈 2ui xixj 𝜏ij xj f i 634 where 𝜏ij is the Reynolds stress tensor 𝜏ij u RANSi u RANSj UiUj uiuj 635 Equation 635 accounts for momentum flux generated by all turbulent fluctuations while the residual stresses in LES Eq 626 exclude the contribution of resolved tur bulent fluctuations Note that the first term in the lefthand side of Eq 634 does not exist in the steady RANS governing equations The URANS governing equations cannot be solved because the Reynolds stresses are unknown To close the system it is required that the Reynolds stresses be approximated in terms of the averaged quantities Under the Boussinesq approximation see Eq 630 in LES the Reynolds stress tensor is 𝜏ij 2𝜈tRANSSij 636 where Sij 12uixj ujxi is the rate of strain tensor of averaged flow field and 𝜈tRANS is the RANS turbulent eddy viscosity to be modeled similar to Eq 631 in LES as 𝜈tRANS lRANS qRANS C𝜇lRANS qRANS 637 k k k k 146 6 Computational Wind Engineering In Eq 637 lRANS and qRANS are the typical length and velocity scales of a turbulent flow respectively and C𝜇 is a nondimensional constant determined in a calibration proce dure A broad selection of closure models of the Reynolds stresses is available 32 including linear eddy viscosity models nonlinear eddy viscosity models and Reynolds stress models Among linear eddy viscosity models the SST Shear Stress Transport model 33 and the SpalartAllmaras SA model 34 are considered capable of predicting reliably flows around bluff bodies with strong adverse pressure gradients and massive flow separation For example SST uses the k𝜔 model 32 for boundarylayer or inner layer flows and the k𝜀 model 35 36 for low shear layer or outer layer flows A blending function is employed for the transition between the two models For details see 37 The spatial and temporal requirements for RANS simulations are much less demand ing than for DNS and LES However RANS simulations should have sufficiently fine grids to capture the change of the averaged flow field especially for nearwall regions characterized by high velocitygradient flow The RANS models typically have two options for the treatment of nearwall flow i resolving the flow called lowRe model and ii using wall functions highRe model In the flow near the wall lowRe RANS models generally require grid resolutions as fine as LES in the direction normal to the wall but much coarser grids in the walltangential streamwise and acrossstream directions than LES The increase in aspect ratios of cells near the wall can therefore lead to a substantial reduction in the total number of cells For the highRe RANS models using typical wall functions the grid closest to a wall should be located in the log layer beyond the viscous sublayer eg 30 z 500 where the upper limit depends on the Reynolds number of the flow so that the wall functions can bridge the gap between the nearwall and the fully turbulent flow region This option can save considerable computational time due to the alleviated grid requirement but the performance can be poor especially for flows around bluff bodies since wall functions are generally developed for relatively simple flows such as flows over flat plates 675 Hybrid RANSLES Simulation URANS models typically perform unsatisfactorily for massively separated flows char acterized by large turbulence scales 16 Such flows can be better simulated by LES However LES simulations of high Reynolds number flows over wallmounted structures are still challenging owing to the prohibitive grid requirements for nearwall regions To alleviate the nearwall grid resolution problem in massively separated flows hybrid RANSLES models have been proposed for example 31 38 These models work in the RANS mode for nearwall flow regions and transition to the LES model for regions away from the wall The nearwall flow is simulated by a less accurate but computation ally more efficient RANS and large turbulent eddies from massively separated flow are resolved by LES at manageable computational cost Detached Eddy Simulations DES 31 are the most widely used hybrid RANSLES model for flows over wallmounted structures at high Reynolds numbers including k k k k 67 Turbulent Flow Simulations 147 0 0 05 10 15 20 0 05 10 15 20 0 05 10 15 20 25 30 35 40 05 z δ z δ x δ x δ x δ 10 0 05 10 0 05 10 a b c Figure 65 Types of grid in boundary layers The dashed line represents the velocity profile 39 Source Reproduced with permission of ANNUAL REVIEWS flows over bluff bodies of interest in wind engineering DES is a nonzonal type2 of model that modifies the original RANS model and includes a transition from RANS to LES The DES SA 31 and DES SST hybrid models 37 are based on the SA RANS 34 and the SST RANS model 33 respectively The computational cost of DES is much lower than for LES but is still higher than RANS Grid generation strategy is much more complicated for DES than for RANS or LES As reported in 16 39 40 the DES model induces grey areas in which the flow is not adequately solved by either pure RANS or pure LES In those areas the turbulence energy modeled in RANS may not be adequately transferred to LESresolved turbulence energy This effect called ModeledStressDepletion MSD may cause premature sepa ration due to inadequate grid spacing GridInduced Separation or GIS 39 As shown in Figure 65 the grid sizes in boundarylayer flow are assumed to be Δx Δy Δz ie Δmax max Δx Δy Δz Δx where the x y and z directions are the alongwind the acrosswind and the normaltowall direction respectively In the DES formulation for grids with Δmax𝛿 1 see Figure 65a the RANS mode is activated in the whole boundary layer If Δmax𝛿 1 Figure 65b c the switch from RANS to LES is activated within the boundary layer thickness This causes unphysical behaviors associated with MSD and GIS Updated versions of DES called Delayed DES DDES 41 and Improved DDES IDDES 42 have been proposed that attempt to improve upon DES For grids with Δmax𝛿 05 to 1 Figure 65b DDES and IDDES prevent LES mode activation For grids with Δmax𝛿 1 Figure 65c IDDES fully enables the LES mode except for wall modeling which is performed in the RANS mode as in WallModeled LES DES perfor mance depends upon type of grid as represented in Figure 65 In particular it has been observed that in some instances the DES performance does not necessarily improve if the grid size is reduced 39 43 2 Another approach to hybrid RANSLES models is a zonal model with distinct zones occupied by pure RANS and pure LES and discontinuous solutions at interfacing boundaries See details in 16 k k k k 148 6 Computational Wind Engineering a 2D SST RANS CD 078 b 2D SST URANS CD 173 c 3D SST URANS CD 124 d SA DES coarse grid CD 116 e SA DES fine grid CD 126 f SST DES fine grid CD 128 SA SpalartAllmaras SST Shear Stress Transport Figure 66 Vorticity isosurfaces around a circular cylinder Re 5 104 experimental drag coefficient CD 115125 39 Source Reproduced with permission of Annual Reviews 676 Performance of Turbulence Models Figure 66 shows visualizations of resolved vortical flow structures around a circular cylinder simulated using various turbulence models 39 As expected 2D steady RANS cannot predict the vortex shedding Figure 66a Note that even 3D steady RANS cannot accurately predict the averaged flow characteristics in such unsteady and separated flows 44 Twodimensional URANS allows the simulation of 2D large eddies associated with vortex shedding but does not capture 3D flow structures Figure 66b 3D URANS captures 3D flow structures but cannot resolve smaller flow structures using finer grids Figure 66c 45 DES predicts 3D flow structures up to finer scales than URANS Figures 66d and e with a sufficiently fine grid it can resolve fine flow structures in the separated flow region Figures 66e and f show that the performance of DES in the LES region does not depend significantly upon the choice of its RANS models ie whether SA or SST 39 68 Verification and Validation Uncertainty Quantification The credibility of CWE simulations depends upon the quality of the physical modeling the competence of the analysts performing the simulations the simulations verification and validation VV and the UQ of the simulation results 46 The analysts depth of understanding of the modeling details and of the simulation results plays a decisive role in the simulation process VV consists of procedures required for assessing the accuracy of simulation results Uncertainty quantification is aimed at identifying characterizing and estimating quantitatively the factors in the analysis that affect the accuracy of the simulation results 46 47 The amount of research into VV and UQ is vast 4853 This k k k k 68 Verification and Validation Uncertainty Quantification 149 section is limited to introducing the reader to a few salient features of their respective procedures Section 681 briefly discusses sources of inaccuracy in CWE simulations Section 682 is a summary description of VV aims and procedures Section 683 is concerned with UQ 681 Sources of Inaccuracy in CWE Simulations Although as shown in Section 683 errors and uncertainties are distinct concepts it will be convenient in this section to refer to both as errors Errors arising in CWE simulation results are typically of four types 11 46 i physical modeling ii discretization iii iteration and iv programinguser errors Physical modeling errors 𝛿model are differences between the behavior of the real phys ical object and its model counterpart 𝛿model pmodel preal 638 where pmodel and preal are the respective response values of interest eg velocity or pres sure They arise from approximations of complex behavior in the governing equations eg approximations inherent in turbulence models effects of computational domain size and boundary conditions and assumptions on fluid properties eg constant air density and temperature Discretization errors 𝛿h are differences between the exact analytical solution of a mathematical model and the exact solution of the models discretized counterpart 𝛿h ph pmodel 639 where ph is the response calculated from the discretized model Discretization error should be estimated for every new type of grid solution scheme or application Among the sources associated with numerical errors the discretization errors are usually the largest and their estimation is the most challenging 46 Iterative errors 𝛿it are differences between the exact and computed solutions of the discretized equations 𝛿it pcomp ph 640 where pcomp is the solution obtained from a computing machine which may entail roundoff errors and convergence errors inherent in iterative methods Roundoff errors resulting from low precision in computer calculations can affect the stability of the solutions In simulations with a stable scheme and negligible roundoff error accumulation the roundoff errors are usually very small compared to other errors 11 The iterationconvergence errors are present because a linearized system of discretized equations is typically solved iteratively In general iterative errors are at least one or two orders of magnitude lower than the discretization errors 11 However if a flow solver uses implicit time integration for unsteady simulations a loose iterative convergence criteria at each time step may lead to significant influence on accuracy of the numerical solution 64 Programinguser errors caused by mistakes or bugs in the software can be classified into two types 46 critical errors by which the software cannot execute a simulation or generate reasonable results and less critical but still nonnegligible errors due to dor mant software faults that may not be easily identified by code verification User errors k k k k 150 6 Computational Wind Engineering are due to blunders or mistakes from users in input preparation for simulation and in postprocessing for output data analysis Human errors generally are not easily detected especially when largescale simulations of complex systems are performed 682 Verification and Validation The objective of verification and validation VV is to establish the credibility of a com putational model by assessing the degree of accuracy of the simulation results 46 The philosophy definition and procedure of VV on modeling and simulation have been developed in practicing communities such as AIAA American Institute of Aeronau tics and Astronautics 50 ASME American Society of Mechanical Engineers 48 49 and DOE Department of Energy 53 Verification is the process of determining that a computational model accurately rep resents the underlying mathematical model and its solution 49 Validation is the pro cess of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model 50 VV processes start with determining the intended uses of the computational model The accuracy require ments for the responses of interest are determined accordingly Verification process addresses the correct implementation of a numerical model in a code and the estima tion of numerical errors in solutions of discretized equations Model validation process employs the verified simulation results and relevant experimental data and assesses the predictive capacity of the model If the agreement between model predictions and experimental outcomes satisfies the accuracy requirement the VV processes end A successful VV can claim that the accuracy of the computational model is adequate for the intended use of the model Otherwise the VV processes are repeated by updating the model and if necessary carrying out additional experiments until the agreement is acceptable For details see 49 Note that the documentation of the VV activities and results serves not only for justifying the current intended use but also for providing informationexperience for potential future uses Verification is limited to estimating numerical errors and is not concerned with the accuracy of physical modeling The verification process can be divided into code verifica tion and solution verification Code verification addresses the correct implementation of the numerical algorithm in the computer code by evaluating the error for a known highly accurate solution referred to as verification benchmark Code verification in gridbased simulations can be performed by a systematic discretization convergence test eg 54 and its convergence to a benchmark solution Practical approaches have been developed for example the method of manufactured solutions 55 to generate analytical solutions required for code verification Code verification is usually performed by code develop ersvendors but should also be performed for specific applications by CWE users of commercialopensource codes 46 48 After the code verification is completed solution verification is conducted The solu tion verification deals with i correctness of the input and output data for a particular solution of a problem of interest and ii numerical accuracy erroruncertainty estima tion for the simulated solution in the discretized time and space domain 46 Numer ical solutions and the errors inherent in them 𝛿h in Eq 639 and 𝛿it in Eq 640 estimated in the solution verification process are considered in the validation process For typical CWE problems numerical errors can be estimated a posteriori for example k k k k 69 CWE versus Wind Tunnel Testing 151 by using multiple simulations with different grid resolutions 47 Solution verification should be performed by CWE users and be required by structural engineers who use CWE simulation results for structural design The interest of CWE users in VV lies in validation of a computational model for the intended use The validation process assesses the accuracy of the computational model by comparison with experimental data quantifies predictive uncertainty in interpola tion or extrapolation of the model and evaluates the acceptability of the model for the intended use 46 56 683 Quantification of Errors and Uncertainties Error and uncertainty are often used interchangeably In particular this is the case in Chapters 7 and 12 for applications unrelated to CWE However in the AIAA VV guide for CFD 50 errors are defined as recognizable deficiencies in all phases or activities of modeling and simulation that are not due to lack of knowledge whereas uncertainties are defined as potential deficiencies in any phase or activity of the modeling process that are due to lack of knowledge Errors can be classified as acknowledged errors and unacknowledged errors Acknowledged errors can be identified and eliminated eg roundoff errors dis cretization errors iterative errors Unacknowledged errors cannot be found or removed eg programming errors improper use of the CWE code Uncertainties can be classified as aleatory and epistemic Aleatory irreducible uncertainties are associated with inherent randomness eg input parameters of a model Epistemic reducible uncertainties are related to a lack of knowledge of or information on a physical model For details see 46 The ASME VV approach 48 provides quantitative evaluations of uncertainties in simulation results by comparison with their counterparts in experiments and employs concepts and definitions of error and uncertainty borrowed from metrology 57 69 CWE versus Wind Tunnel Testing Wind tunnel testing is currently an indispensable tool used i to obtain aerodynamic or aeroelastic data on special structures for which no such data are available and ii to improve standard provisions Its drawbacks include the following i its first costs and the maintenance costs are high ii testing is timeconsuming iii it typically entails violation of the Reynolds number and of other similarity criteria applicable to certain types of special structures eg airsupported structures and iv it is not consistently reliable see Appendix F for highrise building and 58 for lowrise building testing As computer technology and numerical techniques have evolved the prospect of performing CWE simulations has become increasingly attractive given their following potential advantages i ready availability ii relatively low initial and maintenance costs iii relatively fast turnover times iv less restrictive model scale limitations v capability to solve multiphysics problems eg windstructure interaction or rainwind scenarios and vi as is also the case for wind tunnel simulations the fact that errors and uncertainties affecting the estimation of aerodynamic effects have significantly less weight than their wind climatological counterparts see Chapter 12 However CWE is k k k k 152 6 Computational Wind Engineering not yet accepted as a structural design tool because typically its results cannot be used confidently Wind tunnel testing and whenever possible fullscale measurements will still be required for validation purposes until CWE will have evolved into a fully reliable independent tool 610 Best Practice Guidelines Using CWE for selected applications requires the development of appropriate mathe matical models computational grids and domains spatial and temporal discretization schemes solvers turbulence models boundary conditions and convergence criteria capable of being successfully subjected to rigorous VV procedures Best practice guidelines can facilitate the use of such development and cover general applications 59 as well as specific fields such as urban environmental wind 60 61 nuclear power plants eg nuclear reactor safety application 62 dry cask application 63 and structural loads on buildings 8 Best practice guidelines cover a limited number of simulations Therefore it is recommended that VV procedures be applied to simu lations that deviate in any significant aspect from existing simulations covered by the guidelines References 1 Tamura Y and Phuc PV 2015 Development of CFD and applications monologue by a nonCFDexpert Journal of Wind Engineering and Industrial Aerodynamics 144 313 2 Blocken B 2014 50 years of computational wind engineering past present and future Journal of Wind Engineering and Industrial Aerodynamics 129 69102 3 Dagnew AK and Bitsuamlak GT 2013 Computational evaluation of wind loads on buildings a review Wind Structures 16 629660 4 Michalski A Kermel PD Haug E et al 2011 Validation of the computational fluidstructure interaction simulation at realscale tests of a flexible 29 m umbrella in natural wind flow Journal of Wind Engineering and Industrial Aerodynamics 99 4 400413 5 Andre M MierTorrecilla M and Wuchner R 2015 Numerical simulation of wind loads on a trough parabolic solar collector using lattice Boltzmann and finite element methods Journal of Wind Engineering and Industrial Aerodynamics 146 185194 6 HE Volume I Highway structures approval procedures and general design in Design manual for roads and bridges DMRB Highways England HE 2001 7 CEN Eurocode 1 Actions on structures Part 14 Gernal actions Wind actions in EN 199114 ed European Committee for Standardization CEN 2005 8 AIJ Guidbook of recommendation for loads on buildings 2 Windinduced response and load estimationPractical guides of CFD for wind resistant design Tokyo Japan Architectural Institute of Japan 2017 p 434 9 Donea J and Huerta A 2003 Finite Element Methods for Flow Problems 1st ed Chichester UK Wiley k k k k References 153 10 Ferziger JH and Peric M 2002 Computational Methods for Fluid Dynamics 3rd ed New York Springer Verlag 11 Zikanov O 2010 Essential Computational Fluid Dynamics 1st ed Hoboken New Jersey Wiley 12 Issa RI 1986 Solution of the implicitly discretised fluid flow equations by operatorsplitting Journal of Computational Physics 62 4065 13 Courant R Friedrichs K and Lewy H 1967 On the partial difference equations of mathematical physics IBM Journal 11 215234 14 Courant R Friedrichs K and Lewy H 1928 Über die partiellen Differenzengle ichungen der mathematischen Physik Mathematische Annalen 100 3274 15 Kolmogorov A N The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers in Dokl Akad Nauk SSSR 1941 pp 299303 16 Sagaut P Deck S and Terracol M 2013 Multiscale and Multiresolution Approaches in Turbulence 2nd ed London Imperial College Press 17 Tucker PG 2014 Unsteady Computational Fluid Dynamics in Aeronautics 1st ed Dordrecht the Netherlands Springer 18 Spalart PR 2000 Strategies for turbulence modelling and simulations Interna tional Journal of Heat and Fluid Flow 21 252263 19 Voller VR and PortéAgel F 2002 Moores law and numerical modeling Journal of Computational Physics 179 698703 20 Sagaut P 2006 Large Eddy Simulation for Incompressible Flows 3rd ed Berlin Germany SpringerVerlag Berlin Heidelberg 21 Rodi W Constantinescu G and Stoesser T 2013 LargeEddy Simulation in Hydraulics London UK CRC Press 22 Smagorinsky J 1963 General circulation experiments with the primitive equations I The basic experiment Monthly Weather Review 91 99164 23 Boussinesq J 1877 Essai sur la théorie des eaux courantes Mémoires présentés par divers savants à lAcadémie des Sciences 23 1680 24 PortéAgel F Meneveau C and Parlange MB 2000 A scaledependent dynamic model for largeeddy simulation application to a neutral atmospheric boundary layer Journal of Fluid Mechanics 415 261284 25 Lilly DK 1967 The representation of smallscale turbulence in numerical sim ulation experiments In IBM Scientific Computing Symposium on Environmental Sciences ed HH Goldstein 195210 New York Yorktown Heights 26 Deardorff JW 1971 On the magnitude of the subgrid scale eddy coefficient Journal of Computational Physics 7 120133 27 Germano M Piomelli U Moin P and Cabot WH 1991 A dynamic subgridscale eddy viscosity model Physics of Fluids A Fluid Dynamics 3 17601765 28 Lilly DK 1992 A proposed modification of the germano subgrid closure method Physics of Fluids 4 633 29 Choi H and Moin P 2012 Gridpoint requirements for large eddy simulation Chapmans estimates revisited Physics of Fluids 24 011702 30 Piomelli U and Balaras E 2002 Walllayer models for largeeddy simulations Annual Review of Fluid Mechanics 34 349374 31 Spalart P R Jou W H Strelets M and Allmaras S R Comments on the fea sibility of LES for wings and on a hybrid RANSLES approach in Advances in k k k k 154 6 Computational Wind Engineering DNSLES 1st AFOSR International Conference on DNSLES Ruston LA 1997 pp 137147 32 Wilcox D C Turbulence Modeling for CFD DCW Industries 2006 33 Menter F Zonal two equation k𝜔 turbulence models for aerodynamic flows in 23rd Fluid Dynamics Plasmadynamics and Lasers Conference ed American Institute of Aeronautics and Astronautics 1993 34 Spalart P R and Allmaras S R A oneequation turbulence model for aerody namic flows in 30th Aerospace Sciences Meeting and Exhibit Reno NV 1992 pp 122 35 Jones WP and Launder BE 1972 The prediction of laminarization with a twoequation model of turbulence International Journal of Heat and Mass Transfer 15 301314 36 Launder BE and Sharma BI 1974 Application of the energydissipation model of turbulence to the calculation of flow near a spinning disc Letters in Heat and Mass Transfer 1 131137 37 Menter FR Kuntz M and Langtry R 2003 Ten years of industrial experi ence with the SST turbulence model In Turbulence Heat and Mass Transfer 4 ed K Hanjalic Y Nagano and M Tummers 625632 Begell House Inc 38 Speziale CG 1998 Turbulence modeling for timedependent RANS and VLES a review AIAA Journal 36 173184 39 Spalart PR 2009 Detachededdy simulation Annual Review of Fluid Mechanics 41 181202 40 Spalart PR Deck S Shur ML et al 2006 A new version of DetachedEddy Simulation resistant to ambiguous grid densities Theoretical and Computational Fluid Dynamics 20 181 41 Strelets M Detached eddy simulation of massively separated flows in 39th Aerospace Sciences Meeting and Exhibit Reno NV 2001 42 Gritskevich MS Garbaruk AV Schütze J and Menter FR 2012 Development of DDES and IDDES formulations for the k𝜔 shear stress transport model Flow Turbulence and Combustion 88 431449 43 Ke J and Yeo D RANS and hybrid LESRANS simulations of flow over a square cylinder Presented at the 8th International Colloquium on Bluff Body Aerodynam ics and Applications Boston MA 2016 httpswwwnistgovwind 44 Iaccarino G Ooi A Durbin PA and Behnia M 2003 Reynolds averaged sim ulation of unsteady separated flow International Journal of Heat and Fluid Flow 24 147156 45 Shur M Spalart PR Squires KD et al 2005 Threedimensionality in ReynoldsAveraged NavierStokes solutions around twodimensional geometries AIAA Journal 43 12301242 46 Oberkampf W L and Roy C J Verification and Validation in Scientific Comput ing Cambridge UK Cambridge University Press 2010 47 Roache PJ 1997 Quantification of uncertainty in computational fluid dynamics Annual Review of Fluid Mechanics 29 123160 48 ASME Standards for verification and validation in computational fluid dynamics and heat transfer in ASME VV 202009 New York NY American Society of Mechanical Engineers 2009 k k k k References 155 49 ASME Guide for verification and validation in computational solid mechanics in ASME VV 102006 New York NY American Society of Mechanical Engineers 2006 50 AIAA Guide for the verification and validation of computational fluid dynamics simulations AIAAG0771998 American Institute of Aeronautics and Astronau tics Reston Virginia 1998 51 NASA NASA handbook for models and simulations An implementation guide for NSASTD7009 NASAHDBK7009 National Aeronautics and Space Administra tion Washington DC 2013 52 Kaizer J S Fundamental Theory of Scientific Computer Simulation Review NUREGKM0006 Nuclear Regulatory Commission Washington DC 2013 53 Pilch M Trucano T Moya J Froehlich G Hodges A and Peercy D Guide lines for Sandia ASCI verification and validation plans Content and Format Version 20 SAND20003101 Sandia National Laboratory Albuquerque NM 2001 54 Roache PJ 1994 Perspective a method for uniform reporting of grid refinement studies Journal of Fluids Engineering 116 405413 55 Roache PJ 2002 Code verification by the method of manufactured solutions Jour nal of Fluids Engineering 124 410 56 Oberkampf WL and Trucano TG 2008 Verification and validation benchmarks Nuclear Engineering and Design 238 716743 57 JCGM International vocabulary of metrology basic and general concepts and associated terms JCGM 2002012 JCGM 2002008 with minor corrections Joint Committee for Guides on Metrology 2012 58 Fritz WP Bienkiewicz B Cui B et al 2008 International comparison of wind tunnel estimates of wind effects on lowrise buildings testrelated uncertainties Journal of Structural Engineering 134 18871890 59 Casey M and Wintergerste T 2000 ERCOFTAC Special Interest Group on Quality and Trust in Industrial CFD Best Practice Guidelines Brussels Belgium ERCOFTAC European Research Community on Flow Turbulence and Combustion 60 Blocken B 2015 Computational fluid dynamics for urban physics importance scales possibilities limitations and ten tips and tricks towards accurate and reliable simulations Building and Environment 91 219245 61 Franke J Hellsten A Schlünzen H and Carissimo B Best practice guideline for the CFD simulation of flows in the urban environment COST Action 732 COST Brussels Belgium 2007 62 Menter F CFD Best Practice Guidelines for CFD Code Validation for ReactorSafety Applications EVOL ECORA D01 European Commission 5th EURATOM Framework Programme 2002 63 Zigh G and Solis J Computational Fluid Dynamics Best Practice Guidlines for Dry Cask Applications NUREG2152 Nuclear Regulatory Commission Washing ton DC 2013 64 Eça G Vaz L and Hoekstra Iterative errors in unsteady flow simulations Are they really negligible Presented at the 20th Numerical Towing Tank Symposium NuTTS 2017 Wageningen The Netherlands 2017 k k k k 157 7 Uncertainties in Wind Engineering Data 71 Introduction Structural design for wind is affected by errors and uncertainties1 in the measurement and modeling of the micrometeorological wind climatological and aerodynamic factors that determine the wind load Uncertainty quantification is a complex task on which research is ongoing Owing to insufficient information and data it is in many cases necessary to estimate uncertainties not only on the basis of measurements and statistical theory but also by making use of subjective assessments inferences from past practice and simplified structural reliability methods see Appendix E2 To provide context on the use of the uncertainties discussed in this chapter Section 72 presents a simple statistical framework originally developed in 1 that relates uncertainty estimates to the development of safety factors with respect to wind loads called wind load factors The wind load factor specified in the pre2010 versions of the ASCE 7 Standard is larger than unity ASCE American Society of Civil Engineers The 2010 and 2016 versions of the Standard specify a wind load factor equal to unity and to make up for this change specify far longer mean recurrence intervals MRIs of the design wind speeds than their pre2010 counterparts eg 700 years in lieu of 50 years Section 73 discusses the uncertainties considered in this chapter These are used in Chapter 12 to define wind load factors and mean recurrence intervals of design wind effects 72 Statistical Framework for Estimating Uncertainties in the Wind Loads The peak wind effect is a random variable it varies from realization to realization The following approximate expressions commonly hold for the expectation and coefficient of variation CoV ie ratio of the standard deviation to the expectation of the peak 1 For convenience the term uncertainties also applies to errors and uncertainties as defined in Chapter 6 2 The use of far more elaborate and rigorous methods than those developed so far for civil engineering purposes is required by NASA and the Department of Energy for a wide variety of applications Such methods which are beyond the scope of this chapter are discussed in NASAs Handbook for models and simulations available at httpsstandardsnasagovstandardnasanasahdbk7009 and in other documents mentioned in Chapters 6 and 12 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 158 7 Uncertainties in Wind Engineering Data wind effect ppk eg pressure force moment with an Nyear mean recurrence interval ppkN a Ez KdG𝜃m Cppk𝜃mU 2zref N 71 CoVppkN CoV2Ez CoV2Kd CoV2G𝜃m CoV2Cppk𝜃m 4CoV2Uzref N12 72 In Eq 71 the factor a is a constant that depends upon the type of wind effect and the overbar denotes expectation Ez is a surface exposure factor defined by the wind profile and specified in the ASCE Standard the subscript z denotes height above the surface The aerodynamically most unfavorable wind direction is denoted by 𝜃m Kd is a wind directionality reduction factor that accounts for the fact that the direction 𝜃m and the direction of the largest directional wind speeds typically do not coincide The peak aerodynamic coefficient Cppk𝜃m depends upon the area being considered which can be as small as a roof tile or as large as an entire building Once this dependence is taken into account for rigid structures the gust response factor G is unity Flexible structures experience dynamic effects that depend on both wind engineering and structural engineering features The factor G that characterizes dynamic effects is considered in Chapter 12 Uzref N is the wind speed with an Nyear MRI estimated from largest wind speed data regardless of direction The uncertainty in the wind speed Uzref N is due to measurement micrometeorological and probabilistic modeling errors and to the limited size of the data sample on which the estimation is based According to approximate estimates similar to those of 1 CoVEz 016 CoVCppk𝜃m 012 and on the basis of wind speed data at seven locations not exposed to hurricane winds 009 CoVUN 50 years 016 Research reported in 2 suggests that CoVKd 010 Derivation of Eqs 71 and 72 Consider the product p xy of two random variables x and y with means x y fluctuations about the mean x y and variances x2 y2 Then p p p x xy y xy xy yx 73 p x y 74 p2 x 2y2 y 2x2 2x yxy 75 p2 x 2y2 y 2x2 2x yxy 76 If x y are independent the last term in Eq 76 vanishes and CoV2p CoV2y CoV2x 77 However if x y as in the case of the square of the wind speeds it follows from Eq 76 that p2 4 x 2x2 hence the factor 4 in Eq 72 It is easy to see that Eqs 74 and 77 can be extended to a product of any number of mutually independent variables The larger the individual uncertainties in the factors that determine the wind loading the larger the overall uncertainty in the wind effect ppkN being considered and the larger the requisite wind load factor For example for a site at which wind speed data are k k k k 73 Individual and Overall Uncertainties 159 obtained from weather balloon measurements or from wind speeds at locations with poorly defined surface roughness conditions andor from a short extreme wind speed data record the overall uncertainty in the wind effect and therefore the corresponding wind load are greater than for a site at which the wind speed measurements are more reliable Similarly uncertainties in the measurement of aerodynamic pressures can be large if obtained in wind engineering laboratories that use inadequate simulation and measurement techniques Equations 71 and 72 make it possible to consider the effects of individual uncertainties collectively rather than in isolation and enable the estimation of the uncertainty in the overall wind effect as a function of individual uncertainties This allows a rational allocation of resources when considering the reduction of any individ ual uncertainty For example when using public databases of pressure coefficients the lack of data directly applicable to a building with a particular set of dimensions requires the use of interpolations This can result in errors as large as 15 say The reduction of such errors would require the development of databases with larger sets of model dimensions However if the 15 error in the pressure coefficient resulted in an error in the estimation of the design wind effect of only 5 say the expensive development of a database with higher resolution might in practice be considered unnecessary see Section 1242 Structural engineers have pointed out that wind engineering laboratory reports do not provide any indication on the requisite magnitude of the wind load factor see Appendix F or of augmented design mean recurrence intervals consistent with the uncertainties specific to the project at hand Equations 71 and 72 or similar estimates make it possible to depart from the notion that one wind load factor fits all They enable a differentiated approach that accounts albeit approximately for the explicit dependence of the wind load factor on individual uncertainties which may differ for some structures from their typical values The wind engineering laboratory can therefore help to achieve safe structural designs by providing in addition to point estimates uncertainty estimates of relevant aerodynamic and wind climatological features 73 Individual and Overall Uncertainties As noted in Section 71 uncertainty quantification is typically difficult or impossible to achieve rigorously and must therefore be based wholly or in part on subjective assess ments based on consensus among informed professionals in addition to being based on measurements physical considerations and statistical methods 731 Uncertainties in the Estimation of Extreme Wind Speeds LargeScale Extratropical Storms and Thunderstorms It is reasonable to assume that the distributions of extreme wind speeds in largescale extratropical storms and thun derstorms are Extreme Value Type I with parameters that differ at the same site for the two types of storm It is therefore possible to estimate the respective uncertainties by accounting for i measurement errors and ii sampling errors in the estimation of wind speeds for each of the two types of storm For design wind speeds with specified k k k k 160 7 Uncertainties in Wind Engineering Data mean recurrence intervals sampling errors may be determined by using for example Eq 39 If the terrain exposure around the anemometer tower is open measurement errors may be assumed to be relatively small that is in the order of 5 say However if the terrain around the tower is built up the conversion of wind speeds measured at the site to standardized wind speeds ie wind speeds averaged over a specified time interval eg 3 s at a specified elevation eg 10 m above terrain with open exposure the errors can be considerably larger see 18 Errors are likely to be even larger if wind speed measurements are performed using weather balloon data3 Hurricanes Hurricane wind speeds used for structural design are obtained by simu lations that involve the physical modeling of the hurricane wind flow at high altitudes Section 131 and Eq 14 observations of pressure defects radii of maximum rotational wind speeds and storm translation speeds and directions see Section 323 probabilistic models based on observations empirical methods for transforming wind speeds at high altitudes into surface wind speeds and calibration of the physical and probabilistic models against the rare available direct measurements of hurricane wind speeds or against inferences on hurricane wind speeds based on observed hurricane wind damage to buildings and other structures Added to the uncertainties inherent in the physical and probabilistic models used in the simulations are statistical uncertainties due to the relatively small number of hurricane events at various locations on the Gulf and Atlantic coasts In particular available observations may not include the occurrence of abrupt changes of direction of the hurricane translation velocity resulting in the possible failure of engineering models to predict high wind speeds andor storm surge The lack of such observations might explain why according to the ASCE 7 Standard estimated design wind speeds in the New York City area are the same as for example in Arizona or western Massachusetts or the failure to predict hurricane Sandys severity 4 Since rigorous estimates of uncertainties in hurricane wind speeds are in practice not possible it is typically necessary to resort to engineering judgment It is argued in 5 that theoretical models of natural phenomena such as hurricanes or earthquakes while useful should be superseded by prudent risk management considerations that weigh the relatively modest additional costs of conservative design against the costs of potential catastrophic failures Even though in spite of efforts reported in 6 and 7 the rigorous estimation of uncertainties in hurricane wind speeds is difficult if not impossible in the current state of the art it is definitely the case that these uncertainties are greater than their counterparts for extratropical storms note that the estimated uncertainties are considerably smaller in 6 than in 7 732 Uncertainties in the Estimation of Exposure Factors Exposure factors represent ratios between squares of the wind speeds at various eleva tions over suburban terrain or water surfaces and their counterparts at 10 m above the open terrain Wind profiles within cities especially city centers cannot be described in general terms and are simulated in wind tunnels that reproduce to scale the built envi ronment as required for example in 8 Wind tunnel simulations for locations with surface exposure difficult to define tend to reduce the uncertainty in the exposure factor 3 Useful information on uncertainties inherent in weather balloon measurements could be obtained by performing such measurements at a location where reliable surface observations are available k k k k 73 Individual and Overall Uncertainties 161 733 Uncertainties in the Estimation of Pressure Coefficients Errors in the laboratory estimation of pressure coefficients are due to i the violation of the Reynolds number in wind tunnels and to a lesser extent in largescale aerodynamic facilities ii differences between simulated and fullscale atmospheric boundarylayer flows iii laboratory measurement errors iv the estimation of pressure coefficient time history peaks v the duration of the pressure coefficient record and vi possible errors due to blockage Chapter 5 i Reynolds Number Effects Wind tunnel simulations of aerodynamic pressures are typically performed at geometric scales in the order of 1 501 500 and velocity scales of about 1 4 say Since in wind tunnels commonly used for structural engineering appli cations the fluid is air that is the same as for the prototype Reynolds number similarity is typically violated by a factor in the order of 1001000 In some largescale aerody namic facilities the geometric and the velocity scales are in the order of 1 101 50 and 1 11 2 respectively so that the Reynolds number is violated by a factor in the order of 1 101 100 The violation of the Reynolds number can be especially consequential for aerody namic pressures on bodies with rounded shapes As shown in Chapters 4 and 5 this is the case because at the high Reynolds numbers typical of wind flows around build ings the boundary layer that forms at the surface of the body is typically turbulent The turbulent fluctuations transport particles with large momentum from the free flow into the boundary layer thus helping the boundarylayer flow to overcome negative pressure gradients and causing flow separation to occur farther downstream thus reducing the drag on the body with respect to its value at lower Reynolds numbers A remedial mea sure commonly used in wind tunnel simulations is to force the boundarylayer flow to be turbulent by rendering the body surface rougher However the resulting flow still differs from the high Reynolds number flow This contributes to increasing the uncertainty in the pressure coefficients It has been argued that the violation of the Reynolds number is not consequential for flows around bodies with sharp corners since for such bodies flow separation occurs at the corners regardless of Reynolds number This argument is not necessarily borne out by comparisons between fullscale and wind tunnel measurements This has been shown in 9 which reported that peak negative pressure coefficients measured in the wind tunnel can underestimate their prototype counterparts by as much as a 25 see Section 544 In such cases corrections of wind tunnel data based on comparisons between fullscale and laboratory are warranted A systematic effort to develop such corrections remains to be performed Positive pressure coefficients measured in the wind tunnel appear to be adequate however ii Errors in the Simulation of Atmospheric Boundary Layer ABL Flows Wind tun nel simulations of ABL profiles and turbulence are largely empirical see Chapter 5 They depend upon the length of the test section the type of roughness used to retard the flow near the wind tunnel floor and the geometry of and distance between the spires placed at the entrance into the test section to help transform uniform flows into shear flows Such simulations can achieve flows bearing at least a qualitative resem blance between simulated and prototype flows Differences between wind tunnel flows can result in significant differences between the respective pressure coefficient measurements An international roundrobin test k k k k 162 7 Uncertainties in Wind Engineering Data reported in 10 showed that the coefficients of variation of the peak pressure coefficients measured in six reputable wind tunnel laboratories were as high as 1040 On the other hand after the elimination of suspected outliers from results of tests performed by 12 laboratories the respective measurements of pressures on a square cylinder were con sidered to be acceptable provided that the wind profiles and the turbulence intensities did not differ significantly from laboratory to laboratory 11 For wind tunnel tests performed at relatively large geometric scales eg 1 100 for lowrise buildings rather than say 1 500 for tall buildings an additional simulation problem arises the inability to simulate in the wind tunnel the lowfrequency portion of the longitudinal velocity spectra see eg 12 iii Uncertainties Associated with Measurement Equipment A significant con tributor to pressure measurement errors is the calibration of dynamic pressures in tubing systems connecting models to sensors The pressure waves propagating inside a thin circular tube distort the aerodynamic pressures on the model owing to the acoustic and viscothermal effects brought about by fluid action on the tube 13 According to 14 uncertainties associated with measurement equipment are typically approximately 10 iv Statistical Estimation of Pressure Coefficient Peaks Appendix C describes a pow erful peaksoverthreshold method that estimates peak pressure coefficients and their probability distributions An alternative method is discussed in the following Let the pressure coefficient record Cpt for any given direction 𝜃 have length T and be divided into a number n of subintervals epochs of length Tn The peak value of the pressure coefficient in any one epoch i i 1 2 n ie over any one subinterval of length Tn denoted by Cppk iTn forms a data sample of size n It is assumed that the epochs are sufficiently large that their respective peaks are independent and that the data are identically distributed Experience has shown that typically the data Cppk iTn are best fitted by a Type I Extreme Value EV I cumulative distribution function see Eqs 34 and 35 P Cppk T n exp exp Cppk T n 𝜇 𝜎 78 where PCppkTn is the probability that the variate CppkTn is not exceeded during any one epoch of length Tn The probability FrCppk Tn that the variate Cppk Tn is not exceeded during the 1st epoch and the 2nd epoch and the rth epoch is Fr Cppk T n P Cppk T n r exp r exp Cppk T n 𝜇 𝜎 79 Inversion of Eq 79 yields Cppk T n Fr 𝜇 𝜎 ln r 𝜎 ln ln Fr 710 k k k k 73 Individual and Overall Uncertainties 163 Equation 79 shows that Fr is an EV I cumulative distribution function with location parameter 𝜇 𝜎 ln r and scale parameter 𝜎 see Eqs 3436 The expectation of the largest CppkTnFr values over r epochs denoted by CppkTn r is Cppk T n r 𝜇 𝜎 ln r 05772 𝜎 711 see Eq 35a It follows from Eqs 710 and 711 that ln ln FrCppk 05772 712 hence FrCppk exp exp05772 05704 713 Equation 713 may be interpreted as follows Given a large number of realizations in 57 of the cases the observed peak will be lower and in 43 of the cases it will be larger than the expected value The parameters 𝜇 and 𝜎 can be estimated from the sample of data Cppk iTn i 1 2 n by using for example the BLUE estimator or the method of moments Section 333 In applications design peak pressures are currently estimated by substituting in Eq 710 estimated values for the true values of the parameters 𝜇 and 𝜎 and assuming the probability Fr 078 or 08 as specified in 15 p 22 rather than Fr 05704 The use of the probability Fr 08 rather than Fr 05704 is an instance of double counting since it increases in Eq 71 the pressure or force coefficient above its expected value while also accounting in Eq 72 for the deviation of the pressure from its expected value 16 It has been argued that the use of the 078 or 08 value of Fr is consistent with storm durations in excess of 1 hour eg 3 hours However if a storm duration longer than 1 hour were assumed the expected peak corresponding to it should be estimated directly by using in Eq 79 a value of r consistent with that duration Also the assumption that storm durations are longer than one hour would be at variance with US standard practice which follows the convention of 1hour storm durations For a thorough study of peaks of time series of pressure coefficients see 17 v Estimation of Pressure Coefficient Peaks from Short Records In some applications the available records are short This is the case for example for pressure measurements performed in large aerodynamic facilities where operation time is expensive Example 71 Consider a T 90second long record of pressure coefficients at the tap of a roof on a model with length scale 1 8 and velocity scale 1 2 The length of the prototype counterpart of the record is obtained from the condition Tp Lp Lm Um Up Tm 8 1 2 90 seconds 360 seconds Let n 16 The prototype length of each subinterval is then Tp16 36016 225 seconds For the 360second prototype record being considered the mean and standard deviation of the sample consisting of the peak pressures of the 16 subintervals epochs are assumed to be ECppkTn 472 and SDCppkTn 075 respectively to k k k k 164 7 Uncertainties in Wind Engineering Data which there correspond the estimated Type I Extreme Value distribution parameters Eq 35 𝜎 T n 6 𝜋 SD T n 078 075 0585 and 𝜇 T n Cppk T n 05772 𝜎 T n 472 05772 0585 438 The estimated means of the peak CppkT16 225 s r for r 16 and r 160 ie for a 360s and a 3600s long prototype record are Cppk T 16 r 16 𝜇 𝜎 ln r 05772𝜎 438 0585 ln 16 05772 0585 634 Eq 713 and Cppk T 16 r 160 𝜇 𝜎 ln r 05772𝜎 438 0585 ln160 05772 0585 770 The standard deviations of the sampling errors in the estimation of the mean peak CppkT16 225 seconds r can be obtained from Eq 39 Note that in both cases the sample size is n 16 734 Uncertainties in Directionality Factors According to a study reported in 2 uncertainties in the directionality factors may be assumed to be typically in the order of 10 References 1 Ellingwood B Galambos T V MacGregor J G and Cornell C A Development of a probabilitybased load criterion for American National Standard A58 NBS Special Publication 577 National Bureau of Standards Washington DC 1980 2 Habte F Chowdhury A Yeo D and Simiu E 2015 Wind directionality factors for nonhurricane and hurricaneprone regions Journal of Structural Engineering 141 04014208 3 Panofsky HA and Dutton JA 1984 Atmospheric Turbulence Models and Meth ods for Engineering Applications 1e Wiley 4 Yeo D Lin N and Simiu E 2014 Estimation of hurricane wind speed probabil ities application to New York City and other coastal locations Journal of Structural Engineering 140 04014017 5 Emanuel K 2012 Probable cause are scientists too cautious to help us stop cli mate change Foreign Policy Nov 9 2012 httpsforeignpolicycom20121109 probablecause 6 Vickery PJ Wadhera D Twisdale LA Jr and Lavelle FM 2009 US hurricane wind speed risk and uncertainty Journal of Structural Engineering 135 301320 k k k k References 165 7 Coles S and Simiu E 2003 Estimating uncertainty in the extreme value analysis of data generated by a hurricane simulation model Journal of Engineering Mechanics 129 12881294 8 ASCE Wind tunnel testing for buildings and other structures ASCESEI 4912 in ASCE Standard ASCESEI 4912 Reston VA American Society of Civil Engineers 2012 9 Long F Uncertainties in pressure coefficient derived from full and model scale data Report to the National Insititute of Standards and Technology Wind Science and Engineering Research Center Department of Civil Engineering Texas Technical University Lubbock TX 2005 10 Fritz WP Bienkiewicz B Cui B et al 2008 International comparison of wind tunnel estimates of wind effects on lowrise buildings testrelated uncertainties Journal of Structural Engineering 134 18871890 11 Hölscher N and Niemann HJ 1998 Towards quality assurance for wind tunnel tests a comparative testing program of the Windtechnologische Gesellschaft Journal of Wind Engineering and Industrial Aerodynamics 7476 599608 12 Mooneghi MA Irwin PA and Chowdhury AG 2016 Partial turbulence sim ulation method for predicting peak wind loads on small structures and building appurtenances Journal of Wind Engineering and Industrial Aerodynamics 157 47 13 Irwin PA Cooper KR and Girard R 1979 Correction of distortion effects caused by tubing systems in measurements of fluctuating pressures Journal of Wind Engineering and Industrial Aerodynamics 5 93107 14 Diaz P S Q Uncertainty analysis of surface pressure measurements on lowrise buildings MS thesis Civil Engineering University of Western Ontario London Ontario Canada 2006 15 ISO Wind Actions on Structures in ISO 4354 Geneva Switzerland International Standards Organization 2009 16 Simiu E Pintar AL Duthinh D and Yeo D 2017 Wind load factors for use in the wind tunnel procedure ASCEASME Journal of Risk and Uncertainty in Engineering Systems Part A Civil Engineering 3 httpswwwnistgovwind 17 Gavanski E Gurley KR and Kopp GA 2016 Uncertainties in the estimation of local peak pressures on lowrise buildings by using the Gumbel distribution fitting approach Journal of Structural Engineering 142 04016106 18 Masters FJ Vickery PJ Bacon P and Rappaport EN 2010 Toward objective standardized intensity estimates from surface wind speed observations Bulletin of the American Meteorological Society 91 16651681 k k k k 167 Part II Design of Buildings k k k k 169 8 Structural Design for Wind An Overview This chapter starts with a brief history of approaches to the design of structures for wind Section 81 It then presents an overview of two design procedures based on recently developed technology allowing the simultaneous measurement of pressure time histo ries at large numbers of taps placed on wind tunnel models1 Both procedures depend on big data processing and entail iterative computations including dynamics calcula tions that once the wind climatological and aerodynamic data are provided by the wind engineer are most effectively performed by the structural engineer The first of these procedures is called DatabaseAssisted Design DAD and is discussed in Section 82 DAD uses recorded time series of randomly varying pressure coefficients to determine by rigorously accounting for dynamic and directional effects peak demandtocapacity indexes DCIs with specified mean recurrence intervals MRIs for any desired num ber of structural members for details on DCIs see Chapter 13 DAD can be applied to buildings regardless of the complexity of their shape Examples of buildings with complex shapes are the CCTV building the Shanghai World Financial Center and the Burj Khalifa tower The second procedure discussed in Section 83 uses time series of measured pressure coefficients only for the computation of the aerodynamic and inertial forces acting at the building floor levels following which it determines static wind loads used to calculate design DCIs with specified MRIs If the resulting DCIs are close to their counterparts produced by the DAD procedure those loads can be regarded as equiva lent static wind loads ESWLs It can be inferred from Chapter 14 that unlike DAD the procedure for determining ESWLs is typically applicable only to buildings with relatively simple geometries eg buildings with rectangular shape in plan Section 84 briefly compares the DAD and ESWL procedures in particular it discusses the verification of ESWL results against benchmark values obtained by DAD 81 Modern Structural Design for Wind A Brief History Modern structural design for wind emerged in the 1960s as a synthesis of the following developments Modeling of the neutrally stratified atmospheric boundary layer flow including i the variation of wind speeds with height above the ground as functions of upwind surface roughness and ii the properties of atmospheric turbulence 1 These procedures may be inapplicable in the rare cases in which the configuration of the building models does not allow the placement of pressure taps Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 170 8 Structural Design for Wind Probabilistic modeling of extreme wind speeds Modeling of pressures induced on a face of a rectangular building by atmospheric flow normal to that face Frequencydomain modeling of the dynamic alongwind response produced by atmo spheric flow normal to a building face The increase of wind speeds with height above ground was first reported by Helmann in 1917 1 The aerodynamic effects of turbulent shear flows were first researched by Flachsbart in 1932 2 Figure 4312 Flachsbarts work influenced the approach to the 1933 tests of the Empire State Building reported by Dryden and Hill 5 Probabilistic models of extreme values for geophysical applications were developed by Gumbel in the 1940s 6 A pioneering approach to the analytical estimation of the dynamic response of bodies immersed in turbulent flow was developed by Liepmann in 1952 7 A synthesis of these developments was first achieved in the 1960s by Davenport 8 9 a University of Bristol student of the eminent engineer Sir Alfred Pugsley However that synthesis could not account for wind effects induced by vorticity shed in the wake of the struc ture by winds skewed with respect to a building face or affected by the presence of neighboring buildings or for aeroelastic behavior Specialized wind tunnels were there fore developed in the 1960s with a view to simulating the atmospheric boundary layer flow and its aerodynamic dynamic and aeroelastic effects on structures During the 1970s wind tunnel techniques were not sufficiently developed to allow the accurate determination of wind effects for structural design purposes Information on wind effects was based in large part on nonsimultaneous pressures measured at typically small numbers of taps eg six taps for a model that currently accommodates hundreds of pressure taps see Figures 529 and 530 with unavoidable errors that can be significant An improvement in the capability to determine wind effects was achieved in the late 1970s with the development of the high frequency force balance HFFB 10 The HFFB approach used in conjunction with frequencydomain analyses is applied to tall buildings designed to have no unfavorable aeroelastic response under realistic extreme wind loading that is in practice to all welldesigned tall buildings HFFB provides time histories of the effective aerodynamic and dynamic base moments induced by the wind loads Its chief drawback is that it provides no information on the distribution of the wind loads with height since that distribution cannot be inferred from the base moments or shears see eg 11 The loading information needed to calculate the demandtocapacity ratios therefore depended largely on guesswork especially for buildings influenced aerodynamically by neighboring structures Nevertheless the HFFB approach can be useful in the preliminary phase of the design process for the rapid if only qualitative aerodynamic assessment of building configurations orientations and aerodynamic features The HFFB approach is also useful for buildings with facade configurations that do not allow the effective placing of pressure taps From the 1990s on the development of the pressure scanner see Section 57 has rad ically changed the approach to structural design for wind and has rendered the HFFB approach largely obsolete The pressure scanner allows the simultaneous measurement 2 Flachsbart was dismissed by the Nazi authorities for refusing to divorce his Jewish wife 3 and was therefore unable to complete his research Some of his results were rediscovered independently by Jensen in the 1960s 4 k k k k 82 DatabaseAssisted Design 171 of pressures at as many as hundreds of taps and therefore the capture of the pressures variation as a function of time and spatial separation To exploit this new measure ment technology two computerintensive procedures have been developed which are used in conjunction with timedomain analyses Databaseassisted Design and Equiv alent Static Wind Loads ESWL applicable like the HFFB procedure to tall buildings designed to have no unfavorable aeroelastic response under realistic extreme wind load ing Introductions to DAD and ESWL procedures are presented in Sections 82 and 83 respectively 82 DatabaseAssisted Design DAD is a computerintensive technique based on the full use of aerodynamic pressure data for structural design purposes It provides benchmark values against which results of procedures based on ESWLs can be assessed DAD uses timedomain methods which are typically more straightforward transparent and effective than their frequencydomain counterparts Structural design for wind uses two types of wind engineering data i time series of pressure coefficients on a structure measured simultaneously at multiple taps and ii wind climatological data at the building site The task of the wind engineering labo ratory is to deliver these data as well as estimates of the uncertainties inherent in them The tasks of the structural engineer are the following 1 Select the structural system and determine the structures preliminary member sizes based on a simplified model of the wind loading eg a static wind loading taken from standard provisions The structural design so achieved is denoted by D0 2 For the design D0 determine the systems mechanical properties including the modal shapes natural frequencies of vibration and damping ratios as well as the requisite influence coefficients and develop on their basis a dynamic model of the structure PΔ and P𝛿 effects can be accounted for by using for example the geometric stiff ness matrix Chapter 9 3 From the time histories of simultaneously measured pressure coefficients determine the time histories of the randomly varying aerodynamic loads induced at all floor levels by directional mean wind speeds U𝜃 for a sufficient number of speeds U eg 20 m s1 U 80 m s1 say and directions 𝜃 0 𝜃 360 The reference height for the mean wind speeds is typically assumed to be the height of the structure Chapter 10 4 For each of the directional wind speeds defined in task 3 perform the dynamic anal ysis of the structure D0 to obtain the time histories at floor k of i the inertial forces induced by the respective aerodynamic loads and ii the effective windinduced loads FkU𝜃 t applied at the structures center of mass The lateral loading determined in this task consists of the three components acting along the principal axes x y and the torsional axis 𝜗 Chapter 11 5 For each cross section m of interest use the appropriate influence coefficients to determine time series of the DCIs induced by the combination of factored gravity loads and effective wind loads obtained in task 4 The DCIs are the lefthand sides of the design interaction equations and are typically used to size members subjected to k k k k 172 8 Structural Design for Wind more than one type of internal force For example the interaction equations for steel members subjected to flexure and axial forces are 12 If Pr 𝜙pPn 02 Pr 𝜙pPn 8 9 Mrx 𝜙mMnx Mry 𝜙mMny 10 81 If Pr 𝜙pPn 02 Pr 2𝜙pPn Mrx 𝜙mMnx Mry 𝜙mMny 10 82 In Eqs 81 and 82 Pr and Pn are the required and available tensile or compres sive strength Mrx and Mnx are the required and available flexural strength about the strong axis Mry and Mny are the required and available flexural strength about the weak axis 𝜙p and 𝜙m are resistance factors3 The required strengths are based on combinations of wind and gravity effects specified in the applicable code A similar though simpler expression for the DCI is applied to shear forces Additional material on DCIs is provided in Chapter 13 6 For each cross section m of interest construct the response surfaces of the peak com bined effects being sought as functions of wind speed and direction that is for each of the directional wind speeds considered in task 3 determine the corresponding peak of the DCI time series eg Eqs 81 and 82 and construct from the results so obtained the peak DCI response surface The response surfaces are properties of the structure dependent upon its aerodynamic and mechanical characteristics but independent of the wind climate They provide for each cross section of interest the peak DCIs as functions of wind speed and direction Response surfaces are also constructed for peak interstory drift ratios and peak accelerations For details see Chapter 13 7 Use the information contained in the response surfaces and the matrices of direc tional wind speeds at the site to determine by accounting for wind directionality the design DCIs that is the peak DCIs with the specified MRI N for the cross sections of interest For each cross section m the steps required for this purpose are i In the directional wind speed matrix Uij where i and j denote the storm num ber identifier and the wind direction respectively replace the entries Uij by the peak DCIs DCIpk m U Ui 𝜃 𝜃j taken from the response surface for the cross section m ii Transform the matrix DCIpk m Ui 𝜃j so obtained into the vector maxjDCIpk m Ui 𝜃jT where T denotes transpose by disregarding in each row i all entries lower than maxjDCIpk m Ui 𝜃j iii Rankorder the quantities maxjDCIpk m Ui 𝜃j and use nonparametric statistics in conjunction with the mean annual rate of storm arrival 𝜆 to obtain the design DCIs that is the quantities DCIpk m N Chapter 13 and Section A8 Similar operations are performed for interstory drift ratios and accelerations If for the member being considered the design DCI is approximately unity the design of that member is satisfactory from a strength design viewpoint If the uncertainties in the wind velocity andor the aerodynamic data are significantly larger than their 3 Some indexes used in Eqs 81 and 82 are used elsewhere in this book in different contexts in which they are clearly defined k k k k 82 DatabaseAssisted Design 173 typical values on which code requirements are based the design MRIs will exceed the MRIs specified in for example the ASCE 716 Standard and can be determined as in Section 125 13 In general the preliminary design D0 does not satisfy the strength andor service ability design criteria The structural members are then resized to produce a modified structural design D1 This iterative process continues until the final design is satisfac tory If necessary to help satisfy serviceability criteria motion mitigation devices such as Tuned Mass Dampers are used Chapter 16 Tasks 2 through 7 are repeated as necessary until the design DCIs are close to unity to within serviceability constraints Each iteration entails a resizing of the structural members consistent with the respective estimated design DCIs Features of interest of the DAD approach are summarized next The wind engineer performs wind engineering tasks and the structural engineer performs structural engineering tasks The wind engineers tasks are to provide the requisite aerodynamic and wind climatological data with the respective uncertainty estimates These data are used by the structural engineer to determine the stochastic aerodynamic loading and perform the dynamic analyses required to obtain the effective windinduced loading as well as all the subsequent operations resulting in the structures final design Included in these operations is the estimation of the design DCIs interstory drift ratios and building accelerations with the respective specified MRIs consistent with the uncertainties in the aerodynamic and the wind climatological data Chapter 13 This division of tasks is efficient and establishes clear lines of accountability for the wind engineer and for the structural engineer The structural engineers role in designing structures for wind thus becomes similar to the role of the designer of structures for seismic effects whose tasks include performing the requisite dynamic analyses DAD allows higher modes of vibration and any modal shape to be rigorously accounted for Wind effects with specified MRIs obtained by accounting for wind directionality are determined by the structural engineer rigorously and transparently as functions of the properties of the structure inherent in the final structural design The aerodynamics and wind climatological data provided by the wind engineer as well as the operations performed by the structural engineer can be recorded and doc umented in detail allowing the full development of Building Information Modeling BIM for the structural design for wind 14 This feature enables ready traceability and detailed scrutiny of the data by the project stakeholders Owing to currently available computational capabilities the requisite tasks can be readily performed in engineering offices Combined wind effects including DCIs induced by wind loads acting on all building facades as well as by windinduced torsion are determined automatically by using specialized software The software can be accessed via links provided at the end of this chapter Typically satisfactory designs for strength that is designs resulting to within service ability constraints in DCIs close to unity require more than one iteration owing to possibly significant successive changes in member sizes and in the structures dynamic properties As noted earlier once the aerodynamic and wind climatological data as k k k k 174 8 Structural Design for Wind well as estimates of the respective uncertainties are provided by the wind engineer the calculations including all dynamic calculations are performed by the structural engineer This eliminates unnecessary timeconsuming interactions required in ear lier practices between the wind engineering laboratory and the structural engineering office 83 Equivalent Static Wind Loads The ESWL procedure presented in this book is a variant of DAD and like DAD requires the wind engineer to provide wind climatological data at the building site time series of pressure coefficients measured simultaneously at multiple taps and measures of uncer tainties inherent in those data As in the case of DAD once these tasks are completed the ESWLbased design process is fully the responsibility of the structural engineer The ESWL procedure which by definition yields design DCIs that approximate their bench mark counterparts determined by DAD is typically applicable to structures with simple geometries The structural engineers tasks 14 are identical to their counterparts for DAD The subsequent tasks are performed for each of the wind speeds and directions considered in task 3 as follows 4a Determine the static loads FESWL kxp U 𝜃 and acting at the mass center of floor k k 1 2 nf in the direction of the buildings principal axes x y and about the torsional axis 𝜗 where the subscript p p 1 2 pmax identifies distinct wind load ing cases WLCp associated with superpositions of the three EWSL loads and pmax is a function of the number npit of points in time pit used to obtain the peak effects of interest 15 This task is described in detail in Chapter 14 5 For each cross section m of interest calculate the internal forces used to determine its DCI and substitute their expressions into the expressions for the DCIs eg Eq 81 This task requires the use of i the static wind loads determined in task 4a ii the influence coefficients and required to calculate the windinduced internal forces and iii the factored gravity loads and the respective influence coefficients For example is the internal force induced at cross section m by a unit force acting in direction x at the center of mass of floor k The windinduced internal forces are denoted by Their expression is f ESWL mp U 𝜃 nf k1 rmkxFESWL kxp U 𝜃 nf k1 rmkyFESWL kyp U 𝜃 nf k1 rmkϑFESWL kϑp U 𝜃 83 6 The corresponding DCIs denoted by DCImp are obtained by substituting the cal culated internal forces into the expressions for the DCIs For design purposes only the largest of these DCIs is of interest that is DCIRSESWL m U 𝜃 maxpDCIESWL mp U 𝜃 84 The surface representing for each cross section m of interest the dependence of its demandtocapacity index DCIm upon wind speed U and direction 𝜃 is called k k k k 83 Equivalent Static Wind Loads 175 the response surface for the cross section m The superscript RS denotes response surface 7 Use the response surfaces constructed in task 6 the climatological wind speed matrix at the building site Uij and the nonparametric statistical procedure described in detail in Chapter 13 to determine the design peak DCIs with the specified Nyear MRI As was also noted for the DAD procedure depending upon the uncertainties in the aerodynamic and climatological wind speed data as determined by the wind engineering laboratory the design MRI may have to differ from the value specified for example in the ASCE 716 Standard in which case it can be determined as indicated in Chapter 12 If the design DCIs determined in task 7 differ significantly from unity the structures members are resized to create a new design D1 Tasks 27 are then performed on that Calculation of combined gravity and wind effects Demandtocapacity indexes Interstory drift ratios acceler Task 6 Preliminary design Modeling of structure using lumped masses Structural and dynamic properties including 2ndorder effects Dynamic analyses Effective lateral floor loads Displacements Accelerations Response surfaces peak wind effects Design wind effects with specified MRIs Appropriate design Aerodyn pressure coeff database from wind tunnel tests or CFD simulations including uncertainties Load combination cases Directional wind speed database at building site including uncertainties End Redesign yes no Analysis of a full model Modal characteristics Internal forces due to gravity loads Mass Influence coefficients Structural engineers input Wind engineers input Equivalent Static Wind Loads ESWL Determine time series of aerodyn floor loads for specified sets of wind speeds and directions Task 5 Task 4a Task 4 Task 3 Task 2 Task 1 Task 7 Wind load factors or design MRIs Figure 81 Flowchart describing DAD and ESWL procedures 16 k k k k 176 8 Structural Design for Wind design This process is iterated until a structural design is achieved for which in each structural member the design DCI is close to unity to within serviceability constraints As is the case for DAD all calculations are automated The requisite software can be accessed via the link provided at the end of this chapter 84 DAD versus ESWL The ESWL procedure has the same useful features listed for the DAD procedure in Section 82 It has been argued that at least for the time being some structural engi neers may prefer performing the design for wind by using ESWLs However since both the DAD and the ESWL procedures are automated the amount of labor required on the part of the structural engineer is the same regardless of which procedure is used In addition it is worth noting that while design for seismic loads was originally based on static seismic loads structural engineering culture has evolved to the point where this is no longer necessarily the case Design for wind is expected to undergo a similar evolution Given the substitution of static loads for the actual stochastic loads it is appropriate to verify the extent to which the ESWL procedure actually results in structural designs approximately equivalent to those produced by DAD This is achieved by comparing DCIs induced by ESWL and DAD see Chapter 18 The use of peak DCIs obtained by DAD as benchmarks against which DCIs induced by ESWL can be verified is justified by the superior accuracy inherent in the DAD procedure A flowchart describing the sequence of operations leading to the final structural design by the DAD and the ESWL procedures is shown in Figure 81 16 The software DADESWL version 10 a detailed users manual 17 and a tutorial with detailed examples 18 are available for the two procedures at httpswwwnistgovwind References 1 Hellmann G Über die Bewegung der Luft in den untersten Schichten der Atmo sphäre Königlich Preussischen Akademie der Wissenschaften 1917 2 Flachsbart O 1932 Winddruck auf offene und geschlossene Gebäude In Ergeb nisse der Aerodynamischen Versuchanstalt zu Göttingen ed LL Prandtl and A Betz 128134 Munich R Oldenbourg Verlag 3 Plate E Personal comminication 1995 4 Jensen M and Franck N 1965 Model Scale Tests in Turbulent Wind Copen hagen Danish Technical Press 5 Dryden HL and Hill GC Wind pressure on a model of the empire state building Bureau of Standards Journal of Research 10 4 493523 Research Paper 545 April 1933 6 Gumbel EJ 1958 Statistics of Extremes New York Columbia University Press 7 Liepmann HW 1952 On the application of statistical concepts to the buffeting problem Journal of the Aeronautical Sciences 19 793800 8 Davenport AG 1961 The application of statistical concepts to the wind loading of structures Proceedings of the Institution of Civil Engineers 19 449472 k k k k References 177 9 Davenport AG 1967 Gust loading factors Journal of the Structural Division ASCE 93 1134 10 Tschanz T 1982 Measurement of total dynamic loads using elastic models with high natural frequencies In Wind Tunnel Modeling for Civil Engineering Applica tions ed TA Reinhold 296312 Cambridge Cambridge University Press 11 Chen X and Kareem A 2005 Validity of wind load distribution based on high frequency force balance measurements Journal of Structural Engineering 131 984987 12 ANSIAISC Specification for Structural Steel Buildings in ANSIAISC 36010 Chicago Illinois American Institute of Steel Construction 2010 13 Simiu E Pintar AL Duthinh D and Yeo D 2017 Wind load factors for use in the wind tunnel procedure ASCEASME Journal of Risk and Uncertainty in Engineering Systems Part A Civil Engineering 3 04017007 httpswwwnistgov wind 14 ARUP June 14 2017 Building Information Modelling BIM Available wwwarup comservicesbuildingmodelling 15 Yeo D 2013 Multiple pointsintime estimation of peak wind effects on structures Journal of Structural Engineering 139 462471 httpswwwnistgovwind 16 Park S Simiu E and Yeo D Equivalent static wind loads vs databaseassisted design of tall buildings An assessment Engineering Structures submitted https wwwnistgovwind 17 Park S and D Yeo DatabaseAssisted Design and Equivalent Static Wind Loads for Mid and HighRise Structures Concepts Software and Users Manual NIST Technical Note 2000 National Institute of Standards and Technology Gaithersburg MD 2018 Available httpsdoiorg106028NISTTN2000 18 Park S and Yeo D Tutorial for DAD and ESWL Examples of HighRise Building Designs for Wind NIST Technical Note 2001 National Institute of Standards and Technology Gaithersburg MD 2018 Available httpsdoiorg106028NISTTN 2001 k k k k 179 9 Stiffness Matrices SecondOrder Effects and Influence Coefficients For structures with linearly elastic material behavior structural and dynamic analyses can be performed by using stiffness matrices Section 91 and accounting as necessary for secondorder effects eg via geometric stiffness matrices Section 92 Influence coefficients representing wind effects of interest induced by unit loads acting at mass centers along the structures principal axes and unit torsional moments about the cen ters of mass are considered in Section 93 Stiffness matrices geometric stiffness matrices and influence coefficients can be determined by using finite element software Secondorder effects can be determined by a variety of methods other than the geometric stiffness matrix method including the simple moment amplification method 1 Software and user manuals described and accessible via links provided in Chapters 17 and 18 contain modules that perform the requisite calculations 91 Stiffness Matrices To define the stiffness matrix of the linearly elastic structural system of a building with nf floors consider the flexibility matrix a xi1x xi2x xinf x xi1y xi2y xinf y xi1𝜗 xi2𝜗 xinf 𝜗 yi1x yi2x yinf x yi1y yi2y yinf y yi1𝜗 yi2𝜗 yinf 𝜗 𝜗i1x 𝜗i2x 𝜗inf x 𝜗i1y 𝜗i2y 𝜗inf y 𝜗i1𝜗 𝜗i2𝜗 𝜗inf 𝜗 91 which consists of nine component submatrices represented in the righthand side of Eq 91 by their respective ith rows i 1 2 nf for example the entry denoted in Eq 91 by yi 1x yi 2x yinf x represents the matrix y11x y12x y1nf x y21x y22x y2nf x ynf 1x ynf 2x ynf nf x 92 The size of matrix a is 3nf 3nf The terms of matrix a are displacements in the x or y direction or torsional rotations about the mass center of floor i i 12 nf due to a unit horizontal force in the x or y direction or a unit torsional moment about the mass center of floor j j 12 nf and can be obtained by using standard structural analysis Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 180 9 Stiffness Matrices SecondOrder Effects and Influence Coefficients programs For example the term y12x is the y displacement of the mass center of floor 1 due to a unit horizontal force acting at the mass center of floor 2 in direction x The stiffness matrix of the system is the inverse of the matrix a k a1 93 As follows from Eq 93 the product ka is the identity matrix In the matrix k for example the restoring force k1x2y represents the horizontal force in the x direction at the mass center of floor 1 induced by a unit horizontal displacement in the y direction at the mass center of floor 2 For structures with members of known sizes and properties the matrix k is deter mined by using standard finite element software 92 SecondOrder Effects Wind forces induce horizontal displacements that give rise to overturning moments acting at every floor equal to the weight of the floor times the floors horizontal dis placements These overturning moments result in an amplification of the wind effects The study of secondorder effects is concerned with this amplification and its structural and dynamic consequences In linear elastic analysis equilibrium is based on the undeformed geometry of the structure In elastic geometrically nonlinear analysis equilibrium is based on the deformed geometry of the structure while the material behavior is assumed to be elastic in inelastic geometrically nonlinear analysis the equilibrium is based on the deformed geometry and the material behavior is assumed to be inelastic 1 In this book unless otherwise indicated the structural behavior is assumed to be elastic The P P δ Figure 91 PΔ member chord and P𝛿 member curvature effects analysis includes both chord rotation effects due to sway at the member ends ie PΔ effects and member curvature effects ie P𝛿 effects 2 Both effects are illustrated in Figure 91 In Chapter 18 secondorder effects are determined by the geo metric stiffness method in which the total displacements of the structure are obtained by subtracting from the stiffness matrix k a geometric matrix kg developed as shown in 1 The resul tant matrix denoted by ks and henceforth referred to as the effective matrix is softer than the matrix k and replaces the latter in calculations of the structural response to wind and grav ity loading including calculations of influence coefficients and dynamic response In the geometric stiffness method the vari ation of transverse displacements along the members length is commonly approximated by a cubic polynomial An example of the derivation of terms of the matrix kg for a twodimensional beamcolumn member with six degrees of freedom is shown in 2 As is the case for the matrix k for structures with known member sizes and properties the matrix kg can be calculated by using standard finite element software 3 This approach has limitations noted in 2 which appear not to be significant for k k k k References 181 tall buildings subjected to wind loads It is suggested however that the validity of this statement be the object of further research and that an alternative approach to the esti mation of secondorder effects be considered if necessary 93 Influence Coefficients Influence coefficients are used in conjunction with wind and gravity loads acting on the structure to determine internal forces displacements and accelerations induced by those loads Consider the aerodynamic wind load time series FkxU𝜃 t FkyU𝜃 t Fk𝜗U𝜃 t induced along the principal axes and in torsion by wind with mean speed U and direction 𝜃 at reference height zref acting at the center of mass of floor k k 1 2 nf The time series of the internal forces denoted by fmU𝜃 t induced by those load time series at a cross section m can be written as the sum fmU𝜃 t nf k1 rmkxFkxU𝜃 t rmkyFkyU𝜃 t rmk𝜗Fk𝜗U𝜃 t 94 where the influence coefficients rmkx rmky rmk𝜗 are internal forces induced at cross section m by a unit load acting at the center of mass of floor k along the axes x and y and around the vertical axis Similar relations apply to displacements and accelera tions and to gravity loads For any specified wind speed and direction the wind load time series are computed from time histories of pressure coefficients provided by the wind engineer see Chapter 10 References 1 ANSIAISC 2010 Steel Construction Manual 14th ed Chicago IL American Institute of Steel Construction 2 White DW and Hajjar JF 1991 Application of secondorder elastic analysis in LRFD research to practice Engineering Journal American Institute of Steel Construc tion 28 133148 3 Park S and Yeo D 2018 Secondorder effects on windinduced structural behavior of highrise buildings Journal of Structural Engineering 144 doi 101061ASCEST1943541X0001943 httpswwwnistgovwind k k k k 183 10 Aerodynamic Loads Main Structure Secondary Members and Cladding 101 Introduction Aerodynamic loads are based on time series of aerodynamic pressure coefficients mea sured simultaneously at multiple taps on the surfaces of the wind tunnel building model Two main cases are considered in this chapter In the first case the objective is to deter mine for specific structures aerodynamic loads at the center of mass of each floor on main members on secondary members eg purlins and girts and on cladding In the second case the objective is to develop standard provisions on pressure coeffi cients In both cases details of the procedures for determining the loading differ to some extent depending upon whether the pressure taps are placed in orthogonal patterns as is the case for the National Institute of Standards and TechnologyUniversity of Western Ontario NISTUWO database httpswwwnistgovwind 1 or in nonorthogonal patterns as in the Tokyo Polytechnic University TPU database 2 Section 102 discusses orthogonal and nonorthogonal tap placement patterns and the determination of individual tap tributary areas Section 103 is concerned with the determination of aerodynamic loads at floor levels and on main members secondary members and cladding Section 104 describes a method used to develop standard pro visions on pressure coefficients as functions of areas contained within specified zones Section 105 concerns winddriven rain intrusion 102 Pressure Tap Placement Patterns and Tributary Areas Pressure taps may be placed in orthogonal or nonorthogonal patterns Examples of orthogonal pressure tap patterns are shown Figure 101 which shows rectangular trib utary areas of pressure taps represented by cross symbols A nonorthogonal pattern is shown in Figure 102a in which circles indicate pressure tap locations Individual tap tributary areas are conveniently calculated using Voronoi diagrams 5 The diagrams can be derived from Delaunay triangulation 6 which connects a given set of point taps to form triangles that i do not overlap ii cover the entire interior space and iii do not have any tap within the triangles circumcircle The corresponding Voronoi diagram is created by drawing perpendicular bisectors to the sides of the triangles Regions formed from these bisectors contain one tap each and bound an area containing points that are closer to that tap than to any other tap The Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 184 10 Aerodynamic Loads s y s y a b Figure 101 Rectangular tap tributary areas a simple tap array b tap array with varying tap density 3 a b c d Figure 102 Example of nonorthogonal pattern of pressure tap placement and of tributary area assignments 4 Source With permission from ASCE Voronoi MATLAB function 7 can generate both Delaunay triangulation and Voronoi diagrams Figure 102b connects the taps using Delaunay triangulation Figure 102c shows how a Voronoi diagram can be derived from the Delaunay triangulation Figure 102d shows the Voronoi diagram the bounded area created around a tap is the tributary area of that tap 103 Aerodynamic Loading for DatabaseAssisted Design Pressure data on the structures envelope are provided as time series of nondimensional pressure coefficients Cp typically based on the hourly mean wind speed V H at the build ing roof height H Cp p 1 2𝜌V 2 H 101 where p is the net pressure relative to the atmospheric pressure and 𝜌 is the air density 1225 kg m3 for 15C air at sea level k k k k 103 Aerodynamic Loading for DatabaseAssisted Design 185 From the similarity requirement for the reduced frequency nDV where n is the sampling frequency and D is a characteristic dimension of the structure it follows that the prototype time interval Δtp 1np is Δtp Dp Dm Vm Vp Δtm 102 where the subscripts p and m stand for prototype and model respectively DmDp is the geometric scale V mV p is the velocity scale and Δtm is the reciprocal of the sampling frequency nm at model scale Calculations of aerodynamic pressure coefficients based on pressure measurements at taps placed on wind tunnel models require 1 The creation of virtual pressure taps at each edge of the model surface The time series of the pressure coefficients at those taps are obtained by extrapolation from the time histories at the outermost and next to outermost pressure taps Figure 103a This operation is necessary because actual pressure taps cannot be placed at the struc tures edges 2 The generation of a mesh for interpolations between time series of pressure coeffi cients measured at actual taps or estimated at virtual taps Figure 103b Each mesh element has dimensions ΔB ΔH where ΔB B2nB ΔH H2 N B is the building width H is the building height including for buildings with parapets the parapet height in which case the height of the uppermost mesh element is equal to the height of the parapet nB is the number of pressure taps in each pressure tap row B H Pressure Taps Real Pressure Taps Virtual ΔH ΔB Nth Floor Roof N1th Floor Top N2th Floor 2nd Floor 1st Floor Mesh a Virtual taps at edges b Mesh for interpolating pressures HNh ϑ Real or virtual tap Center of mesh cell c Floor forces on lumped mass system yn y x xn θn Interpolation Calc floor loads Extrapolation Figure 103 a Actual and virtual pressure tap locations b Interpolation mesh on model surface and points of application of wind forces obtained by interpolation at centers of mesh cells h floor height c Wind forces applied at floorlevel lumped masses k k k k 186 10 Aerodynamic Loads a Tributary area for Nth floor shaded area b Tributary area for typical floors 1 to N1 shaded area h LumpedMass HNh ΔH ΔB Figure 104 Tributary areas for calculation of floor wind loads and N is the number of floors Software described and applied in Chapter 18 offers the option of carrying out the interpolations by any of three methods supported by MATLAB An alternative method is described in 8 3 For multistory buildings time series of floor wind loads are applied at the floor cen ters of mass They are determined as functions of time series of pressure coefficients obtained by interpolation of the respective tributary areas Figure 104 and of the mean wind speeds at the elevation of the top of the building The wind loads consist at each floor of forces acting along the two principal directions of the structure and a torsional moment Figure 103c It is typically assumed that pressure coefficients do not depend significantly upon Reynolds number and are therefore identical for the model and the prototype However for the design of cladding a more conservative approach may be adopted to account for the fact that wind tunnel simulations may underestimate peak suctions as shown in Figure 524 104 Peaks of Spatially Averaged Pressure Coefficients for Use in Code Provisions 1041 Pressures Within an Area A Contained in a Specified Pressure Zone Standards specify pressures applicable to areas of various sizes A contained in specified zones eg middle edge or corner zones of roofs or walls within which it is assumed for practical design purposes that the pressures are uniform Except for an area A covering the entire area of the zone being considered the number of areas of specified size A within a zone exceeds unity To develop standard provisions on pressure coefficients the following steps are required 9 1 Identify all areas of size A within the zone see Section 1042 2 Determine the tributary area Bl of each tap l contained in the zone 3 For each of the areas A determine its intersections al with the tap tributary areas Bl For example let the four rectangles shown in Figure 105 represent tap tributary areas B1 B2 B3 B4 and let the area A of interest be the shaded area of Figure 105 The intersections of area A with the areas Bl l 1 2 3 4 are denoted by al l 1 2 3 4 k k k k 104 Peaks of Spatially Averaged Pressure Coefficients for Use in Code Provisions 187 Figure 105 Intersection of four pressure tap tributary areas cells with shaded area A 9 4 For each wind direction 𝜃j and for each of the areas of size A within the zone being considered obtain the time history pA t 𝜃j l plt 𝜃jal A 103a l al A 103b where plt 𝜃j is the time history of the pressure induced by wind with direction 𝜃j at the tap contained in area Bl 5 Determine for each of the areas A and for each of the directions 𝜃j the peak of the time history pA t 𝜃j using for example the procedure described in Appendix C Alternatively the procedure described in httpswwwnistgovwind may be used This requires the partitioning of the record into equal segments and the creation of a sample of peak values consisting of the peak of each of the segments The largest of all those peaks is the pressure being sought for codification purposes 6 Divide that pressure by the dynamic pressure 12𝜌 maxjU2zref 𝜃j to obtain the corresponding pressure coefficient CpA t For compliance with ASCE 7 Standard requirements the pressure coefficients CpA t are rescaled to be consistent with 3second peak gust wind speeds and are reduced via multiplication by a directionality reduction factor see Section 135 1042 Identifying Areas A Within a Specified Pressure Zone Pressure Taps with Rectangular Tributary Areas The summation process in Eq 103a is simplest when the cells representing the tributary areas of the taps are rectangular Figure 101a Special consideration must be given to areas A in edge and corner zones since such areas generally do not coincide with cell boundaries see eg Figure 105 and to cases in which grids of different densities merge as indicated by arrows in Figure 101b To see how various areas of size A are determined within a specified zone with area larger than A consider the sixcell zone with orthogonal tap placement shown in Figure 106 9 We seek the number of distinct rectangles with areas A within that zone The areas A may consist of one cell or of rectangular conterminous aggregates of two three four or six cells There are six possible rectangular areas consisting of one cell each The cell on the upper left corner is denoted Aa To Aa is added a cell in the downward ydirection the twocell rectangle so obtained is denoted Aa2 With this step the lower boundary of the zone is reached therefore no additional cell can be added in direction y To the cell selected in step Aa one cell is added in direction s rightward The twocell rectangle so obtained is denoted Aa31 In a next step denoted Aa32 an additional cell is added again in direction s rightward Thus two additional rectangles have been created in step Aa3 With step Aa32 the rightmost boundary of the zone has been reached so further expansion in the direction s is not possible Next step Ab consists of adding to the cell selected in step Aa via expansion in both k k k k 188 10 Aerodynamic Loads Aa Initialize at upper left Ba Initialize Ca Initialize Ca2 Expand in y Ca3 Expand in s Cb Expand in y and s Fa Initialize Aa2 Expand down in y Aa31 Aa3 2 Expand right in s Ab Expand in y and s Ab2 Expand in y impossible Ba2 Expand in y impossible Ba31 Ba3 2 Expand in s Bb Expand in y and s impossible Ea Initialize Ea2 Expand in y Ab3 Expand in s Da Initialize Da2 Expand in y impossible Da3 Expand in s Figure 106 Sixcell zone with orthogonal tap placement 9 directions y downward and s rightward Thus a rectangle consisting of four cells is created Expansion in the direction y downward is attempted in step Ab2 but is not possible Step Ab3 consists of expanding in the s direction rightward which results in a sixcell rectangle All possibilities of expansion from the single cell selected in step Aa being exhausted one proceeds to the next initial cell direction rightward step Ba The procedure is repeated until all possible initial cells have been used Figure 106 shows six rectangles formed by one cell seven rectangles formed by two cells two rectangles formed by three cells two rectangles formed by four cells and one rectangle formed by six cells for a total of eighteen rectangles If the cells are rectangles of unit area for the zone of area 6 the following numbers of pressure time series result 6 with area A 1 7 with area A 2 2 with area A 3 2 with area A 4 and 1 with area A 6 In this example areas A have aspect ratios ranging from 1 to 3 We need to calculate the peak average pressure coefficient for each of the 6 onecell areas for each of the seven twocell areas and so forth To limit the number of combinations for large zones the aspect ratio of the rectan gles formed by the aggregation of cells is limited to four at most This aspect ratio covers many practical units of components and cladding and allows consideration of long nar row zones along the edges of roofs and walls The number of area combinations increases very quickly with the size of the grid for example a 19 8 grid produces 4290 areas of aspect ratio 4 whereas a subset 16 7 of the same grid produces 2351 such areas If the zone being studied overlaps areas of different tap density the coarser density is used overall and full and partial cell areas in the highdensity region are combined as needed Figure 107a shows a portion of a zone with two grid densities To conform with the coarser grid the two rows of three cells at the bottom of the figure are transformed into two rectangles each Figure 107b k k k k 104 Peaks of Spatially Averaged Pressure Coefficients for Use in Code Provisions 189 a b Figure 107 Combination of areas with different tap densities Example 101 Results are shown for Building 7 open country exposure of the NISTUWO database data set jp1 httpswwwnistgovwind 1 The building Figure 108a was modeled at a 1 100 scale and data were collected for 100 seconds at 500 Hz its fullscale width length and height are 122 m 40 ft 191 m 625 ft and 122 m 40 ft respectively and its roof slope is 48 The peak averaged pressure coefficients were rescaled to be consistent with ASCE 710 Standard 10 3second peak gust wind speeds Figure 108b shows that ASCE 710 specifications in which peak wind pressure coefficients are denoted by GCp underestimate negative pressures over almost all of the areas within the corner zone by factors of up to 23 These results and a thorough study in 11 confirm the finding that negative pressures specified in the ASCE 710 Standard tend to be strongly unconservative Pressure Taps with Polygonal Tributary Areas To produce intersections of tap trib utary areas with rectangular areas A contained within a specified zone each building façade and roof surface is swept in small discrete steps by overlaid rectangles with area A The first set of rectangles with area A have sides equal to the horizontal and vertical distances between adjacent taps that is the smallest possible useful rectangles In the subsequent sets the sizes of the rectangles are progressively increased horizontally ver tically and both horizontally and vertically until the largest rectangle is determined by the dimensions of the facade Stepwise offsetting of each of those sets of rectangles by amounts equal to the smallest distances between taps ensures that no rectangular area A for which the averaged pressure coefficient needs to be calculated is missed Example 102 Consider the wall represented in Figure 102 Let the smallest horizon tal and vertical distances between taps be 2 m Two sets of rectangular areas A are shown in the figure a set consisting of a 2 2 m grid Figure 109a and a set consisting of a 2 4 m grid Figure 109c Figure 109b shows 2 2 m rectangles with 1 m offset in the x and y directions Figure 109d shows rectangles with dimensions 2 4 m and 1 m offset in the x direction In Figure 109 all the shaded areas contain pressures As the rectan gles borders cross outside a building surface partial or incomplete elements created by such borders are neglected They are represented in Figure 109 as blank areas within the façades Table 101 lists grids with minimum sizes A 2 2 m and maximum sizes A 3 3 m with no offset with 1 m offsets and with 2 m offsets k k k k 190 10 Aerodynamic Loads a b 60 0 10 20 30 40 0 10 20 30 40 50 Zaxis yaxis xaxis 0 20 40 4 35 3 25 2 15 1 05 0 05 1 1 10 100 GCp Area ft2 ASCE zone 2 Max Min ASCE zone 2 Figure 108 a Building 7 corner with pressure taps 1 ft2 00929 m2 b Peak of averaged pressure time histories Source After 9 The method just described was programmed using MATLAB 7 to process build ings available in TPUs lowrise building pressures database specifically case numbers 13108 2 The wind tunnel tests of lowrise buildings without eave were performed at a length scale of 1100 velocity scale of 13 ie a 3100 time scale for suburban ter rain At a reference height of 01 m the turbulence intensity was 025 and the test wind k k k k 104 Peaks of Spatially Averaged Pressure Coefficients for Use in Code Provisions 191 offset offset a b c d Figure 109 Example of superposed rectangular surfaces with areas A 2 m 2 and A 2 m 4 m with no offsets and with 1 m offsets offsets 4 Source With permission from ASCE Table 101 Grid areas and offset combinations Grid size Offset x m y m x direction y direction 2 2 0 0 2 2 0 1 2 2 1 0 2 2 1 1 2 3 0 0 2 3 0 1 2 3 0 2 2 3 1 0 2 3 1 1 2 3 1 2 3 2 0 0 3 2 0 1 3 2 1 0 3 2 1 1 3 2 2 0 3 2 2 1 3 3 0 0 3 3 0 1 3 3 0 2 3 3 1 0 3 3 1 1 3 3 1 2 3 3 2 0 3 3 2 1 3 3 2 2 k k k k 192 10 Aerodynamic Loads Wind Wind D 2 5 3 6 4 1 1 5 3 2 6 4 H0 H0 H R B y x x z D B θ θ β β y Figure 1010 TPU lowrise building showing geometric parameters Source From 2 Courtesy of Professor Y Tamura Tokyo Polytechnic University velocity was 74 m s1 which corresponds to a 22 m s1 mean hourly wind speed at a 10 m height in full scale Wind pressure timehistory data were recorded at 500 Hz for 18 seconds or 18 1003 seconds 10minute full scale An example of such a building is shown in Figure 1010 TPUs aerodynamic database incorporates a moving average calculation for the pres sure time series data Denoting the data sampling interval by Δt and the net pressure above ambient at tap i at time t by pit 𝜃 TPU defines the pressure denoted by pi t 𝜃 at tap i at time t as pi t 𝜃 avgpit Δt 𝜃 pit 𝜃 pit Δt 𝜃 104 An example of a building from the TPU database is shown in Figure 1010 Consider building TP1 case 61 of the database B 16 m D 24 m H0 12 m roof slope 48 see Figure 1010 Figure 1011 shows the Voronoi diagram applied to that building the pressure taps are indicated by circles bounded by the polygons that define the tributary areas With the tributary areas in place the overlaid rectangleoffset combinations can then be specified The smallest overlaid rectangle was chosen as 2 2 m based on the minimum 2 m tap spacing the largest was chosen to be 7 7 m The rectangles were incremented from 2 2 m up to 7 7 m by increments of 05 m and were offset in increments of 05 m in the x and y directions Based on these rectangleoffset combinations the total number of combinations is 9801 each rectangle having an aspect ratio of 35 or less ASCE 716 Commentary limits the aspect ratio of areas relevant to the design of components and cladding to 3 The process by which peaks of average pressures are calculated as functions of areas within codespecified zones involves the use of Boolean algebra and the MATLAB function Polybool and is repeated for all available tested wind directions 𝜃 0 15 30 45 60 75 and 90 For additional details on the method and its application to the assessment of ASCE 7 Standard provisions see 4 in which it is noted that the TPU tap and wind direction resolution are lower than their NISTUWO counterparts particularly for buildings for which ASCE 7 Standard edge zones and corner zones have small dimensions Nevertheless no significant differences were found between wind loads on main structural members based on 1 on the one hand and 2 on the other 12 k k k k References 193 Surface 4 Surface 6 Surface 5 Surface 1 Surface 3 Surface 2 Figure 1011 Tributary areas achieved using the Voronoi diagram 4 Source With permission from ASCE 105 Aerodynamic Pressures and WindDriven Rain Recent advances in the area of winddriven rain water intrusion include the develop ment of fullscale testing under conditions simulating i atmospheric boundary layer hurricane force winds and ii up to 760 mm h1 rain simulated by continuous spray ing of water through a plumbing system with spray nozzles 13 The frontal area of the wind and winddrivenrain field simulated in 13 exceeded 30 m2 Measurements were performed of the amount of water intruded through nailed and through selfadherent heavy and light secondary water barriers and of internal and external aerodynamic pres sures induced by the wind flow Tests of specimens with different slopes showed that the severity of the intrusion increases as the roof slope decreases Additional testing described in 14 was conducted using records of tropical cyclone winddriven rain data as a basis for the development of the target parameters considered in the simulation including raindrop size distribution The results of the tests were used to propose enhancements to simplified test protocols specified in current standards For additional material on rain water intrusion due to directly impinging rain drops and surface runoff see 15 and references quoted therein References 1 Ho T Surry D and Morrish D NISTTTU Cooperative Agreement Wind storm Mitigation Initiative Wind Tunnel Experiments on Generic Low Buildings BLWTSS202003 Boundary Layer Wind Tunnel Laboratory University of Western Ontario London Canada 2003 k k k k 194 10 Aerodynamic Loads 2 Tamura Y Aerodynamic Database of LowRise Buildings Global Center of Excel lence Program Tokyo Polytechnic University Tokyo Japan 2012 3 Main J A and Fritz W P DatabaseAssisted Design for Wind NIST Building Sci ence Series 180 National Institute of Standards and Technology Gaithersburg MD 2006 4 Gierson ML Phillips BM Duthinh D and Ayyub BM 2017 Windpressure coefficients on lowrise building enclosures using modern windtunnel data and Voronoi diagrams ASCEASME Journal of Risk and Uncertainty in Engineering Systems Part A Civil Engineering 3 04017010 5 Voronoi G 1908 New applications of continuous parameters to the theory of quadratic forms Journal of Pure and Applied Mathematics 133 133 97178 doi101515crll190813397 6 Delaunay B 1934 On the empty sphere In Memory of Georges Voronoi Bul letin of the USSR Academy of Sciences Section Mathematics and Natural Sciences 6 793800 7 MATLAB MATLAB documentation 2014b The Mathworks Inc 2014 8 Uematsu Y Kuribara O Yamada M et al 2001 Windinduced dynamic behavior and its load estimation of a singlelayer latticed dome with a long span Journal of Wind Engineering and Industrial Aerodynamics 89 16711687 9 Duthinh D Main J A and Phillips B M Methodology to Analyze Wind Pressure Data on Components and Cladding of LowRise Buildings NIST TN 1903 National Institute of Standards and Technology Gaithersburg MD 2016 httpswwwnist govwind 10 ASCE Minimum design loads for buildings and other structures ASCESEI 710 in ASCE Standard ASCESEI 710 Reston VA American Society of Civil Engineers 2010 11 Gavanski E Gurley KC and Kopp GA 2016 Uncertainties in the estimation of local peak pressures on lowrise buildings by using the Gumbel distribution fitting approach Journal of Structural Engineering 142 11 04016106 12 Hagos A Habte F Gan Chowdhury A and Yeo D 2014 Comparisons of two wind tunnel pressure databases and partial validation against fullscale measure ments Journal of Structural Engineering 149 04014065 13 Bitsuamlak GT Gan Chowdhury A and Sambare D 2009 Application of a fullscale testing facility for assessing winddrivenrain intrusion Building and Environment 44 24302441 14 Baheru T Gan Chowdhury A Bitsuamlak G et al 2014 Simulation of winddriven rain associated with tropical storms and hurricanes using the 12fan Wall of Wind Building and Environment 76 1829 15 Baheru T Gan Chowdhury A Pinelli JP and Bitsuamlak G 2014 Distribution of winddriven rain deposition on lowrise buildings direct impinging raindrops versus surface runoff Journal of Wind Engineering and Industrial Aerodynamics 133 2738 k k k k 195 11 Dynamic and Effective WindInduced Loads 111 Introduction Unlike seismic loads which consist of forces of inertia wind loads consist of sums of applied aerodynamic forces and forces of inertia Rigid structures are by definition struc tures for which windinduced forces of inertia are negligible Flexible structures are defined as structures for which the windinduced forces of inertia are significant The forces of inertia are due to resonant amplification effects A wellknown example of resonant amplification is the effect on a bridge of a military formation marching in lockstep at a frequency equal or close to the bridges fundamental frequency of vibra tion The effects of successive steps are additive a first step causes a deflection whose maximum is reached when the second step strikes The second step causes an additional deflection and subsequent steps keep adding to the response The randomly fluctuating wind loading can be represented as a sum of harmonic components see Appendices B and D Windinduced resonant amplification effects are caused by harmonic loading components with frequencies equal or close to the natural frequencies of vibration of the structure The forces of inertia are yielded by dynamic analyses based on secondorder ordinary differential equations of motion in accordance with Newtons second law The analyses can be performed by solving the equations of motion in the frequency domain or in the time domain The use of the frequency domain approach was predominant in the 1960s primarily because it does not require the direct solution of the differential equations instead the latter are converted to algebraic equations via Fourier transformation see Appendix D The development of pressure scanners allows the simultaneous wind tunnel mea surement of pressure time histories at large numbers of taps mounted on the external surfaces of rigid building models Inherent in the measurements is phase information on pressure fluctuations and therefore information on the extent to which the pressures acting at different points on the structure are in or out of phase that is the extent to which those pressures are mutually coherent see Figure 427 for an illustrative anima tion It is currently a routine task to obtain via simple weighted summations of pressure time histories the time histories of the windinduced forces acting on the structure see Chapter 10 Once those time histories are determined it is again a routine matter to solve numerically the equations of windinduced motion of the structure in the time domain Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 196 11 Dynamic and Effective WindInduced Loads The purpose of this chapter is to present the basic theory that governs the multidegreeoffreedom behavior of structural systems assumed to be linearly elastic Section 112 discusses the simple case of the linear singledegreeoffreedom system The multidegreeoffreedom case is considered in Section 113 The solution of the structures equations of motion yields the forces of inertia induced by the wind loading as well as the structures displacements and accelerations Section 114 concerns for any specified direction of the wind speed the determination of the corresponding effective wind loads defined as the sum of the aerodynamic and inertial loads In the High Frequency Force Balance HFFB approach dynamic response calculations are performed partly by the structural engineer and partly by the wind engineer This practice is left over from the late 1970s when dynamic calculations were performed in the frequency domain to avoid computations involving the solution of differential equations of motion The drawbacks of this practice include i difficulties in the estima tion of combined wind effects ii the lack of information on the distribution of the wind loads with height which prevents the realistic determination of wind effects in structural members iii the impossibility of determining the dynamic response in higher modes of vibration and iv the need to resort to correction factors to compensate with varying degrees of approximation for the errors due to the assumptions that the shape of the fundamental modes of vibration in sway are linear and that the shape of the fundamen tal torsional mode is independent of height These drawbacks are especially significant for buildings affected aerodynamically by neighboring buildings The advances in com putational capabilities achieved in the twentyfirst century render the HFFB approach obsolete in the sense in which for example the moment distribution method is obso lete This is the case for detailed final design purposes although the use of the HFFB approach for rapid preliminary design purposes remains warranted 112 The SingleDegreeofFreedom Linear System xt Ft M B A Figure 111 Singledegree offreedom system The system of Figure 111 consists of a particle of mass M concentrated at point B of a member AB with linear elastic behavior and negligible mass The particle is subjected to a force Ft The displacement xt of the mass m is opposed by i a restoring force kx supplied by the elastic spring inherent in the member AB and ii a damping force c dxdt c x1 where k is the systems stiffness ie the magnitude of the restoring force corresponding to a unit displacement x of the mass M and c is the damping coefficient Both k and c are assumed to be constant The inverse of the systems stiffness k is referred to as the flexibility of the system ie the systems displacement corresponding to a unit restoring force Newtons second law states that the product of the particles mass by its acceleration Mx is equal to the total force applied to the particle The equation of motion of the system is then Mx c x kx Ft 111 1 Here and elsewhere in the book the dot denotes differentiation with respect to time that is x dxdt k k k k 113 TimeDomain Solutions for 3D Response of MultiDegreeofFreedom Systems 197 With the notations n1 kM2𝜋 and 𝜁1 c2 kM where n1 denotes the frequency of vibration of the oscillator2 and 𝜁1 is the damping ratio ie the ratio of the damping c to the critical damping ccr 2 kM beyond which the systems motion would no longer be oscillatory Eq 111 becomes x 2𝜁12𝜋 n1 x 2𝜋 n12x Ft M 112 For structures 𝜁1 is typically small in the order of 1 We note for future reference that the product of the systems stiffness and flexibility is k 1k 1 and that the sys tems kinetic energy and strain energy are T 12 M x2 and V kx dx 12 kx2 An alternative derivation of the equation of free vibrations of the undamped and unforced system ie of the system with c 0 and Ft 0 can be obtained from the systems Lagrangian L T V 113 where T is the total kinetic energy and V is the potential energy eg strain energy of the system For the system under consideration L 1 2M x2 1 2kx2 114 From Lagranges equations d dt L qi L qi 0 115 where the generalized coordinate qi x i 1 it follows that Mx kx 0 116 113 TimeDomain Solutions for 3D Response of MultiDegreeofFreedom Systems Figure 112 Torsional deformation of MeyerKiser building in 1926 Miami hurricane Source From 1 In general the dynamic response to wind of flexible buildings with linearly elastic behavior entails translational motions sway along their principal axes and torsional motions about the buildings elastic center The torsional motions are due to the eccentricity of the aerodynamic and inertial forces with respect to the elastic center An example of torsional deformations induced by wind is shown in Figure 112 The systems equations of free vibration are obtained by following steps analogous to those that led for the single degreeoffreedom system to Eq 116 2 The quantity 2𝜋n is called circular frequency and is commonly denoted by 𝜔 k k k k 198 11 Dynamic and Effective WindInduced Loads 1131 Natural Frequencies and Modes of Vibration The total kinetic energy of a structure with nf masses eg nf floors is T 1 2 nf n1 mn x2 n mn y2 n In 𝜗2 n 117 where xn yn are the displacements of the mass mn in the x and y directions respectively 𝜗n is the torsional rotation of the nth mass about its elastic center and nf is the total number of masses The total strain energy of the system is V 1 2qTkq 118 qT x1 x2 xnf y1y2 ynf 𝜗1 𝜗2 𝜗nf 119 where T denotes tranpose and k is the stiffness matrix see Section 91 For the freely vibrating structure the displacements of the mass center and the tor sional rotation about the mass center at the elevation zi of the ith floor form a vector wt of dimension 3nf Its terms are denoted as follows w1t x1t w2t x2t wnf t xnf t wnf 1t y1t wnf 2t y2t w2nf t ynf t w2nf 1t 𝜗1t w2nf 2t 𝜗2t w3nf t 𝜗nf t 1110 The equations of motion of the undamped freely vibrating system M wt kwt 0 1111 are obtained from the Lagrange equations Eq 115 In Eq 1111 M is a diagonal matrix of the floor masses for sway motions or mass moments of inertia for tor sional motions Equation 1111 are coupled owing to the crossterms of the matrix k Assume solutions of the form wt A cos𝜔t 𝜑 1112 where A is a vector to be defined subsequently Substitution of these solutions in Eq 1111 yields M𝜔2 kA 0 1113 Equation 1113 is a system of linear homogeneous equations in the unknowns A1 A2 Anf that is k11 M1𝜔2A1 k12A2 k1nf A3nf 0 k21A1 k22 M2𝜔2A2 k2 nf A3nf 0 k3nf 1A1 k3nf 2A2 k3nf 3nf M3nf 𝜔2A3nf 0 1114 For Eq 1114 to have nonzero solutions the determinant of the coefficients of the unknowns A must vanish This condition yields a 3nf degree equation in 𝜔2 called the k k k k 113 TimeDomain Solutions for 3D Response of MultiDegreeofFreedom Systems 199 Figure 113 First four normal modes of a cantilever beam characteristic equation Its 3nf roots are called eigenvalues The rankordered frequen cies 𝜔1 𝜔2 𝜔3nf are called the systems natural frequencies of vibration To each of the 3nf eigenvalues there corresponds an eigenvector with 3nf components obtained from Eq 1114 which defines a natural or normal mode of vibration The eigenvec tors corresponding to the 3nf eigenvalues 𝜔 form a 3nf 3nf matrix 𝜙 For i j the vectors 𝜙i and 𝜙j can be shown to be mutually orthogonal with respect to mass or mass moments of inertia weighting that is 3nf k1 𝜙ik𝜙jkMk 0 i j 1115 The free vibrations with their normal modal shapes and associated frequencies are properties of the structural system independent of the loads The first four normal modes along one of the principal axes of a continuous cantilever beam are shown in Figure 113 1132 Solutions of Equations of Motion of Forced System The equations of motion of the forced system are M wt kwt Ft 1116 where Ft is the vector of the wind forces torsional moments with components Fx1t Fx2t Fxnf t Fy1t Fy2t Fynf t M𝜗1t M𝜗2t M𝜗nf t acting at the centers of mass of floors 1 2 nf The variables wt can be written as wt 𝜙𝜉t 1117 where 𝜙 is the matrix consisting of the 3nf eigenvectors 𝜙j j 12 3nf and the coefficients 𝜉 t called generalized coordinates indicate what fraction of each mode enters the given deflection pattern Substitution of Eq 1117 into Eq 1116 yields M𝜙 𝜉t k𝜙𝜉t Ft 1118 k k k k 200 11 Dynamic and Effective WindInduced Loads Premultiplication of Eq 1118 by 𝜙T where the superscript T denotes transpose yields 𝜙TM𝜙 𝜉t 𝜙Tk𝜙𝜉t 𝜙TFt 1119 Owing to the orthogonality of the eigenvectors Eq 1119 to which modal viscous damping terms proportional to the modal damping ratios 𝜁m are added can be written as Mm 𝜉mt 2Mm𝜔m𝜁m 𝜉t Mm𝜔2 m𝜉t 𝜙TFtm m 1 2 3nf 1120 In Eq 1120 the quantities Mm 𝜙TM𝜙m and the quantities in righthand side of Eq 1120 are called the mth mode generalized masses and generalized forces respectively It follows from the unforced equation of motion of the system that Mm𝜔m 2 𝜙Tk𝜙m Once Eq 1120 are solved numerically the physical coordinates wt ie the coordinates x1t x2t xnft y1t y2t ynft 𝜗1t 𝜗2t 𝜗nft are given by Eq 1117 which can be written as wt mmax m1 𝜙m𝜉mt 1121 where mmax is the highest mode that contributes significantly to the response Accel erations wt are obtained by differentiating Eq 1121 twice the second derivatives of the generalized coordinates being known once Eq 1120 are solved The requisite numerical calculations are performed using software that outputs directly the natural frequencies and modes of vibration of the structure and the forces of inertia induced by the wind loading being considered The total timedependent windinduced forces acting on the structure consist of the sums of the applied aerodynamic forces and the inertial forces associated with the struc tures dynamic response If tuned mass dampers are used to reduce the magnitude of the dynamic response they can be viewed as additional masses connected to the structure by springs and dampingproducing devices for details see Chapter 16 114 Simultaneous Pressure Measurements and Effective Windinduced Loads One of the inputs to Eq 1120 is the vector Ft of the applied aerodynamic loads Eq 1116 Figure 114 is an example of the placement of taps used to obtain time histories of simultaneously windinduced pressure coefficients The applied aerodynamic forces and torsional moments induced by wind with speeds Uzref 𝜃w where 𝜃w is the wind direction and zref is the reference height act at the locations of the pressure taps and are obtained from measured time histories of aero dynamic force coefficients Their resultants acting at the mass centers of each floor or group of floors are obtained as indicated in Section 103 and are added algebraically to their inertial counterparts thus yielding the effective windinduced lateral forces and torsional moment at the center of mass of each floor These are used in conjunction with influence coefficients Section 93 to determine internal forces and their weighted combinations this forms the basis on which the buildings structural members are sized as shown in subsequent chapters k k k k Reference 201 SOUTHWEST SOUTHEAST NORTHEAST NORTHWEST Figure 114 Example of pressure tap arrangement on the facades of a building model Source Courtesy of Dr I Venanzi University of Perugia and Dr G Bartoli University of Florence Reference 1 Schmit FE 1926 The Florida hurricane and its effects Engineering News Record 97 624627 k k k k 203 12 Wind Load Factors and Design Mean Recurrence Intervals 121 Introduction A 2004 landmark report by Skidmore Owings and Merrill Appendix F noted the absence in wind engineering laboratory reports of information or guidance pertaining to wind load factors The latter depend upon uncertainties in the micrometeorological aerodynamic and wind climatological parameters that govern structural design These uncertainties can differ in some cases from those on which standard provisions are based The purpose of this chapter is to discuss the development for such cases of appropriate wind load factors or if wind load factors are specified to be equal to unity of appropriately augmented mean recurrence intervals MRI of design wind effects see Section 125 Attempts to develop design criteria applicable to structural systems have been unsuc cessful owing in large part to difficulties arising in the reliability analysis of statically indeterminate structures For this reason strength design criteria are generally focused on individual structural members see Appendix E In modern codes factors assuring that probabilities of failure are acceptably low differ according to whether they apply to loads or resistances and are called load and resistance factors respectively hence the term load and resistance factors design or LRFD Load factors depend upon the type of load eg dead live snow wind loads and are defined as the quantities by which nominal loads or load effects need to be multiplied to obtain the design loads Their magnitude is so calibrated that the resulting structural designs are comparable to proven designs based on past practices The calibration is of necessity imperfect owing to the variety of materials construction techniques and design procedures used in past practices However a feature of past practices that was preserved in LRFD is the choice of MRIs of nominal wind effects which are approximately 50 or 100 years a choice largely based on engineering judgment and experience The load factor that multiplies the nominal Nyear wind load is called the Nyear wind load factor and is denoted by 𝛾wN In pre2010 versions of the ASCE 7 Standard the wind load factor was specified to be approximately 16 In the 2010 and 2016 versions to simplify the design process the wind load factor was specified to be unity However to compensate for the reduction of the load factor from 16 to 1 and achieve design wind effects approximately equal to those implicit in pre2010 ASCE 7 Standard require ments the MRIs of the design wind speeds associated with a wind load factor equal to unity were augmented from 50 or 100 to 700 and 1700 years respectively In addition Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 204 12 Wind Load Factors and Design Mean Recurrence Intervals wind speeds with a 3000year MRI are currently specified for structures classified as essential such as for example fire and police stations Uncertainties in quantities that determine windinduced effects on rigid structures were discussed in Chapter 7 For flexible structures uncertainties in the dynamic fea tures of the structure come into play as well and are discussed in Section 122 The definition of the wind load factor is introduced in Section 123 which also discusses the calibration of the wind load factor with respect to past practice Section 124 pro vides examples of the dependence of the wind load factor upon uncertainties specific to particular design situations The examples show that the uncertainty in the wind speeds dominates the other individual uncertainties Section 125 concerns the use of augmented design MRIs in lieu of products of wind load factors larger than unity by nominal wind loads or wind effects1 122 Uncertainties in the Dynamic Response The dynamic response of the structure depends upon its dynamic properties natural frequencies modal shapes and damping ratios The uncertainty in the dynamic response can in principle be estimated approximately by performing Monte Carlo sim ulations of the response based on assumed probability distributions of the structures dynamic properties In practice the estimation of the uncertainty must be performed largely on the basis of engineering judgment and experience by accounting for among others the cracking behavior of reinforced concrete and the behavior of joints in some types of steel structures It is suggested that for flexible structures the assumption CoVG 012 used in the development of the wind load factor specified in earlier versions of the ASCE 7 Standard is reasonable for preliminary calculations based on Eq 72 1 It was mentioned in Chapter 7 that NASA and the Department of Energy require the use of far more elaborate approaches to uncertainty quantification than are currently available for civil engineering purposes see for example httpsstandardsnasagovstandardnasanasahdbk7009 which provides technical information clarification examples processes and techniques to help institute good modeling and simulation practices in NASA As a companion guide to NASASTD7009 the Handbook provides a broader scope of information than may be included in a Standard and promotes good practices in the production use and consumption of NASA modeling and simulation products NASASTD7009 specifies what a modeling and simulation activity shall or should do in the requirements but does not prescribe how the requirements are to be met which varies with the specific engineering discipline or who is responsible for complying with the requirements which depends on the size and type of project A guidance document which is not constrained by the requirements of a Standard is better suited to address these additional aspects and provide necessary clarification As indicated in 1 the NASA Jet Propulsion Laboratory at the California Institute of Technology is pursuing Quantification of Margins and Uncertainty QMU technology to enable certification of models and simulations for extrapolation to poorlytestable conditions and provides a formalism for establishing credibility of a digital twin that would predict system performance under difficulttotest conditions Among the tools used in QMU are Sandias Sierra Mechanics and Multiphysics tools on models and simulations Sandias DAKOTA uncertainty analysis tool the ASME VV 102006 Guide for Verification and Validation in Computational Solid Mechanics the AIAA Guide for the Verification of Computational Fluid Dynamics Simulation and Department of Energy and Defense Guidelines and Recommended Practices These tools and recommended practices are outside the scope of this chapter but it may be anticipated that adapting them or their principles for civil engineering applications will be considered in the future k k k k 123 Wind Load Factors Definition and Calibration 205 123 Wind Load Factors Definition and Calibration The design peak wind effect with a 50year MRI is defined as ppk desN 50 years ppkN 50 years1 𝛽 CoVppkN 50 years 121 where ppkN 50 years is the expectation and CoVppkN 50 years is the coefficient of variation of the peak wind effect ppk with a 50year MRI For codification purposes the factor 𝛽 is determined by calibration with respect to past practice and consensus among expert practitioners the value 𝛽 2 suggested in 2 appears to be reasonable and is adopted here for illustrative purposes The quantity 𝛾wN 50 years 1 𝛽 CoVppkN 50 years 122 is the wind load factor by which the nominal expected peak wind effect with MRI N 50 years must be multiplied to yield the design peak wind effect Therefore ppk desN 50 years 𝛾wN 50 years ppkN 50 years 123 Example 121 For a rigid building with the notations of Eq 72 let CoVEz 012 CoVKd 01 CoVG𝜃m 005 say where 𝜃m defines the most unfavorable wind direction CoVCppk𝜃m 012 and CoVUN 50 years 012 see 2 It follows from Eq 72 that CoVppkN 0315 and with 𝛽 20 𝛾w 163 as calculated in Eq 124 𝛾w 1 2 0122 012 0052 0122 4 012212 163 124 This is approximately the value adopted in the ASCE 705 Standard 3 4 The following excerpt from 2 pp 6 7 illustrates the problems arising in the calibra tion of the factor 𝛽 with respect to past practice reliability with respect to wind loads appears to be relatively low compared to that for gravity loads at least according to the methods used for structural safety checking in conventional design2 These are methods which are simpli fied representations of real building behavior and they have presumably given satisfactory performance in the past It was decided to propose load factors for combinations involving wind loads that will give calculated 𝛽 values which are comparable to those existing in current practice and not to attempt to raise these values to those for gravity loads by increasing the nominal loads or the load fac tors for wind loading Based on the information given here the profession may well feel challenged 1 to justify more explicitly by analysis or test why current simplified wind calculations may be yielding conservative estimates of loads resistances or safety 2 to justify why current safety levels for gravity loads are higher than necessary if indeed this is true 3 to explain why lower safety levels are appropriate for wind visavis gravity loads or 4 to agree to raise the wind loads or load factors to achieve a similar reliability as that inherent in gravity loads While the authors feel that arguments can be cited in favor and against all four options they decided that this report is not the appropriate forum for what should be a professionwide debate 2 According to those methods the factor 𝛽 for gravity loads is 30 rather than 20 2 k k k k 206 12 Wind Load Factors and Design Mean Recurrence Intervals 124 Wind Load Factors vs Individual Uncertainties Equations 71 72 122 and 123 show that the uncertainty in the peak wind effect and therefore the magnitude of the wind load factor depend upon the individual uncertain ties that appear in the righthand side of Eq 72 This section examines for various cases of interest the degree to which the influence of an individual uncertainty on the magnitude of the wind load factor is significant Except as otherwise noted the individ ual uncertainties being considered are assumed to be those of Example 121 1241 Effect of Wind Speed Record Length Assume that the length of the record of the largest yearly wind speeds to which there corresponds the value CoVUN 50 years 012 is 30 years That value is due to measurement and sampling errors for which the CoVs are assumed to be 007 and 01 respectively It was seen that to the CoVs of the uncertainties considered in Example 121 there corresponds a wind load factor 𝛾w 163 Assume now that the record length on the basis of which the sampling errors in the estimation of the 50year speed was esti mated was only 10 years as may be the case for remote locations for which few reliable meteorological measurements are available Since the standard deviation of the sam pling error is approximately proportional to the reciprocal of the square root of the sample size see Eq 39 the coefficient of variation characterizing the sampling errors may be assumed to be 3 times larger than for the case in which the record length is 30 years Therefore CoVUN 0072 01 3212 0187 Instead of 𝛾w 16 it follows from Eq 122 that the estimated wind load factor is 𝛾w 1 2 0122 012 0052 0122 4 0187212 185 125 The ratio between the wind load factors based on the 10year wind speed record and the 30year record of wind speeds all other uncertainties being unchanged is approximately 114 This is in part a consequence of the multiplication of CoVUN by the factor 4 see Eq 72 owing to which the contribution to the wind load factor of the uncertainty in the wind speed dominates the contributions of the other individual uncertainties 1242 Effect of Aerodynamic Interpolation Errors Large sets of aerodynamic pressure data used for databaseassisted design cannot cover all possible model dimensions and roof slopes For this reason interpolations based on databases with limited numbers of models are typically necessary in the design pro cess According to calculations reported in 5 such interpolations entail errors that depending upon the number of models in the database can have CoVs as large as 015 say Accounting for this CoV in the expression for the load factor 𝛾w 1 2 0122 012 0052 0122 0152 4 012212 170 126 rather than 163 that is the increase in the estimated value of the wind load factor in this example is approximately 5 This result suggests that the number of models in large aerodynamic databases does not necessarily have to be increased unless the CoVs of the interpolation errors in the estimation of the pressure coefficients exceed 15 say k k k k 124 Wind Load Factors vs Individual Uncertainties 207 1243 Number of Pressure Taps Installed on Building Models The lower the number of taps placed on the model the larger will be the errors in the estimation of the wind effects Figures 529 and 530 show the vast difference between the numbers of taps typically used before and after the development of pressure scanners to determine wind loads For strength design purposes useful assessments of the extent to which the number of pressure taps installed on the building model is adequate by modern standards can be made by comparing base shears and moments obtained by highfrequency force balance measurements to their counterparts based on pressure time histories at the taps 6 or in some cases by comparing wind effects based on all the available taps on the one hand and on say half the number of taps on the other 1244 Effect of Reducing Uncertainty in the Terrain Exposure Factor Adhoc wind tunnel testing that reproduces to scale the built environment of the struc ture being designed has the advantage of reducing the uncertainty in the terrain expo sure factor Because the wind tunnel simulation of the atmospheric flow is imperfect the reduction may be relatively modest from CoVEz 012 as in 2 to CoVEz 005 say For a rigid structure this would result in a less than 3 reduction of the wind load factor from 𝛾w 163 see Eq 124 to 𝛾w 159 𝛾w 1 20052 0102 0052 0122 4 012212 159 127 1245 Flexible Buildings Flexible structures experience dynamic effects that may be expressed in terms of a dynamic response factor G According to 2 typically CoVG 012 This value is based on early studies of uncertainties in the alongwind response 7 In some instances natural frequencies modal shapes and modal damping ratios are dependent upon factors that are difficult to quantify and on which relatively few reliable data exist For example estimates of the extent to which cracking of concrete influences the structures stiffness characteristics may still be affected by significant uncertainties For these reasons the coefficients of variation of the uncertainty in the dynamic effects may be larger than 012 Nonzero values of CoVG increase the coefficient of variation of the peak wind effect and will therefore result in wind load factors larger than their rigid structure coun terparts This explains the quest by structural engineers for adhoc wind load factors applicable to tall buildings see Appendix F Assuming for example that CoVG 012 and that the other individual uncertainties affecting the wind load factor have the values used in Eq 124 𝛾w 1 20122 012 0122 0122 4 012212 167 128 rather than 163 for the rigid structure case If in addition the length of the wind speed record is 10 years and CoVUN 0187 as in Section 1241 𝛾w 188 1246 Notes 1 Except for the uncertainty in the wind speed individual uncertainties in the quan tities that determine wind effects typically have relatively small or negligible effects k k k k 208 12 Wind Load Factors and Design Mean Recurrence Intervals on the magnitude of the wind load factors This fact should be considered before significant resources are devoted to efforts to reduce these uncertainties 2 The magnitude of the wind load factor can be affected significantly by uncertainties in the wind speeds that are larger than those typically assumed in standards This is especially true of hurricane wind speeds for which estimates of uncertainties are difficult to determine reliably 3 Wind load factors are larger for flexible buildings than for rigid buildings and the joint effect of uncertainties in the dynamic response and of larger than typical uncer tainties in the wind speeds can result in large increases in the wind load factors Standard provisions on the wind tunnel procedure should clearly indicate this fact 4 It was shown that typical uncertainties in pressure coefficients obtained in wind tun nel tests have relatively minor effects on the magnitude of the wind load factor This suggests that the use of Computational Wind Engineering simulations to obtain esti mates of pressure coefficients should be acceptable for practical purposes as long as the CoVs of the uncertainties in those estimates are lower than say 15 125 Wind Load Factors and Design Mean Recurrence Intervals ASCE 705 Standard and earlier versions specified a typical MRI of the design wind speed N 50 years and a wind load factor 𝛾w 16 Later versions instead specify no wind load factor ie a wind load factor 𝛾w 1 and an augmented MRI N1 of the design wind speed such that the design wind loads are approximately the same in the earlier and the current standards Since wind effects determined in accordance with conventional provisions of the ASCE 7 Standard are proportional to the square of the wind speeds U this condition yields the relation U2N1 𝛾wU2N 129 For N 50 years 100 years and 𝛾w 16 this relation was assumed to yield N1 700 years 1700 years These values correspond to typical probability distributions of extreme wind speeds Since those probabilities can depend fairly strongly on geographical location the values N1 of the MRIs of the design wind speeds and wind effects may turn out to differ in some cases significantly from 700 or 1700 years Example 122 Let the mean EU and the standard deviation SDU of the extreme annual wind speed sample be 59 and 641 mph respectively The 50 and 700years wind speeds are estimated by Eq 37b to be 756 and 889 mph respectively The design wind effect is ppk des c𝛾wU2N 50 years cU2N1 years where c is a coefficient that reflects the relation between wind effect and the square of the wind speed Therefore from Eq 129 UN1 years 𝛾w 12 U 50 years k k k k References 209 where N can be estimated using Eqs 35 and 37 For 𝛾w 16 UN1 years 956 mph rather than 889 mph and N1 2700 years rather than N1 700 years as specified in the ASCE 716 Standard For 𝛾w 15 UN1 years 926 mph and N1 1500 years This example shows that like the wind load factors the MRIs of the design wind effects should be specified by accounting for the wind climate statistics and the specific uncer tainties in the micrometeorological wind climatological aerodynamic directionality and dynamic features of the structure of interest In light of this example it appears that the validity of the neat correspondence suggested in 8 figure 3 between MRIs and factors 𝛽 see Eq 123 is not warranted References 1 Peterson L D Quantification of margins and uncertainties for modelinformed flight system qualification presented at the NASA Thermal and Fluids Analy sis Workshop Hampton VA 2011 httpskisscaltecheduworkshopsxTerra presentations1petersonpdf 2 Ellingwood B Galambos T V MacGregor J G and Cornell C A Development of a probabilitybased load criterion for American National Standard A58 NBS Spe cial Publication 577 National Bureau of Standards Washington DC 1980 https wwwnistgovwind 3 ASCE Minimum design loads for buildings and other structures in ASCE Standard ASCESEI 705 Reston VA American Society of Civil Engineers 2005 4 Simiu E Pintar AL Duthinh D and Yeo D 2017 Wind load factors for use in the wind tunnel procedure ASCEASME Journal of Risk and Uncertainty in Engineer ing Systems Part A Civil Engineering 3 doi101061AJRUA60000910 httpswww nistgovwind 5 Habte F Chowdhury AG Yeo D and Simiu E 2017 Design of rigid structures for wind using time series of demandtocapacity indexes application to steel portal frames Engineering Structures 132 428442 6 Dragoiescu C Garber J and Kumar K S The use and limitation of the pres sure integration technique for predicting windinduced responses of tall buildings in European and African Conference on Wind Engineering Florence Italy 2009 pp 181184 7 Vickery B J On the reliability of gust loading factors in Technical Meeting Con cerning Wind Loads on Buildings and Structures Washington DC 1970 httpswww nistgovwind 8 McAllister TP Wang N and Ellingwood BR Riskinformed mean recurrence intervals for updated wind maps in ASCE 716 Journal of Structural Engineering 144 5 06018001 doi101061ASCEST1943541X0002011 k k k k 211 13 Wind Effects with Specified MRIs DCIs InterStory Drift and Accelerations 131 Introduction Wind directionality is accounted for in different ways depending upon whether the wind climatological data consist of directional or nondirectional wind speeds as defined in Section 322 If the design is based on directional wind speeds as is commonly the case for designs using wind tunnel test results wind effects with specified mean recurrence intervals MRIs are determined by accounting explicitly for the dependence of both the wind speeds and the wind effects upon direction For the databaseassisted design approach this requires 1 The use of matrices of directional wind speeds Uij provided by the wind engineer ing laboratory where Uij is the mean wind speed at top of building in storm event i i 1 2 ns from direction 𝜃 𝜃j j 1 2 nd based on a sample of mea sured or simulated directional wind speeds The number ns of storms for which wind speeds are available in the matrix Uij must be sufficiently large to allow the reliable estimation by nonparametric statistics of wind effects with the required MRI If as is the case for the ASCE 710 and ASCE 716 Standards the MRI is 700 years or larger Monte Carlo simulations are used to meet this requirement see 1 and Sections 337 and A8 2 The development for each type of wind effect of interest eg base shear base moment internal force peak demandtocapacity index DCI displacement acceleration of time series RU 𝜃 t representing the dependence of the wind effect R upon the wind speed U the direction 𝜃 and the time t The length T of the time series RU 𝜃 t is equal to the length of the time series of pressure coefficients provided by the wind engineering laboratory However the peak value of RU 𝜃 t that is maxtRU 𝜃 t henceforth denoted as RpkU 𝜃 can be determined for time series with any specified length T1 T by using for example the procedure described in Section 733 v or the procedure in Appendix C The response surface is the threedimensional plot of RpkU 𝜃 as a function of wind speed U and direction 𝜃 Section 132 3 The transformation of the directional wind speed matrix Uij into the matrix RpkUij This is accomplished by substituting the quantities RpkUij for the wind velocities Uij in the matrix Uij Sections 1331 and 1332 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 212 13 Wind Effects with Specified MRIs DCIs InterStory Drift and Accelerations 4 The transformation of the matrices RpkUij into vectors Ri maxjRpkUij by disregarding in each windstorm i all wind effects RpkUij lower than the largest wind effect maxjRpkUij occurring in that storm Section 1333 5 The application to the ns rankordered quantities Ri of the nonparametric sta tistical estimation procedure in Section A9 for regions with one or two types of windstorm to obtain the wind effects RN with the specified MRI Section 134 If the design is based on nondirectional wind speeds which is the case if directional wind speed data are not available the design is based on pressure coefficients and wind speeds with the respective most unfavorable directions which typically do not coin cide It follows from the assumed linear dependence of the mean wind loads upon the square of the nondirectional wind speeds that the MRI of the wind loads is the same as the MRI of the wind speeds However a correction factor smaller than unity called wind directionality factor is applied to the wind effect to account approximately for the noncoincidence of the most unfavorable pressure and wind directions Section 135 Material on DCIs and on interstory drift and accelerations is provided in Sections 136 and 137 respectively A method for estimating directionality effects developed in the 1970s and still being used by some wind engineering laboratories is described in Section B6 which discusses the reasons why the method is impractical and prone to yielding inadequate estimates of the wind effects being sought In addition that method is viewed by structural engineers as lacking transparency as indicated in Appendix F An alternative method proposed in 2 is also being used by some laboratories in spite of the fact that it can yield uncon servative results In practical applications operations covered by this chapter can be performed by using software for which links are provided in Chapters 17 and 18 132 Directional Wind Speeds and Response Surfaces Once the wind engineering laboratory provides the requisite aerodynamic and wind cli matological data as affected by terrain exposure at the structures site the structural engineers first step toward determining peak wind effects RN where N denotes the specified MRI is to develop response surfaces that is threedimensional plots repre senting the dependence of peak wind effects RpkUij upon wind speed and direction A response surface is constructed for each wind effect of interest An example of response surface for a peak DCI involving the axial force and bending moment at a given member cross section is shown in Figure 131 In general owing to nonlinearities inherent in resonance effects andor column insta bility the ordinates of the DCI response surfaces are not proportional to the squares of the wind speeds The wind effect of interest must therefore be determined separately for each wind speed and direction The response surfaces are properties of the structure independent of the wind climate As shown subsequently they are used in a simple nonparametric statistical procedure that yields peak wind effects with any specified MRI Response surfaces for DCIs are developed as follows Consider the time series of the effective forces FkxU 𝜃 t FkyU 𝜃 t Fk𝜗U 𝜃 t t time induced by wind with k k k k 133 Transformation of Wind Speed Matrix into Vectors of Largest Wind Effects 213 DCIm pkU θ 360 270 180 90 θ deg U ms 0 1020304050607080 0 05 1 15 2 Figure 131 Response surface for the peak demandtocapacity index of a cross section m as a function of wind speed and direction speed U from direction 𝜃 and acting at the center of mass of floor k 1 2 nf in the directions of the buildings principal axes and in torsion respectively The internal forces fmU 𝜃 t induced by the effective wind forces at any given cross section m can be written as fmU 𝜃 t nf k1 rmkx FkxU 𝜃 t nf k1 rmky FkyU 𝜃 t nf k1 rmk𝜗 Fk𝜗U 𝜃 t 131 where rmkx rmky rmk𝜗 are influence coefficients see Section 93 Equation 131 is then used to obtain time series of demandtocapacity DCIs at cross sections m denoted by DCImU 𝜃 t in which it is recalled that effects of factored gravity loads are also accounted for A similar approach is used for displacements and accelerations 133 Transformation of Wind Speed Matrix into Vectors of Largest Wind Effects 1331 Matrix of Largest Directional Wind Speeds In the following we focus on the DCIpk m Uij induced in cross section m by the wind speed Uij acting in storm i from direction j at the building site Similar approaches can be used for any other wind effects Consider for illustrative purposes the 3 4 matrix of wind speeds in m s1 Uij 34 𝟒𝟓 32 44 37 39 36 𝟓𝟏 42 44 35 𝟒𝟔 132 at the site of the structure See Section 323 for wind speed data that are available or that can be developed by simulation from such data Under the convention inherent k k k k 214 13 Wind Effects with Specified MRIs DCIs InterStory Drift and Accelerations in the notation Uij the 3 4 matrix corresponds to three storm events and four wind directions that is i 1 2 3 and j 1 2 3 4 For example the wind speed that occurs in the second storm event from the third direction is U23 36 m s1 The entries in the wind speed matrix could be for example mean hourly speeds at the elevation of the top of the structure with direction j over terrain with suburban exposure In the matrix of Eq 132 the largest wind speeds in each of the three storms are indicated in bold type 1332 Transformation of Matrix Uij into Matrix of DemandtoCapacity Indexes DCIpk m Uij Transform the matrix Uij into the matrix DCIpk m Uij by substituting for the quantities Uij the ordinates DCIpk m Uij of the cross section m response surface Assume that these quantities are DCIs and that the result of this operation is the matrix DCIpk m Uij 070 𝟏𝟎𝟐 080 068 083 083 𝟏𝟎𝟏 091 𝟏𝟎𝟕 098 096 074 133 1333 Vector DCImi maxjDCIpk m Uij The directional wind effects induced by the wind speeds occurring in storm i depend upon the wind direction j It is only the largest of those wind effects that is DCImi maxj DCIpk m Uij i 1 2 3 that are of interest from a design viewpoint These largest wind effects shown in bold type in Eq 133 form a vector 102 101 107T where T denotes transpose Note that DCImi is not necessarily induced by the speed maxjUij For example DCIm3 107 is not induced by the speed maxjU3j U34 46 m s1 but rather by the speed U31 42 m s1 The components of the vector DCImi constitute the sample of the largest peak wind effects occurring in each of the ns storm events in this example i 1 2 3 ns 3 The estimation of the response with any specified MRI is based on this sample used in conjunction with the mean annual rate of occurrence of the storms see Section 134 134 Estimation of Directional Wind Effects with Specified MRIs The peak wind effects DCImN where N denotes the specified MRI in years could in principle be determined by using parametric statistics This would entail the fitting of a cumulative distribution function CDF to the sample DCImi i 1 2 ns The variate DCIm with an MRI Nf where Nf is the number of average time intervals between successive storms corresponds in the example of Section 133 to a CDF ordinate P 1 1Nf However the designer is interested in the variate DCIm with an MRI N in years rather than in average time intervals between successive storms Since the mean annual rate of storm arrival is 𝜆 N Nf 𝜆 For example if the storms being considered are k k k k 135 NonDirectional Wind Speeds Wind Directionality Reduction Factors 215 tropical cyclones it is typically the case that 𝜆 1 stormyear so N Nf The converse is true for the case 𝜆 1 stormyear Therefore the variate DCIm with an MRI N DCIm N corresponds to the ordinate P 1 1𝜆N of the CDF fitted to the data sample DCImi i 1 2 ns A drawback of parametric statistics for this type of application would be that few stud ies have been performed on and little is known about the best fitting types of probability distribution of the various wind effects as opposed to wind speeds If as is the case for the ASCE 716 Standard the MRIs of interest are large eg 3003000 years the uncertainty inherent in the choice of the best fitting type of probability distribution may entail significant probabilistic modeling errors It is therefore prudent to use the nonparametric approach The application of nonparametric statistics requires the development by the wind engineering laboratory of synthetic directional wind speed samples from measured directional wind data The development entails three phases In the first phase the measured directional wind speeds are processed by the wind engineer so that they are micrometeorologically consistent with the wind speeds used in the development of the directional aerodynamic pressure time series In the second phase the directional wind speed data so obtained are fitted to Extreme Value Type I distributions see Chapters 3 and Appendix A which are widely accepted as appropriate for the probabilistic description of extreme wind speeds A probability distribution is fitted to the wind speeds from each direction j When doing so it may be assumed for practical purposes that wind speeds from different directions are for practical purposes mutually indepen dent provided that the respective azimuths do not differ by less than 10 say In the third phase the Extreme Value Type I distributions are used to develop by Monte Carlo simulation see Section A8 the requisite large sets of directional extreme wind speed data 1 These sets are provided to the structural engineer by the wind engineering laboratory The structural engineer can then use the simulated extreme wind speed data as input to software subroutines for the estimation of wind effects with specified MRIs This approach is implemented in the software presented in Chapters 17 and 18 which uses the procedure of Section A91 for regions with a single type or Section A92 for regions with two types of storm hazard eg synoptic storms and thunderstorms 135 NonDirectional Wind Speeds Wind Directionality Reduction Factors If dynamic effects are negligible design wind loads W stdN are typically based in stan dards on nondirectional sets of pressure or force coefficients Cp maxjCpj where Cpj is the peak directional force or pressure coefficient corresponding to wind direction j j 1 2 jmax eg jmax 16 wind speeds with an Nyear MRI UN estimated from the nondirectional wind speeds data Ui maxjUij where Uij is the largest directional wind speed from direc tion j during storm event i defined for the appropriate terrain exposure height above ground and averaging time k k k k 216 13 Wind Effects with Specified MRIs DCIs InterStory Drift and Accelerations a factor Kd that accounts for directionality effects Therefore WstdN a Kd Cp U2N 134 where a is a constant The wind directionality factor Kd is defined as the ratio of cal culated wind effects WdirN and WndN that account and do not account for wind directionality respectively Kd WdirN WndN 135 where the numerator and the denominator are estimated respectively from the data Wi dir a maxjCpjUij 2 136 and Wi nd a maxjCpj maxj Uij2 137 i 1 2 ns the indexes dir and nd stand for directional and nondirectional It is clear that typically W i nd W i dir and that Kd 1 Example 131 Consider the directional wind speed matrix Uij of Eq 132 Assume that the directional aerodynamic coefficients Cpj are 07 08 12 and 06 for directions j 1 2 3 and 4 respectively It can be easily verified that the entries in Table 131 are smaller for column 2 than for column 1 For the simplified estimated value of W stdN to be reasonably correct it is required that the directionality factor in Eq 134 be approximately equal to the ratio W dirN W ndN According to the ASCE 7 Standard this is the case for typical buildings if Kd 085 reflecting the fact that the climatologically and aerodynamically most unfa vorable wind directions typically do not coincide Calculations reported in 3 indicate that the use of this value in design is typically reasonable although for hurricaneprone regions it is prudent to use the value Kd 09 Table 131 Comparison of nondirectional and directional wind load estimates 1 2 I maxj Cj maxjUij 2 m2 s2 nondirectional maxjCj Uij 2 m2 s2 directional 1 12 452 2430 08 452 1620 j 2 2 12 512 3121 12 362 1561 j 3 3 12 462 2539 08 442 1549 j 2 k k k k 137 InterStory Drift and Floor Accelerations 217 136 DemandtoCapacity Indexes This section is a brief presentation of material on DCIs for steel and reinforced concrete buildings The DCIm is a measure of the degree to which the strength of a structural cross section m is adequate In general the index is defined as a ratio or sum of ratios of the required internal forces to the respective available capacities A DCI larger than unity indicates that the design of the cross section being considered is inadequate The general expression for the DCIs used in design is DCIPMt f Put 𝜙pPnt Mut 𝜙mMnt 1 138 DCIVTt f Vut 𝜙vVnt Tut 𝜙tTnt 1 139 where the symbols P M V and T represent compressive or tensile strength flexural strength shear strength and torsional strength respectively the subscripts u and n indi cate required and available strength respectively and 𝜙i resistance factors i p m v t corresponding to axial flexural shear and torsional strength respectively The available strength is specified by the AISC Steel Construction Manual 4 for steel structures and the ACI Building Code Requirements for Structural Concrete 5 or other documents For details on the application of 4 and 5 in the context of this book see 6 7 137 InterStory Drift and Floor Accelerations The approach to determining wind effects with specified MRIs considered in Sections 132134 is applicable in particular to interstory drift ratios and floor accelerations The timeseries of the interstory drift ratios at the kth story dkxt and dkyt corre sponding to the x and yprincipal axis of the building are Figure 132 dkxt xkt Dky𝜗kt xk1t Dk1y𝜗k1t hk 1310a dkyt ykt Dkx𝜗kt yk1t Dk1x𝜗k1t hk 1310b Figure 132 Position parameters at floor k for interstory drift and accelerations Column line of interest Dkx Dky xk yk ϑk k k k k 218 13 Wind Effects with Specified MRIs DCIs InterStory Drift and Accelerations where xkt ykt and 𝜗kt are the displacements and rotation at the mass center at the kth floor Dkx and Dky are distances along the x and yaxis from the mass center on the kth floor to the point of interest on that floor and hk is the kth story height The timeseries of the resultant acceleration at floor k akrt is yielded by the expres sion akrt xkt Dky 𝜗kt2 ykt Dkx 𝜗kt2 1311 where accelerations xkt ykt and 𝜗kt of the mass center at the kth floor pertain to the x y and 𝜗 ie rotational axis and Dkx and Dky are the distances along the x and y axis from the mass center to the point of interest on the kth floor Figure 132 References 1 Yeo D 2014 Generation of large directional wind speed data sets for estima tion of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 2 Simiu E and Filliben JJ 2005 Wind tunnel testing and the sectorbysector approach to wind directionality effects Journal of Structural Engineering 131 11431145 httpswwwnistgovwind 3 Habte F Chowdhury A Yeo D and Simiu E 2015 Wind directionality factors for nonhurricane and hurricaneprone regions Journal of Structural Engineering 141 04014208 4 ANSIAISC 2010 Steel Construction Manual 14 ed American Institute of Steel Construction 5 ACI 2014 Building Code Requirements for Structural Concrete ACI 31814 and Commentary Farmington Hills MI American Concrete Institute 6 Yeo D Databaseassisted design for wind Concepts software and example for of highrise reinforced concrete structures NIST Technical Note 1665 National Institute of Standards and Technology Gaithersburg MD 2010 httpswwwnistgovwind 7 Park S Yeo D and Simiu E Databaseassisted design and equivalent static wind loads for mid and highrise structures concepts software and users manual NIST Technical Note 2000 National Institute of Standards and Technology Gaithersburg MD 2018 httpswwwnistgovwind k k k k 219 14 Equivalent Static Wind Loads 141 Introduction This chapter presents a procedure for determining equivalent static wind loads ESWLs on mid and highrise buildings A similar procedure is presented in 1 and 2 and is demonstrated in a case study in Chapter 18 The tasks performed by using ESWLs commonalities and differences between those tasks and the tasks performed by using DatabaseAssisted Design DAD and compar ative ESWL and DAD features were considered in Sections 83 and 84 Section 142 describes a procedure for determining ESWLs It follows from the description of that procedure that ESWLbased designs are typically limited to buildings with simple geometries For structures with complex geometries riskconsistent designs require the use of the more computerintensive and more accurate DAD procedure Like DAD the ESWL procedures presented in this chapter and in 1 2 are user friendly transparent readily subjected to effective public scrutiny and easily integrated into Building Information Modeling BIM systems Also like DAD ESWL renders obsolete the High Frequency Force Balance HFFB practice wherein analyses of windinduced dynamic effects are performed by the wind engineer in the absence of information of the distribution of the wind loads with height In contrast to HFFB ESWL allows iterative structural designs to be readily performed with no timeconsuming backandforth interactions between the wind and the structural engineer For structures with relatively simple shapes wind effects calculated by using ESWL approximate reasonably closely their DAD counterparts The latter may serve as reliable benchmarks against which ESWL calculations can be verified However the ESWL procedure can be less effective if wind speeds from a direction that is unfavorable from a structural point of view are dominant Also for structures with complex shapes the ESWL procedure may be inapplicable 142 Estimation of Equivalent Static Wind Loads Earlier approaches to the estimation of ESWLs are described in 35 This section describes an approach to structural design that typically induces in structural mem bers DCIs approximately equal to their counterparts obtained by using DAD As noted earlier like other ESWL approaches the approach presented in this section is applicable only to structures with relatively simple shapes Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 220 14 Equivalent Static Wind Loads 2nd 1st nf th nf1th nf th nf1th h h h All floor heights h Mbyt maxtFnfxt maxtF2xt maxtF1xt maxtMbyt maxtFnf1x t h h h h 2nd 1st a b Fnfxt ESWL Fnfx αMyt1 maxt Fnfx t ESWL F2x αMyt1 maxt F2x t ESWL F1x αMyt1 maxt F1x t ESWL ESWL ESWL ESWL h F1x 2F1x nf 1Fnf1x nfFnfx maxt Mby t αMyt1 Mb0y ESWL Fnf1x αMyt1 maxt Fnf1x t F2xt F1xt Fnf1x t Figure 141 Lumped mass structure with fluctuating wind loads acting in the direction x a fluctuating wind loads in DAD and b equivalent static wind loads in ESWL Typically owing to imperfect spatial correlations peak windinduced loads at different floors do not occur at the same times We first consider the simple though physically unrealistic case in which the wind loads are assumed to act on the structure only in the direction x of a principal axis of the building We denote the effective ie aerodynamic plus dynamic randomly fluctuating load at floor k by Fkxt where k 1 2 nf see Figure 141 Assume for the sake of simplicity that all floors have height h The sum of the moments of the loads maxtFkxt with respect to the building base is Mb0y hmaxt F1xt 2 maxt F2xt nf maxt Fnf xt 141 Owing to the imperfect spatial correlation between any pair of timedependent floor loads the peak of the actual base moment Mbyt induced by the effective loads Fkxt is maxtMbyt Mb0y 142 Denote by FESWL kx the ESWL acting at the floor k In order for the static loads FESWL kx to produce a peak base moment maxtMbyt the peak floor loads maxtFkxt are multiplied by a reduction coefficient 𝛼Myt1 such that FESWL kx 𝛼Myt1maxtFkxt 143a 𝛼Myt1 maxtMbyt Mb0y 143b where t1 is the time of occurrence of the peak base moment maxtMbyt The ESWLs FESWL kx determined as described here are acceptable for design purposes if they induce k k k k 142 Estimation of Equivalent Static Wind Loads 221 in each structural member DCIs approximately equal to the peak DCIs induced by the fluctuating loads The equivalence of static and fluctuating forces must apply to the internal forces fmt at all cross sections m within the structure where fmt rm1x F1xt rm2x F2xt rmnf x Fnf xt 144 m 1 2 mmax identifies the cross section being considered and rmkx k 1 2 nf are influence coefficients that is the loads FESWL kx must satisfy the system of equations maxtfmt rm1xFESWL 1x rm2xFESWL 2x rmnf xFESWL nf x m 1 2 mmax k 1 2 nf 145 Since mmax nf Eq 145 cannot be satisfied exactly and in certain cases even approximately In reality loads induced by wind with given velocity U𝜃 do not act along direction x only as was assumed for simplicity in Eqs 141145 Rather they act simultaneously along the structures principal axes x and y and about the vertical torsional axis 𝜗 In addition during any one storm the structure is subjected to winds from all directions 𝜃 with each of the velocities U𝜃 inducing three simultaneous loads along the axes x y and about the axis 𝜗 It is shown in Chapter 18 that if directional wind effects are accounted for equations analogous to Eq 145 can in practice be satisfied to within an approximation in the order of 10 or less This is attributed to the fact that for some wind directions those equations overestimate while for other directions they underestimate the wind effects being sought However if the extreme wind climate is dominated by winds with direction unfavorable from a structural point of view for some members the approximation may be in the order of 20 or more If a member experiences effects of three simultaneous fluctuating loads an approx imate estimate of the peak of the combined effects induced in the member by those loads can be obtained by the following approach Three wind loading cases WLCs are considered In the first WLC denoted by WLC1 the peak effect induced by the first load called the WLC1 principal load is added to the effects induced by the second and third loads called WLC1 companion loads at the time t1 of occurrence of that peak In the second third WLC case denoted by WLC2 WLC3 the peak effect induced by the second third load called the WLC2 WLC3 principal load is added to the effects induced by the first and third second loads called WLC2 WLC3 companion loads at the time t2 t3 of occurrence of that peak Of the three WLCs only the WLC producing the largest wind effect is retained for design purposes By applying this approach to the problem at hand we have FESWL kx 𝛼princ Myt1 maxtFkxt 146 where 𝛼princ Myt1 maxtMbyt Mb0y Mbyt1 Mb0y 147 t1 is the time of occurrence of the peak of Mbyt and Mb0y is the base moment induced by the loads maxtFkxt k 1 2 nf The superscript princ indicates that the k k k k 222 14 Equivalent Static Wind Loads reduction factor 𝛼princ Myt1 applied to the loads maxtFkxt acting in the x direction cor responds to the peak value of the base moment Mby We rewrite Eq 146 in the form FESWL kx 𝛼princ Myt1 maxtFnf xt maxtF2xt maxtF1xt 148 For the companion loads we have FESWL ky 𝛼comp Mxt1 maxtFnf yt maxtF2yt maxtF1yt 149 where 𝛼comp Mxt1 Mbxt1 Mb0x 1410 and FESWL k𝜗 𝛼comp M𝜗t1 maxtFnf 𝜗t maxtF2𝜗t maxtF1𝜗t 1411 where 𝛼comp M𝜗t1 Mb𝜗t1 Mb0𝜗 1412 The procedure just described is based on the pointintime PIT estimator of the peak of a sum of random time series A similar but more reliable estimator was devel oped in 6 and is based on the multiple pointsintime MPIT estimator illustrated in Figure 142 The MPIT approach makes use of rankordered peaks in each time series of base moments and base torsion Let the number of largest values of time series Mbx be npit 4 see the upper four circle symbols in Figure 142a Denote the times of occurrence of these values by tj j 1 2 3 4 The moments Mbxtj called princi pal components are combined with the values Mbytj and Mb𝜗tj see x symbols in Figure 142b and c called companion components The same procedure is used for the lowest negative values of Mbx Next the procedure is used for the npit 4 peak posi tive values and the peak negative values of Mby and finally for Mb𝜗 The total number of WLCs is then 4 2 3 24 It is shown in the case study of Chapter 18 that the accu racy of the estimated DCIs ie the degree to which the DCIs obtained by ESWL are k k k k 142 Estimation of Equivalent Static Wind Loads 223 0 1000 2000 3000 4000 5000 6000 7000 Time s 0 1000 2000 3000 4000 5000 6000 7000 Time s 0 1000 2000 3000 4000 5000 6000 7000 Time s 5 0 5 107 107 106 Mbx Nm 8 6 4 2 0 Mby Nm 5 0 5 Mb Nm Principal components Companion components a b c Figure 142 Effective base moment components and wind load cases WLCs k k k k 224 14 Equivalent Static Wind Loads close to the peak DCIs obtained by DAD improves as the number npit of points in time increases For additional details that further explain the accuracy of ESWLs estimated by the approach presented in this chapter see Sections 83 and 84 DAD and ESWL computations can be performed by using respectively the DAD and the ESWL option of the DADESWL version 10 software see Chapter 18 References 1 Park S Yeo D and Simiu E Databaseassisted design and equivalent static wind loads for mid and highrise structures concepts software and users manual NIST Technical Note 2000 National Institute of Standards and Technology Gaithersburg MD 2018 httpswwwnistgovwind 2 Park S Simiu E and Yeo D Equivalent static wind loads vs databaseassisted design of tall buildings An assessment Engineering Structures submitted https wwwnistgovwind 3 Boggs D and Lepage A 2006 Wind Tunnel Methods vol 240 Special Publication 125142 Farmington Hills MI American Concrete Institute 4 Garber J Browne M T L Xie J and Kumar KS Benefits of the pressure integra tion technique in the design of tall buildings for wind 12th International Conference on Wind Engineering ICWE12 Cairns 2007 5 Huang G and Chen X 2007 Wind load effects and equivalent static wind loads of tall buildings based on synchronous pressure measurements Engineering Structures 29 26412653 6 Yeo D 2013 Multiple pointsintime estimation of peak wind effects on structures Journal of Structural Engineering 139 462471 httpswwwnistgovwind k k k k 225 15 WindInduced Discomfort in and Around Buildings 151 Introduction It is required that structures subjected to wind loads be sufficiently strong to perform adequately from a structural safety viewpoint For tall buildings the designer must also take into account windrelated serviceability requirements meaning that structures should be so designed that their windinduced motions will not cause unacceptable discomfort to the building occupants Windinduced discomfort is also of concern in the context of the serviceability of out door areas within a built environment Certain building and open space configurations may give rise to relatively intense local wind flows It is the designers task to ascertain in the planning stage the possible existence of zones in which such flows would cause unacceptable discomfort to users of the outdoor areas of concern Appropriate design decisions must be made to eliminate such zones if they exist The notion of unacceptable discomfort may be defined as follows In any given design situation various degrees of windinduced discomfort may be expected to occur with certain frequencies that depend upon the features of the design and the wind climate at the location in question The discomfort is unacceptable if these frequencies are judged to be too high Statements specifying maximum acceptable frequencies of occurrence for various degrees of discomfort are known as comfort criteria In practice reference is made to a suitable parameter various values of which are associated with various degrees of discomfort In the case of windinduced structural motions the relevant parameter is the building acceleration at the top floors In criteria pertaining to the serviceabil ity of pedestrian areas the parameter employed is an appropriate measure of the wind speed near the ground at the location of concern It is therefore necessary to assign max imum probabilities of exceedance to the parameters corresponding to various degrees of discomfort Verifying the compliance of a design with requirements set forth in a given set of comfort criteria involves two steps First an estimate must be obtained of the wind velocities under the action of which the parameter of concern will exceed the critical values specified by the comfort criteria Second the probabilities of exceedance of those velocities must be estimated on the basis of appropriate wind climatological informa tion The design is regarded as adequate if the probabilities so estimated are lower than the maximum acceptable probabilities specified by the comfort criteria The development of comfort criteria for the design of tall buildings is discussed in Section 152 Comfort criteria for pedestrian areas are considered in Sections 153155 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 226 15 WindInduced Discomfort in and Around Buildings 152 Occupant WindInduced Discomfort in Tall Buildings 1521 Human Response to WindInduced Vibrations Studies of human response to mechanical vibrations have been conducted predomi nantly by the aerospace industry Because the frequencies of vibration of interest in aerospace applications is relatively high usually 135 Hz the usefulness of these stud ies to the structural engineer is generally limited Nevertheless results obtained for high frequencies have been extrapolated to frequencies lower than 1 Hz 1 as shown in Table 151 Results of experiments aimed at establishing perception thresholds for periodic motions of 006702 Hz suggested that about 50 of the subjects reported perception thresholds of 106 g respectively 2 According to 3 for frequencies of 01025 Hz perception thresholds vary between 06 and 03 g respectively It is noted in 4 that creaking noises that occur during building motions tend to increase significantly the feeling of discomfort and should be minimized by proper detailing Based on the results reported in 2 5 proposed a simple criterion that limits the average number of 1 g accelerations at the top occupied floor to at most 12 per year On the basis of interviews with building occupants it was tentatively suggested in 6 that The return periods for storms causing an rms horizontal acceleration at the building top that exceeds 05 g shall not be less than 6 years The rms shall represent an average over the 20min period of the highest storm intensity and be spatially averaged over the building floor The first step in verifying the compliance of a design with requirements set forth in comfort criteria is the estimation for each wind direction of the speeds that would induce the acceleration levels of interest Databaseassisted design methods can be used for obtaining plots of wind speed versus accelerations for the wind velocities that induce critical building accelerations An example of such a plot is shown in Figure 151 If tuned mass dampers are used to reduce building motions the accelerations can be esti mated by methods mentioned in Chapter 16 The second step is the estimation of the frequency of occurrence of accelerations higher than the critical value specified in the comfort criteria The frequency may be defined as the mean number of days per year during which the maximum wind speeds exceed the values corresponding to the plot of Figure 151 This information can be obtained from wind speed data typically available Table 151 Proposed correspondence between degrees of user discomfort and the accelerations causing them Degree of discomfort perceived Accelerations as percentages of discomfort from the acceleration of gravity g Imperceptible 12 g Perceptible 12112 g Annoying 1125 g Very Annoying 515 g Intolerable 15 g k k k k 153 Comfort Criteria for Pedestrian Areas Within a Built Environment 227 NW SW SE S N W NE Building perimeter 15 ms 20 ms 30 ms 25 ms E σ σ Figure 151 Wind speeds inducing critical building accelerations in the United States in the public domain see Section 323 For details on research concerning human discomfort due to building motions see for example 7 153 Comfort Criteria for Pedestrian Areas Within a Built Environment The problem of windinduced discomfort in pedestrian areas is not new Figure 152 For the sake of its historical interest we reproduce in Figure 153 a note by the great naturalist Buffon describing the flow changes occurring upwind of a tower for which it offers a charming but no longer tenable explanation A translation of the note follows On reflected wind I must report here an observation which it seems to me has escaped the atten tion of physicists even though everyone is in a position to verify it It seems that reflected wind is stronger than direct and the more so as one is closer the obsta cle that reflects it I have experienced this a number of times by approaching a tower that is about 100 feet high and is situated on the north end of my garden in Montbard When a strong wind blows from the south up to thirty steps from the tower one feels strongly pushed after which there is an interval of five of six steps where one ceases to be pushed and where the wind which is reflected by the tower is so to speak in equilibrium with the direct wind After this the closer one k k k k 228 15 WindInduced Discomfort in and Around Buildings Figure 152 The Gust Lithograph by Marlet Source Photo Bibliothèque Nationale de France approaches the tower the more the wind reflected by it is violent It pushes you back much more strongly than the direct wind pushed you forward The cause of this effect which is a general one and can be experienced against all large build ings against sheer cliffs and so forth is not difficult to find The air in the direct wind acts only with its ordinary speed and mass in the reflected wind the speed is slightly lower but the mass is considerably increased by the compression that the air suffers against the obstacle that reflects it and as the momentum of any motion is composed of the speed multiplied by the mass the momentum is con siderably larger after the compression than before It is a mass of ordinary air that pushes you in the first case and it is a mass of air that is once or twice as dense that pushes you back in the second case 1531 Wind Speeds Pedestrian Discomfort and Comfort Criteria Observations of wind speeds on people and calculations involving the rate of working against the wind suggest that the following degrees of discomfort are induced by wind speeds V at 2 m above ground averaged over 10 min1 h V 5 m s1 onset of discomfort V 10 m s1 definitely unpleasant V 20 m s1 dangerous 8 According to 8 if mean speeds V occur less than 10 of the time complaints about wind conditions are unlikely to arise If such speeds occur between 10 and 20 of the time complaints might arise For frequencies in excess than 20 remedial mea sures are necessary An alternative set of comfort criteria proposed in 9 is shown in Table 152 k k k k 154 Zones of High Surface Winds Within a Built Environment 229 Figure 153 Facsimile of note on reflected wind Source From Histoire Naturelle Générale et Particulière Contenant les Epoques de la Nature Par M le Comte de Buffon Intendant du Jardin et du Cabinet du Roi de lAcadémie Française de celle des Sciences etc Tome Treizième A Paris De lImprimerie Royale 1778 A more elaborate view of pedestrian comfort that accounts for local climate character istics other than wind speeds including thermal characteristics is discussed in 10 154 Zones of High Surface Winds Within a Built Environment 1541 Wind Effects Near Tall Buildings As noted in 8 high wind speeds occurring at pedestrian level around tall buildings are in general associated with the following types of flow 1 Vortex flows that develop near the ground Figures 154 and 155 2 Corner streams Figure 154 3 Air flows through ground floor openings connecting the windward to the leeward side of a building Figure 154 or crossflows from the windward side of one building to the leeward side of a neighboring building Figure 156 k k k k 230 15 WindInduced Discomfort in and Around Buildings Table 152 Comfort criteria for various pedestrian areas Criterion Area Description Limiting Wind Speed Frequency of Occurrence 1 Plazas and Parks Gusts to about 6 m s1 10 of the time about 1000 h yr 1 2 Walkways and other Gusts to about 12 m s1 1 or 2 times a month about 50 h yr 1 areas subject to pedestrian access 3 All of the above Gusts to about 20 m s1 About 5 h yr 1 4 All of the above Gusts to about 25 m s1 Less than 1 h yr 1 Wind direction Vortex flow Through flow Corner streams H B h C L W B A Figure 154 Regions of high surface winds around a tall building Source By permission of the Director Building Research Establishment UK Copyright Controller of Her Britannic Majestys Stationery Office HMSO The flow visualization of Figure 155 was obtained by injecting smoke into the airstream The flow patterns in the immediate vicinity of the windward face are consistent with the fact that pressures are highest at roughly twothirds of the height of the taller building that is the air flows from zones of higher to zones of lower pressure Part of the air deflected downward by the building forms a vortex and thus sweeps the ground in a reverse flow area A marked vortex flow in Figure 154 Another part is accelerated around the building corners and forms jets that sweep the ground near the building sides areas B marked corner streams in Figure 154 If an opening connecting the windward to the leeward side is present at or near the ground level part of the descending air will be sucked from the zone of relatively low pressures suctions on the leeward side A throughflow will thus sweep the area C in Figure 154 Throughflows of this type have caused serious discomfort to users of the MIT Earth Sciences Building in Cambridge Massachusetts a structure about 20 stories in height Crossflows between pairs of buildings are caused by similar pressure differences as shown in Figure 156 k k k k 154 Zones of High Surface Winds Within a Built Environment 231 Figure 155 Wind flow in front of a tall building wind blowing from left to right Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO Main wind direction Local wind direction Low buildings Figure 156 Crossflow between two tall buildings Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO k k k k 232 15 WindInduced Discomfort in and Around Buildings 1542 Wind Speeds at Pedestrian Level in a Basic Reference Case 8 The pattern of the surface wind flow within a site depends in a complex way upon the relative location the dimensions the shapes and certain of the architectural features of the building of interest upon the roughness and the topographical features of the terrain around the site and upon the possible presence near the site of one or several tall buildings To study the surface flow under conditions significantly different from those depicted schematically in Figures 154 and 156 it may be necessary to conduct wind tunnel tests or perform Computational Wind Engineering simulations eg 11 12 either of which can provide useful if approximate information However for suburban built environments that retain a basic similarity with the config urations of Figures 154156 and in which the height of the buildings does not exceed 100 m or so information based on aerodynamic studies reported in 8 is useful for the prediction of surface winds in a wide range of practical situations The surface winds depend upon the dimensions H W L and h defined in Figure 154 and are expressed in terms of ratios VV H where V and V H are mean speeds at pedestrian level and at elevation H respectively In certain applications it is useful to estimate the ratio VV 0 where V 0 is the mean speed at 10 m above ground in open terrain The ratios VV 0 can be obtained as follows V V0 V VH VH V0 151 Approximate ratios V HV 0 corresponding to suburban built environments suggested in 8 are listed in in Table 153 In the material that follows the wind direction is assumed to be normal to the building face unless otherwise stated Speeds in Vortex Flow V A and V H denote the maximum mean wind speed at pedestrian level in zone A and the mean speed at elevation H respectively Figure 154 Approxi mate ratios V AV H are given in Figure 157 as functions of WH for various rations LH and for the ranges of values Hh shown The height h corresponded in the model tests to typical heights of suburban buildings 716 m It is noted that as the building becomes slenderer as the ratio WH decreases the ratio V AV H decreases Typical examples of the variation of V A with individual variables are shown in Figure 158 If the distance L between the lowrise and highrise building is small the vortex cannot penetrate effectively between the buildings and V A is small If L is very large or if h is very small the vortex that forms upwind of the tall building will be poorly organized and weak V A will therefore be relatively low If h approaches the value of H the taller building will in effect be sheltered and the speed V A will thus be low It is noted that the ratio V AV H is in the order of 05 for a range of practical situations Speeds in Corner Streams Figure 159 shows the approximate dependence of the ratio V BV H upon Hh where V B is the largest mean speed at pedestrian level in the zones swept by the corner stream and V H is the mean speed at elevation H Examples of the Table 153 Approximate ratios VHV0 H m 20 30 40 50 60 70 80 90 100 V HV 0 073 082 089 094 099 104 108 111 114 k k k k 154 Zones of High Surface Winds Within a Built Environment 233 01 02 VAVH 03 04 05 06 07 0 0 01 02 WH VAVH 03 04 05 06 07 0 0 01 02 VAVH 03 04 05 06 07 0 0 10 15 WH 20 05 10 15 20 01 02 WH VAVH 03 04 05 0 0 05 10 15 20 05 10 WH LH025 8 Hh2 LH05 8 Hh2 LH 10 8 Hh 2 LH 20 4 Hh 2 15 Figure 157 Ratios VAVH 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO variation of V B with the variables H L W h and 𝜃 are given in Figure 1510 The speed V B varies weakly with the angle 𝜃 between the mean wind direction and the normal to the building face However the orientation of the corner streams and hence the position of the point of maximum speed V B may depend significantly upon the direction of the mean wind Information about the wind speed field around the corner of a wide building model H 04 m W 04 m is given in Figure 1511 The wind speed decreases rather slowly within a distance from the building corner approximately equal to H The ratio YD2 where Y is defined as in Figure 1511 and D is the building depth provides an approx imate measure of the position of the corner stream Measured values of this ratio for various values of H and of WD2 are shown in Figure 1512 It is noted that the ratio V BV H is approximately 095 for a range of practical situations Speeds in a ThroughFlow Let V C denote the maximum mean wind speed through a ground floor passageway connecting the windward and the leeward side of a building Figure 154 Figure 1513 shows the approximate dependence of the ratio V CV H upon the parameter Hh Examples of the variation of V C with H W L h and 𝜃 are given in Figure 1514 The data of Figures 1513 and 1514 are based on tests in which the entrances to the passageways were sharpedged If the edges of the entrance are rounded k k k k 234 15 WindInduced Discomfort in and Around Buildings 0 0 01 02 03 04 05 H m a W m c L m b 06 07 08 02 03 04 05 06 07 08 H 04 m W 04 m h 01 m θ 0 H 04 m L 04 m h 01 m θ 0 L 04 m W 04 m h 01 m θ 0 1 2 VA msec 3 4 0 0 01 1 02 03 04 05 2 VA msec 3 4 h m d H 04 m W 04 m L 04 m θ 0 0 0 01 1 02 03 04 05 2 VA msec 3 4 0 0 01 1 2 VA msec 3 4 Figure 158 Examples of the variation of VA with individual parameters 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO 3 4 5 6 Hh 7 8 0 01 02 03 04 05 VBVH 06 07 08 09 10 11 WH 05 05 LH Figure 159 Ratios VBVH 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO k k k k 154 Zones of High Surface Winds Within a Built Environment 235 00 01 02 03 04 05 06 07 08 H m h m 1 2 3 VB msec 4 5 6 00 01 02 03 04 05 1 2 3 VB msec 4 5 6 L 04 m W04 m h 01 m θ 0 L 03 m H 04 m W04 m θ 0 W m L m 00 01 02 03 04 05 1 2 3 VB msec 4 5 6 00 01 02 03 04 05 06 07 08 1 2 3 VB msec 4 5 6 L 03 m H 04 m h 01 m θ 0 H 04 m W 04 m h 01 m θ 0 Wind angle θ 0 90 45 0 45 90 1 2 3 VB msec 4 5 6 H 04 m W 04 m h 0 m Figure 1510 Examples of the variation of VB with individual parameters 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO to form a bellmouth shape the speeds V C are reduced with respect to those data by as much as 25 It is noted that the ratio V CV H is approximately 12 for a range of practical situations As noted earlier the approximate validity of the information provided in Figures 157 1514 is limited to buildings with regular shape in plan and heights of 100 m or less 1543 Case Studies Case 1 Model of a Building in Utrecht The Netherlands 13 The model of a building with height H 80 m width W 80 m depth D 22 m Hh 80 and LH 05 is shown in plan in Figure 1515 Contours of ratios VV H shown in Figure 1515 for south and north winds were obtained in wind tunnel tests 13 Ratios V AV H and V BV H are about 065 at the centerline of the building and 090 respectively versus the values 060 and 100 from Figures 157 and 159 a reasonably good agreement Note that the vortex flow is asymmetrical and contains regions in which the ratios VV H are as high as 08 k k k k 236 15 WindInduced Discomfort in and Around Buildings 0 02 375 ms 35 ms 325 ms 3 ms VB 01 03 04 05 06 m 0 01 02 03 m Wind direction Model building in plan W Y X Figure 1511 Surface wind speed field in a corner stream 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO 0 0 05 2 10 08 m 06 m 04 m Y constant W 03 m 02 m H 15 4 6 WX YX 8 10 1 2 Figure 1512 Empirical curve YX versus WX 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO k k k k 154 Zones of High Surface Winds Within a Built Environment 237 0 01 02 03 04 05 06 07 08 VCVH 09 10 11 12 13 14 15 4 3 5 6 Hh 7 8 WH 05 01LH Figure 1513 Ratios VCVH 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO Case 2 Model of Place Desjardins Montreal 14 Figure 1516 shows a 1 400 model of a design considered for a development in Place Desjardins Montreal Tests were con ducted only for the predominant wind direction shown in the Figure 1517 Surface flow patterns were observed by using thread tufts taped to the model surfaces a wool tuft on the end of a handheld rod and a liquid mixture of kerosenechalk china clay sprayed over the horizontal surfaces of the model As the wind blows over the model the mix ture is swept away from the highspeed zones and accumulates in zones of stagnating flow After the evaporation of the kerosene the white accumulations of chalk indicate zones of low speeds while areas that are dark indicate zones surface winds are high Wind speed measurements were made in the latter zones The numbers in Figure 1517 are ratios of mean wind speeds at the locations shown to the mean speed V 1 at 18 m above ground at the northwest corner of the development The percentages in the figure represent turbulent intensities and the arrows show the direc tion of the wind component that was measured by the probe The quantities not between k k k k 238 15 WindInduced Discomfort in and Around Buildings 00 0102030405 a 060708 H m 1 2 3 VC ms 4 5 6 7 8 00 01 1 02 03 b 04 W m c L m 2 3 VC ms 4 5 6 7 8 00 01 1 02030405060708 2 3 VC ms 4 5 6 7 8 L 03 m W 04 m h 01 m θ 0 L 03 m H 04 m h 01 m θ 0 0 0 0 45 04 m 005 m 90 45 90 1 2 3 4 5 6 7 0 01 1 02 03 d e 04 h m 2 3 VC ms VC ms 4 5 6 7 8 L 03 m H 04 m W 04 m θ 0 H 04 m W 04 m h 01 m θ 0 Wind angle θ Width of opening H 04 m W 04 m h 0 Figure 1514 Examples of the variation of VC with individual parameters 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO parentheses correspond to measurements made in the absence of a projected 50story tower near the southwest corner of the development Results of measurements made with the tower in place are shown in parentheses The presence of the tower changed the ratios of the wind speeds at locations 8 and 10 to the wind speed V 1 from 311 to 296 to approximately 338 and 248 respectively k k k k 154 Zones of High Surface Winds Within a Built Environment 239 6 2 7 8 9 N 5 4 3 1 H31 m 10 Figure 1515 Plan view Case 1 Source After 13 The results just listed correspond to the case of the uncovered mall In the absence of the tower covering the mall reduced the mean wind speeds by a factor of almost three at location 8 however there was no reduction at location 10 with the tower in place it reduced the mean wind speeds by a factor of five at location 8 and a factor of 167 at location 10 Case 3 Commerce Court Plaza Toronto 15 16 A wind tunnel model and a plan view of the Commerce Court project in Toronto are shown in Figures 1518 and 1519 Sur face flow patterns obtained by smoke visualization are shown for two wind directions in Figures 1520 and 1521 15 Ratios VV H where V and V H are mean wind speeds are 27 and 240 m above ground were obtained from measurements in the wind tunnel and after the completion of the structures on the actual site The results of the measurements are shown in Figure 1522 as functions of wind direction for locations 1 through 7 The agreement between wind tunnel and fullscale values is generally acceptable although differences of 30 50 and larger can be noted in certain cases After the completion of the Commerce Court Plaza conditions were found to be particularly annoying on windy days for pedestrians walking from the relatively pro tected zone north of the 32story tower into the flow funneled through the passageway 23 see Figure 1519 Wind tunnel tests indicated that the provision of screens at the ground level as shown in Figure 1523a would result at locations 2 5 and 6 in reductions of undesirable mean speeds in the order of 40 However the placement of screens was rejected for architectural reasons Instead potted evergreens about 3 m high were placed as shown in Figure 1523b This reduced the mean speeds by about 20 at locations 2 10 at location 5 and 33 at location 6 Case 4 Shopping Center Croydon England 8 An office building 75 m tall 70 m wide and 18 m deep adjoins a shopping center 75 m long A passageway 12 m wide and 37 m k k k k 240 15 WindInduced Discomfort in and Around Buildings Figure 1516 Place Desjardins model Source Courtesy of the National Aeronautical Establishment National Research Council of Canada high underneath the building connects the shopping center on the west side of the build ing to the street on the east side Figure 1524 The complex was designed and built without a roof over the shopping mall After the completion of the building complex it became apparent that remedial measures were necessary to reduce wind speeds in the passageway and the shopping mall The ground level wind flow was investigated in the wind tunnel first for the complex as initially built with the mall not covered and then with various arrangements of roofs over the mall and screens within the passage way Ratios VV H measured in the wind tunnel V and V H are the mean speeds at 18 and 75 m above ground respectively are shown in Figure 1524 for three cases For the k k k k 154 Zones of High Surface Winds Within a Built Environment 241 14 166 13 469 115 465 207 625 142 580 280 419 316 580 140 950 100 503 100 503 311 414 338 340 296 310 248 331 415 278 243 377 135 384 192 500 266 518 090 837 132 555 Wind direction 345 372 382 361 9 7 N 6 5 8 10 12 11 4 3 2 1 Figure 1517 Wind speed ratios and turbulence intensities place Desjardins Source Courtesy of the National Aeronautical Establishment National Research Council of Canada complex as first built the highest values of the ratio VV H were 068 in the vortex flow zone and 101 in the throughflow zone The provision of a full roof over the mall but of no screens within the passageway reduced considerably pedestrian level speeds caused by west winds However with east winds the flow was trapped under the roof and the wind speeds within the mall were for this reason high as shown in Figure 1524 the speeds were also high at the east entrance of the passageway A solid roof close to the tall building followed by a partial roof over the rest of the mall and a screen obstructing 75 of the passageway area resulted in a significant reduction of surface winds The solution actually applied which proved effective was to provide i a full roof over the entire mall and ii screens with 75 blockage in the passageway k k k k 242 15 WindInduced Discomfort in and Around Buildings Figure 1518 Commerce Court model Source Reprinted from 15 with permission from Elsevier 155 Frequencies of Ocurrence of Unpleasant Winds 1551 Detailed Estimation Procedure Let V 0V 𝜃 denote the wind speeds at 10 m above ground in open terrain that induce pedestrian winds V blowing from direction 𝜃 at a given location in a built environment The frequency of occurrence of wind speeds larger than V denoted by f V is approxi mately f V n i1 f V0 i 152 where f V0 i are the frequencies of occurrence in open terrain of winds with speeds larger than V 0V 𝜃i and the directions 𝜃i 𝜋 n 𝜃 𝜃i 𝜋 n where 𝜃i 2𝜋in i 1 2 n In practical applications a 16point compass is sometimes used so that n 16 k k k k 155 Frequencies of Ocurrence of Unpleasant Winds 243 145 CIBC building 32 stories CIBC tower 54 stories H240 m 5 Stories 14 Stories 4 5 7 1 Building north True north 6 3 2 36 m 70 m Figure 1519 Plan view Commerce Court Source Reprinted from 15 with permission from Elsevier CIBC tower CIBC bldg N Office bldg Office bldg Up Wind Vortex Figure 1520 Surface wind flow pattern Commerce Court east wind Source Courtesy of Professor A G Davenport k k k k 244 15 WindInduced Discomfort in and Around Buildings CIBC tower CIBC bldg N Office bldg Office bldg Vortex Wind 10 m 10 m Up Figure 1521 Surface wind flow pattern Commerce Court southwest wind Source Courtesy of Professor A G Davenport To obtain f V it is necessary first to estimate the values of V 0V 𝜃i From wind clima tological data it is then possible to estimate the frequencies f V0 i The speed V 0V 𝜃i can be written as V0V 𝜃i 1 VVH𝜃i V0𝜃i VH𝜃iV 153 The ratios V 0𝜃iV H𝜃i characterize the site micrometeorologically For standard roughness conditions in open terrain these ratios depend upon the elevation H and upon the roughness conditions upwind of the site The ratios VV H𝜃i are obtained from wind tunnel tests Consider for example all threehour interval observations in a year 8 obs day1 365 days 2920 obs and assume that 58 of these observations represent north northwesterly NNW winds with speeds in excess of 6 m s1 The frequency of occurrence of such wind can then be estimated as f 6 1 582920 2 It is desirable to base frequency estimates on several years of data In some applications it may be of interest to estimate frequencies for individual seasons or for a grouping of seasons Also data for times not relevant from a pedestrian comfort viewpoint eg between 11 pm5 am may in some cases be eliminated from the data set k k k k Location 1 14 V 0675 VH Location 2 V 0675 VH Location 3 Building north True north 145 V 0675 VH Location 5 V 0675 VH Location 6 V 0675 VH Location 7 V 0675 VH Location 4 V 0675 VH 0450 0450 0450 0450 0225 0450 0450 0450 0225 0225 0225 0225 0225 0225 4 5 Full scale Wind tunnel 7 1 6 2 3 Figure 1522 Surface mean wind speeds at the Commerce Court Plaza Commerce Court Source Reprinted from 15 with permission from Elsevier k k k k 246 15 WindInduced Discomfort in and Around Buildings 5 Stories 54 Stories 32 Stories Porous wind screen at ground Street Street Trees a Street Solid wind screen at ground Street N Street 14 Stories b Figure 1523 Remedial measures at Commerce Court a screens b trees Source After 17 1552 Simplified Estimation Procedure A simplified version of the procedure just presented is suggested in 8 for built environ ments similar in configuration to the basic reference case Figure 154 In this version the aerodynamic information used rather than being a function of wind direction is limited to the results given for example in Figures 157 159 and 1513 The ratios k k k k 155 Frequencies of Ocurrence of Unpleasant Winds 247 As first built With full roof and no screen East wind With partial roof and 75 screen 026 048 021 007 045 017 017 052 017 044 061 023 052 067 043 056 063 047 078 071 053 101 088 059 As first built With full roof and no screen West wind With partial roof and 75 screen 072 032 040 036 028 019 049 027 023 065 025 023 068 020 028 065 019 019 057 024 017 053 049 023 Figure 1524 Model test results Croydon 8 Source By permission of the Director Building Research Establishment UK Copyright Controller of HMSO V HV 0 of mean wind at elevation H in the in the built environment to mean wind at 10 m above ground in open terrain may be taken from Table 152 or determined as shown in Section 236 The requisite climatological information consists of the of the frequen cies of occurrence of all winds with speeds in excess of various values V 0 regardless of their direction According to 8 this simplified procedure provides generally reli able indications on the serviceability of pedestrian areas in a built environment of the type represented in Figure 154 provided that it is used in conjunction with the comfort criteria proposed in 8 To illustrate the procedure proposed in 8 consider the case of a building complex for which H 70 m W 50 m L 35 m and h 10 m For these notations see Figure 154 From Figures 157 and 159 V AV H 06 and V BV H 095 where V A and V B are the highest mean speeds in the vortex and corner flow respectively For H 70 m V HV 0 104 Table 152 so V AV 0 063 and V BV 0 100 Given the requisite wind speed data it is possible to estimate the frequencies of winds V A 5 and V B 5 m s1 In order for V A 5 V 0 5063 8 m s1 For V B 5 m s1 V 0 5100 500 m s1 The frequency of 5 m s1 winds depends upon the local wind climate If that frequency exceeds 20 according to the comfort criterion of 8 see Section 1521 the wind conditions at the site are unacceptable k k k k 248 15 WindInduced Discomfort in and Around Buildings References 1 Chang FK 1973 Human response to motions in tall buildings Journal of the Structural Division 99 12591272 2 Chen PW and Robertson LE 1972 Human perception thresholds of horizontal motion Journal of the Structural Division 92 16811695 3 Goto T Human Perception and Tolerance of Motion Monograph of Council on Tall Buildings and Urban Habitat PC 1981 pp 817849 4 Reed J W Windinduced motion and human discomfort in tall buildings Research Report No R7142 Department of Civil Engineering MIT Cambridge 1971 5 Feld L 1971 Superstructure for 1350 ft World Trade Center Civil Engineering ASCE 41 6670 6 Hansen RJ Reed JW and Vanmarcke EH 1973 Human response to windinduced motion of buildings Journal of the Structural Division 99 15891605 7 Lamb S and Kwok KCS 2017 The fundamental human response to windinduced building motion Journal of Wind Engineering and Industrial Aero dynamics 165 7985 8 Penwarden A D and Wise A F E Wind environment around buildings Build ing Research Establishment Report Department of the Environment Building Research Establishment Her Majestys Stationery Office London 1975 9 Apperley L W and Vickery B J The prediction and evaluation of the ground level environment in the Fifth Australian Conference on Hydraulics and Fluid Mechanics University of Canterbury Christchurch New Zealand 1974 10 Wu H and Kriksic F 2012 Designing for pedestrian comfort in response to local climate Journal of Wind Engineering and Industrial Aerodynamics 104106 397407 11 Mochida A and Lun IYF 2008 Prediction of wind environment and ther mal comfort at pedestrian level in urban area Journal of Wind Engineering and Industrial Aerodynamics 96 14981527 12 LlagunoMunitxa M BouZeid E and Hultmark M 2017 The influence of building geometry on street canyon air flow validation of large eddy simulations against wind tunnel experiments Journal of Wind Engineering and Industrial Aero dynamics 165 115130 13 Poestkoke R Windtunnelmetingen aan een model van het Transitorium II van de Rijksuniversiteit Utrecht Report No TR72110L National Aerospace Laboratory NLR The Netherlands 1972 14 Standen N M A wind tunnel study of wind conditions on scale models of place Desjardins Montreal Laboratory Technical Report No LTRLA101 National Research Council of Canada National Aeronautical Establishment Ottawa 1972 15 Isyumov N and Davenport AG 1975 Comparison of fullscale and wind tun nel wind speed measurements in the Commerce Court Plaza Journal of Wind Engineering and Industrial Aerodynamics 1 201212 k k k k References 249 16 Davenport A G Bowen C F P and Isyumov N A study of wind effects on the Commerce Court project Part II wind environment at pedestrian level Engi neering Science Research Report No BLWT370 University of Western Ontario Faculty of Engineering Science London Canada 1970 17 Isyumov N and Davenport AG The ground level wind environment in builtup areas in Proceedings of the Fourth International Conference on Wind Effects on Buildings and Structures London 1975 Cambridge University Press Cambridge 1977 pp 403422 k k k k 251 16 Mitigation of Building Motions Tuned Mass Dampers 161 Introduction Tuned mass dampers TMDs are the most commonly used devices for reducing tall structure accelerations and interstory drift due to translation and torsion Generally TMD effects are not taken into account in strength calculations The TMD was invented in 1909 by Frahm and was originally used in mechanical engi neering systems Since the 1970s TMDs have been used to mitigate building motions Examples of buildings in which TMDs were used include the Citicorp Center New York City the John Hancock tower Boston equipped with dual TMDs designed to control both drift and torsional motions and the Taipei 101 tower For additional examples see 1 Basic TMD theory was developed in 2 TMDs consist of one or more masses in the order of 2 of the total mass of the structure added to and interacting dynamically with the structure through springs and damping devices The structures motion is reduced by the forces of inertia due to the motion of the TMDs A schematic view of a TMD operating on the top floor of the Citi corp Center building is shown in Figure 161 The mass of the TMD consists in this case of a 400ton concrete block bearing on a thin oil film The TMD structural stiffness is provided by pneumatic springs that can be tuned to the actual frequency of vibration of the building as determined experimentally in the field The damping is provided by hydraulic shock absorbers The system included failsafe devices to prevent excessive travel of the concrete block 3 Descriptions and theory applicable to buildings are pre sented in 1 for various types of TMD including translation and pendular TMDs placed at or near the top of the building TMD pairs placed at opposite sides of the top building floor designed to reduce torsional motions and TMDs installed at several elevations tuned to reduce motions in more than one mode of vibration Dampers that produce forces of inertia due to fluid motion have also been used Early contributions to the design of TMDs for building motion control were made in 3 and 4 For recent devel opments on multidegreeoffreedom system TMDs under random excitation see 5 which provides comprehensive references Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 252 16 Mitigation of Building Motions Overtravel snubber reaction buttresses Slip bearing surface Pressure balanced hydrostatic slip bearings 12 Anti yaw linkage Over travel snubbers Control console Boom connection to mass block Nitrogen charged spring fixture NS control actuator EW spring fixture 400 ton concrete mass block 30 square EW control actuator Fluid reservoir Hydraulic power supply Pump motor control center Figure 161 Tuned mass damper system Citicorp Center New York City Source Courtesy of MTS Systems Corp Minneapolis 162 SingleDegreeofFreedom Systems Figure 162 is a TMD schematic in which m c k and md cd kd are the mass viscous damping coefficient and spring constant of the structure idealized as a singledegreeoffreedom system SDOF and of the TMD respectively Assume that the forcing function pt in Figure 162 is harmonic The equations of motion of the system are m x c x k x p sin Ωt cd xd kd xd 161 mdx xd cd xd kdxd 0 162 where x is the displacement of the SDOF system and xd is the displacement of the TMD with respect to the SDOF system The solutions of Eqs 161 and 162 are harmonic and have amplitudes X p k H x p kd Hd 163ab where the dynamic amplification factors also known as mechanical admittance func tions of the structure and of the TMD are denoted by H and Hd H 𝛽2 d 𝛽22 2𝜁d𝛽𝛽d2 D Hd 𝛽2 D 164ab D 𝛽2 d𝛽2𝛾 1 𝛽2𝛽2 d 𝛽2 4𝜁𝜁d𝛽d𝛽22 4𝜁𝛽𝛽2 d 𝛽2 𝜁d𝛽d𝛽1 𝛽21 𝛾212 165 k k k k 162 SingleDegreeofFreedom Systems 253 Figure 162 Schematic of a damped system equipped with a damped tuned mass damper k kd cd m md x x xd pt c see eg 2 In Eqs 164ab and 165 the following nondimensional parameters are used 𝛽 Ω 𝜔 Ω km 𝛽d 𝜔d 𝜔 kdmd km 166ab 𝜁 c 2 km 𝜁d cd 2 kdmd 167ab An optimal design of TMD should consider the largest acceptable levels of the response of the structure and the TMD that is H and Hd Figure 163 shows for a given set of 𝛾 𝛽d and 𝜁 values and for several values of 𝜁d the dependence of the dynamic amplification factor H upon the nondimensionalized excitation frequency 𝛽 For 𝜁d 0 the amplifi cation factor H of the structure has two separate peaks as does the amplification factor Hd of the TMD As 𝜁d increases up to 𝜁d opt ie approximately 009 see Eq 169 the ordinates of the two peaks of the factors H and Hd decrease As 𝜁d increases further the two peaks of H and Hd merge into one peak For H that peak increases as 𝜁d approaches unity whereas for Hd the peak continues to decrease As shown in Figure 163 if the ratio 𝛽 of the excitation frequency to the natural frequency 𝜔 of the structure is con tained in the interval 085115 the TMD reduces the response by amounts that depend upon that ratio As explained in 1 p 247 because of the dependence of D upon 𝜁 no analytical expressions can be obtained for the optimal tuning frequency ratio 𝛽d opt and optimal damping 𝜁d opt as functions of the mass ratio 𝛾 Numerical calculations are therefore resorted to The reader is referred to 1 for plots of the calculated optimal values of H and Hd as functions of mdm for various values of 𝜁 The optimal values of the parameters 𝛽d and 𝜁d as functions of mdm and 𝜁 can be obtained from the following expressions based on curve fitting schemes proposed in 6 𝛽d opt 1 05 mdm 1 mdm 1 2𝜁2 1 2375 1034 mdm 0426 mdm𝜁 mdm 3730 16903 mdm 20496 mdm𝜁2 mdm 168 𝜁d opt 3𝛾 81 mdm1 05mdm 0151𝜁 0170𝜁2 0163𝜁 4980𝜁2mdm 169 k k k k 254 16 Mitigation of Building Motions H a H β 08 0 5 10 15 20 25 30 085 09 095 1 105 11 115 No TMD ζd 0 ζd 002 ζd 009 ζd 015 ζd 1 12 Hd b Hd β ζd 0 ζd 002 ζd 009 ζd 015 ζd 1 0 08 085 09 095 1 105 11 115 12 20 40 60 80 100 120 140 160 180 200 HdH c HdH ζd 0 ζd 002 ζd 009 ζd 015 ζd 1 β 08 0 2 4 6 8 10 12 14 16 18 20 085 09 095 1 105 11 115 12 Figure 163 Dynamic amplification factor H and Hd as functions 𝛽 with various values of 𝜁d mdm 003 𝛽d 097 𝜁 002 For a singledegreeoffreedom linear oscillator with no TMD the largest possible value of the mechanical admittance function is HSD 1 2𝜁SD 1 𝜁2 SD 1610 where 𝜁SD denotes the oscillators damping ratio For 𝜁SD in the order of 002 say HSD 1 2𝜁SD 1611 Similarly the equivalent damping for the mass m provided to the system described by Eqs 161 and 162 can be written as 𝜁e 1 2Hopt 1612 k k k k 163 TMDs for MultipleDegreeofFreedom Systems 255 Example 161 Following 1 p 251 it is assumed that the damping ratio is 𝜁 002 and that the dynamic amplification factor Hopt and the ratio between the amplitudes of the TMD and the structure are limited by the inequalities Hopt 7 1613 Hd opt Hopt 6 1614 that is Hd opt 42 From 1 figure 528 it follows that for 𝜁 002 the required ratio mdm 003 From 1 figure 529 it follows that Eq 1614 is satisfied The value 𝛽d opt the stiffness kd and the optimal damping ratio 𝜁d opt are then obtained from Eqs 168 166b and 169 respectively The equivalent damping provided by the TMD is 12Hopt 007 163 TMDs for MultipleDegreeofFreedom Systems Figure 164 shows a twodegreeoffreedom 2DOF system The equations of motion of masses m1 m2 and md are 1 m1x1 c1 x1 k1x1 k2x2 x1 c2 x2 x1 p1 1615 m2x2 c2 x2 x1 k2x2 x1 kdxd cd xd p2 1616 md xd cd xd kdxd md x2 1617 Expressing x1 and x2 in terms of model shapes and generalized coordinates x1 x2 𝜙11 𝜙12 𝜙21 𝜙22 q1 q2 or x 𝚽 q 1618ab where 𝚽 is the modal matrix and q is the generalized coordinate vector Based on the orthogonality of natural modes Eqs 1615 and 1616 are transformed into the uncou pled equations of a SDOF structure m j qj c j qj k j qj 𝜙j1p1 𝜙j2p2 cd xd kdxd for j 1 2 1619 k1 c1 m1 x1 p1t c2 cd m2 md k2 kd x2 x2 xd p2t Figure 164 TwoDOF system with tuned mass damper k k k k 256 16 Mitigation of Building Motions where the modal mass stiffness and damping matrices are defined as m j 𝚽T j M𝚽j 1620 c j 𝚽T j C𝚽j 1621 k j 𝚽T j K𝚽j 1622 In Eqs 16201622 the jth modal vector of 𝚽 is 𝚽j 𝜙1j 𝜙2j 1623 Consider the case of a TMD designed to control the first modal response ie j 1 If the external forcing frequency is close to 𝜔1 k1m1 the response in the first mode dominates Equations 1618ab then yield x2 𝜙21 q1 and q1 x2 𝜙21 1624 Substitution of Eq 1624 into Eq 1619 in which j 1 yields m 1ex2 c 1e x2 k 1ex2 p 1e cd xd kdxd 1625 where the equivalent mass damping stiffness and force matrices are m 1e m 1 𝜙2 21 1626 k 1e k 1 𝜙2 21 1627 c 1e 𝛼k 1e 1628 p 1e 𝜙11p1 𝜙12p2 𝜙21 1629 Equation 1628 is derived under the assumption that damping is proportional to the stiffness 1 Equations 1625 and 1617 have the same form as Eqs 161 and 162 respec tively Therefore with appropriate changes of notation the solutions discussed in Section 162 are also applicable to Eqs 1625 and 1617 For details and a numerical example see 1 Reference 5 presents a frequencydomain approach to the optimization of TMDs installed at several levels of multipledegreeoffreedom structures subjected to wind loads defined by their power spectral densities References 1 Connor J and Laflamme S 2016 Tuned mass damper systems In Structural Motion Engineering 199285 Springer International Publishing 2 Den Hartog JP 1956 Mechanical Vibrations 4th ed New York McGrawHill 3 McNamara RJ 1977 Tuned mass dampers for buildings Journal of the Structural Division ASCE 103 17851798 k k k k References 257 4 Luft RW 1979 Optimal tuned mass dampers for buildings Journal of the Struc tural Division ASCE 105 27662772 5 Lee CL Chen YT Chung LL and Wang YP 2006 Optimal design theories and applications of tuned mass dampers Engineering Structures 28 4353 6 Tsai HC and Lin GC 1993 Optimum tunedmass dampers for minimizing steadystate response of supportexcited and damped systems Earthquake Engineering Structural Dynamics 22 957973 k k k k 259 17 Rigid Portal Frames Case Study 171 Introduction Conventional methods for determining wind loads on rigid structural systems as defined by the analytical method of the ASCE 7 Standard 1 involve the use of tables and plots contained in standards and codes Wind effects determined by such methods can differ from those consistent with laboratory measurements by amounts that can exceed 50 2 3 This is due in part to the severe data storage limitations inherent in conventional standards in which vast amounts of aerodynamic data varying randomly in time and space are reduced to a far smaller number of enveloping timeinvariant data In addition for lowrise buildings of the type covered by 1 the specified wind loads referred to in the standard as pseudoloads do not account for i the distance between frames which affects the spatial coherence of the aerodynamic pressures impinging on the frames and ii the structural systems actual member sizes and therefore the influence coefficients used in the structural calculations Lastly the ASCE 7 provisions are based on wind tunnel experiments conducted in part between three and four decades ago with obsolete pressure measurement technology no archived records of pressure measurements and numbers of building geometries and pressure taps lower by more than one order of magnitude than those of current aerodynamic databases 4 In contrast in the DAD approach pseudoloads are replaced by close approximations to the actual loads This chapter presents an application of the DAD approach to the design of portal frames wherein timedomain methods allow wind effects to be calculated by using large numbers of stored time series of measured pressure coefficients and wind effect com binations are performed objectively and rigorously via summations of time series The DAD approach accounts naturally for the imperfect spatial coherence of pressures act ing at different points of the structure examples of which are shown in the visualization of Figure 427 Software for the application of the DAD approach to rigid structures was first developed for frames of simple gable roof buildings in 5 This chapter presents an updated version of this approach and a case study reported in 6 that calculates peak demandtocapacity indexes DCIs directly used by the structural engineer to size structural members of gable roof building frames The aerodynamic pressure coefficients used in 5 and 6 were taken from the NISTUWO database 7 Results based on the NISTUWO and the Tokyo Polytechnic University TPU database Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 260 17 Rigid Portal Frames 8 the largest available to date were found in 9 to yield comparable results Calculations reported in 6 confirmed this conclusion Checking the adequacy of a member cross section consists of ascertaining that subject to possible serviceability and constructability constraints its DCI is close to unity If the DCI does not satisfy this condition the cross section is redesigned The member properties based on this iteration process can then be used to recalculate the influence coefficients by which revised wind loads are transformed into wind effects and to check the adequacy of the resulting DCIs Since the capacity of members in compression is determined by stability considera tions their DCIs depend nonlinearly upon axial load and are therefore not proportional to the squares of the wind speeds For this reason to estimate wind effects with the req uisite mean recurrence intervals it is necessary to produce DCI response surfaces see Section 132 The estimation of the peak DCIs from DCI time series can be performed by a multiplepointsintime method based on observed peak values 10 An alternative approach to the estimation of peaks based on rigorous statistical methods and capable of producing error estimates is presented in Appendix C The peak DCI response surfaces are properties of the structure independent of the wind climate and depend upon the structures terrain exposure aerodynamic behavior structural system and member sizes The response surfaces are used in conjunction with nonparametric statistics to estimate peak DCIs with any specified mean recurrence intervals MRIs Sections 134 and A9 Since the design MRIs specified in 1 are in the order of hundreds or thousands of years the use of nonparametric statistics requires the wind speed data sets to be commensurately large Databases of simulated hurricane wind speeds that meet this requirement are available see Section 323 and Monte Carlo simulations can be performed to develop large wind speed data sets from smaller sets of measured data 11 The results obtained in the case study presented in this chapter confirm the existence of significant errors in the estimation of wind effects by the ASCE 7 Standard envelope procedure The requisite software and a detailed users manual are available in 12 The DAD procedure as used in this chapter is typically applicable to any low or midrise buildings in addition to simple buildings with gable roofs portal frames and bracing parallel to the ridge Depending upon the preferences of the user alternative methods for the estimation of time series peaks the interpolation of results based on buildings with dimensions different from those of the building of interest and the esti mation of secondary effects may be used in lieu of the methods employed in 6 172 Aerodynamic and Wind Climatological Databases Aerodynamic databases are developed by wind engineering laboratories and contain time histories of simultaneously measured pressure coefficients at large numbers of taps Figure 171 shows a building model with the locations of the taps Pressure coefficient timehistory databases for oneofakind structures are obtained in adhoc wind tunnel tests rather than from preexisting databases Climatological databases are also developed by wind engineering laboratories They typically consist of directional or nondirectional extreme wind speeds that account for the buildings directional terrain exposure and cover periods in the order of typically k k k k 173 Structural System 261 Figure 171 Wind tunnel model of an industrial building Source Courtesy of the Boundary Layer Wind Tunnel Laboratory University of Western Ontario tens of years of measured data or as many as thousands of years of synthetic data as well as providing the mean rate of arrival of storm events per year Section A64 Directional wind speed data Uij i 1 2 ns j 1 2 nd are typically presented in the form of ns nd matrices in which ns is the number of storm events and nd is the number of wind directions eg nd 16 nondirectional wind speed data sets are vectors with components Ui i 1 2 ns where Ui largest wind speed in storm i regardless of direction see Chapter 13 The climatological database considered in the case study presented in this chapter consisted of directional hurricane wind speeds generated by Monte Carlo simulations for 999 storm events and 16 wind directions Section 323 173 Structural System The structural system being considered consists of equally spaced momentresisting steel portal frames commonly used in lowrise industrial buildings Figure 172 Roof and wall panels form the exterior envelope of the buildings and are attached to purlins and girts supported by the frames Bracing is provided in the planes of the exterior walls parallel to the ridge The coupling between frames due to the roof diaphragms is neglected The purlins and girts are attached to the frames by hinges The purlins and girts act as bracings to the outer flanges and the inner flanges are also braced The fol lowing limitations are imposed i The taper should be linear or piecewise linear and ii the taper slope should typically not exceed 15 13 k k k k 262 17 Rigid Portal Frames Stiffeners Bracings Roof Panel Purlins Rafter Girts Column Wall Panel Figure 172 Schematic of the structural system 174 Overview of the Design Procedure The sizing of the structural members requires calculations of the respective peak DCIs The DCIs pertaining to axial forces and bending moments at any cross sections of the frames are determined using Eqs 81 and 82 A similar simpler equation pertains to shear forces 14 The wind forces acting along the axis parallel to the ridge and the torsional moment about the structures elastic center are resisted by secondary bracing members hence only the wind forces due to winds normal to the buildings ridge contribute to the frame DCIs Therefore for the application at hand the quantities with subscript y in Eqs 81 and 82 need not be considered The time histories of the internal forces in the expres sions for the DCIs are computed as sums of factored load effects due to wind loads and gravity loads Design for strength requires considering the following five LRFD load combination cases 1 Case 1 14D Case 2 12D 16 L 05Lr Case 3 12D 16Lr 05W Case 4 12D 10W 05Lr Case 5 09D 10W where D Lr and W denote dead load roof live load and wind load respectively The dead load includes both superimposed dead load and frame selfweight The superim posed dead load and roof live load are assumed to be uniformly distributed on the roof k k k k 175 Interpolation Methods 263 surface They impose forces on the frame through the framepurlin connections in the vertical downward direction Selfweights are determined by dividing the frames into sufficiently large numbers of elements The directional wind speeds matrix see Section 1331 and the mean annual rate of storm arrival were assumed to be those listed for Miami milepost 1450 in Figure 31 The member capacities are determined as specified in 13 14 To comply with AISC requirements on secondorder effects a firstorder analysis method can be used that accounts for geometric imperfections 6 14 The axial capacity of a member in com pression is the smaller of the calculated inplane and outofplane buckling capacities computed by the method of successive approximations described in 15 Equations 81 and 82 and their shear force counterpart maintain the phase rela tionship among the axial force bending moments and shear force hence they result in DCIs that rigorously reflect the actual combined wind effects The preliminary design of the structure starts with an informed guess as to the struc tural systems member sizes that is with a preliminary design denoted by D0 to which there corresponds a set of influence coefficients denoted by IC0 The wind loads applied to this preliminary design are taken from the standard or code being used For the case study presented here the loads used for the preliminary design were obtained from the ASCE 7 Standard 1 As performed in 6 the next step is the calculation of the peak DCIs with the speci fied mean recurrence interval N inherent in the design D0 see Chapter 13 Unless those DCIs are close to unity the cross sections are modified This results in a new design D1 for which the corresponding set of influence coefficients IC1 is calculated A new set of DCIs is calculated based again on the wind loads taken from the standard The pro cedure is repeated until a design Dn is achieved such that the effect of using a new set of influence coefficients ICn 1 is negligible that is until the design Dn 1 is in prac tice identical to the design Dn At this point the procedure is applied by using instead of the ASCE 7 Standard wind pressures wind pressures based on the time histories of the pressure coefficients taken from the aerodynamics database This results in a design Dn 2 to which there corresponds a set of influence coefficients ICn 2 and a new set of DCIs The cross sections are then modified and the calculations are repeated until the DCIs are close to unity Typically this will be the final design Dfinal although the user may perform an additional iteration to check that convergence of the DCIs to unity has been achieved to within constructability and serviceability constraints For the struc tural system considered in this chapter the approach just described was found to yield the requisite results faster than the alternative approach in which the loads based on the aerodynamic database are used to determine the designs D1 through Dn 1 This is due to the fact that load estimates specified in 1 for the type of structure depicted in Figure 171 are less unrealistic than those specified in 1 for other types of structure 175 Interpolation Methods For the databases with large numbers of data measured on models with different dimen sions to be of practical use simple and reliable interpolation schemes need to be devel oped that enable the prediction of wind responses for building dimensions not available in the databases This issue was addressed in among others Refs 5 6 16 k k k k 264 17 Rigid Portal Frames The interpolation scheme presented in detail in 6 produces responses of the building of interest that unless the interpolations are performed between buildings with signifi cantly different dimensions differ from the actual responses by amounts in the order of 510 It is shown in Section 1242 that even larger errors are typically inconsequential from a structural design viewpoint 176 Comparisons Between Results Based on DAD and on ASCE 7 Standard This section presents results of comparisons between 700year i DADbased DCIs involving axial forces and moments denoted by DCIPM and DCIs involving shear forces denoted by DCIV to their counterparts based on the ASCE 710 Standard Chapter 28 Additional sets of comparisons are reported in 6 Unless otherwise specified the assumed frame spacing was 76 m Results are shown for the first interior frame The frame supports were assumed to be pinned and all the calculations were conducted for the enclosed building enclosure category 1761 Buildings with Various Eave Heights For buildings with various eave heights Figure 173 shows ratios of DCIPMs based on DAD to their counterparts based on the ASCE 710 Standard Chapter 28 The buildings had the following dimensions B 244 m L 381 m roof slope 48 and H 49 m 73 m and 98 m In most cases represented in Figure 173 the DCIs are underestimated by the ASCE 710 Standard provisions especially for suburban exposure 4 08 09 1 11 12 13 14 15 16 17 18 6 8 10 DCIDADDCIASCE Open Terrain Eave Height m 4 08 09 1 11 12 13 14 15 16 17 18 6 8 10 Suburban Eave Height m Knee Pinch Ridge Figure 173 DCIDADDCIASCE as a function of eave height k k k k References 265 Knee Pinch Ridge 0 1 15 2 25 3 48 10 14 Roof Slope deg 20 266 30 DCIDADDCIASCE 0 1 15 2 25 3 48 10 14 Roof Slope deg 20 266 30 Open Terrain Suburban Figure 174 DCIDADDCIASCE as a function of roof slope 1762 Buildings with Various Roof Slopes For buildings with different roof slopes Figure 174 shows ratios between DCIPMs com puted by using DAD and the ASCE 710 Standard Chapter 28 The buildings have the following dimensions B 244 m L 381 m H 73 m and roof slope 48 140 and 267 Owing to a strong discontinuity of the pressure coefficient variation at roof slopes of about 22 interpolations cannot be performed between wind effects on roofs with slopes lower than 22 on the one hand and larger than 22 on the other 17 References 1 ASCE Minimum design loads for buildings and other structures ASCESEI 716 in ASCE Standard ASCESEI 716 Reston VA American Society of Civil Engineers 2017 2 Coffman BF Main JA Duthinh D and Simiu E 2010 Wind effects on lowrise buildingsdatabasedassisted design vs ASCE 705 standard estimates Journal of Structural Engineering 136 744748 3 Pierre LMS Kopp GA Surry D and Ho TCE 2005 The UWO contribution to the NIST aerodynamic database for wind loads on low buildings part 2 Compar ison of data with wind load provisions Journal of Wind Engineering and Industrial Aerodynamics 93 3159 4 Davenport A G Surry D and Stathopoulos T Wind loads on lowrise buildings Part 1 The Boundary Layer Wind Tunnel University of Western Ontario London Ontario Canada 1977 k k k k 266 17 Rigid Portal Frames 5 Main J A and Fritz W P DatabaseAssisted Design for Wind Concepts Soft ware and Examples for Rigid and Flexible Buildings NIST Building Science Series 170 National Institute of Standards and Technology Gaithersburg MD 2006 httpswwwnistgovwind 6 Habte F Chowdhury AG Yeo D and Simiu E 2017 Design of rigid structures for wind using time series of demandtocapacity indexes application to steel portal frames Engineering Structures 132 428442 7 NIST Dec 18 2017 NISTUWO aerodynamic database Available httpswww nistgovwind 8 TPU Dec 18 2017 TPU aerodynamic database Available httpwindarcht kougeiacjpsystemengcontentscodetpu 9 Hagos A Habte F Chowdhury A and Yeo D 2014 Comparisons of two wind tunnel pressure databases and partial validation against fullscale measurements Journal of Structural Engineering 140 04014065 10 Yeo D 2013 Multiple pointsintime estimation of peak wind effects on structures Journal of Structural Engineering 139 462471 httpswwwnistgovwind 11 Yeo D 2014 Generation of large directional wind speed data sets for estimation of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 12 Habte F Chowdhury A G and Park S The Use of DemandtoCapacity Indexes for the Iterative Design of Rigid Structures for Wind NIST Technical Note 1908 National Institute of Standards and Technology Gaithersburg MD 2016 https wwwnistgovwind 13 Kaehler R C White D W and Kim Y D Frame Design Using WebTapered Members American Institute of Steel Construction 2011 14 ANSIAISC Specification for Structural Steel Buildings in ANSIAISC 36010 Chicago IL American Institute of Steel Construction 2010 15 Timoshenko S and Gere JM 1961 Theory of Elasticity Stability 2nd ed McGrawHill 16 Masters F Gurley K and Kopp GA 2010 Multivariate stochastic simulation of wind pressure over lowrise structures through linear model interpolation Journal of Wind Engineering and Industrial Aerodynamics 98 226235 17 Stathopoulos T Personal communication 2007 k k k k 267 18 Tall Buildings Case Studies1 181 Introduction Tall buildings can be designed by using the DatabaseAssisted Design DAD option or the related Equivalent Static Wind Loads ESWL option of the DADESWL v 10 software Both options are available at httpswwwnistgovwind A users manual 1 provides detailed guidance on the use of the software and its application to several examples including steel and reinforced concrete building examples The purpose of this chapter is to introduce the reader to that software and illustrate the application of its two options Section 182 briefly discusses an approach to per forming a structures preliminary design and outlines the subsequent iterative use of DADESWL to perform the final design Section 183 lists the contributions of the wind engineering laboratory to the design process Section 184 is an introduction to the soft ware Section 185 briefly presents the application of the DAD approach and of the ESWL approach to the structural design of a 47story steel building The software is also applicable to the design of midrise buildings via the simple device of using as input appropriately large values of the natural frequencies of vibration in the fundamental sway and torsional modes and disregarding higher modes 182 Preliminary Design and Design Iterations The structural design process starts with the development of a preliminary design This entails the choice of a structural system for the building being considered eg moment frames the geometry and morphological features of which must be consistent with architectural and other nonstructural design requirements The member sizes of the preliminary system are initially guessed at by the structural designer on the basis of experience This will produce a system that typically will not meet strength and service ability requirements It is therefore advisable to redesign the structural system produced by the structural engineers educated guesses by using for the wind loading simple mod els specified for example in the ASCE 7 Standard for buildings of all heights The new design so obtained is referred to here as design D0 The structural engineer must check the adequacy of design D0 that is whether it satisfies the specified strength and serviceability when subjected to realistic rather 1 Major contributions to this chapter by Dr Sejun Park are acknowledged with thanks Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 268 18 Tall Buildings than simplified wind loads The information inherent in design D0 and the data provided by the wind engineering laboratory are used by the structural engineer in the DADESWL software to determine the members demandtocapacity indexes DCIs interstory drift ratios and accelerations with the respective specified design mean recurrence intervals Chapter 13 For strength design it is required that no member DCI exceed unity or be significantly less than unity except as required by serviceability constraints If as is typically the case for design D0 these requirements are not satisfied the members cross sections need to be modified and the software is applied iteratively to successive designs D1 D2 until a satisfactory final design is achieved 183 Wind Engineering Contribution to the Design Process Realistic wind loads must be based on the following information provided by the wind engineering laboratory 1 Aerodynamic data consisting of pressure coefficient time series obtained simulta neously at multiple taps on the façades of the building model either from adhoc wind tunnel tests or in the future by adhoc Computational Wind Engineering CWE simulations or from databases such as 2 3 The following prototype data are required Elevation of the reference wind speed usually the elevation of the top of the building wind directions sampling rate number of sampling points and coor dinates defining the location of the taps on the building façades 2 Wind climatological data consisting of q matrices nsq ndq of directional wind speed data and the respective rates of storm arrival of up to q types of storm see Section A9 where depending upon the wind climate q 1 eg synoptic storms only q 2 eg hurricanes and thunderstorms or q 3 eg hurricanes noreasterns and thunderstorms The nsq rows correspond to a number nsq of storms see Sections 1331 323 and 4 the ndq columns correspond to say ndq 1636 wind directions The matrix entries are mean wind speeds averaged over say 3060 minutes at the location of the empty preconstruction building site and the elevation of the reference wind speed see item 1 3 Measures of uncertainty in the pressure coefficients and the directional wind speeds to be used in procedures for producing estimates of wind load factors or of aug mented design mean recurrence intervals of the wind effects of interest see Chapters 7 and 12 The contribution of the wind engineering laboratory to the design process is com pleted once the information described here is delivered to the structural engineer The same information is used with no modification for the analysis of each of the iterative designs 184 Using the DADESWL Software For a structure with given mechanical properties the DADESWL software is used by the structural engineer to determine the effects of interest induced by combinations of k k k k 184 Using the DADESWL Software 269 i gravity loads and ii wind loads based on the aerodynamic and wind climatological information provided by the wind engineering laboratory This section provides a sum mary description of the DADESWL software based on the detailed description avail able in 1 1841 Accessing the DADESWL Software DADESWL v 10 can be accessed via the website httpswwwnistgovwind The standalone executable version of DADESWL requires installation of MCRIn stallerexe which is available on the main page The website includes among others the input files for the examples described in Section 185 1842 Project Directory and its Contents It is recommended that a directory named DADESWL with the structure shown in Figure 181 be created for each project on the users local drive The directory saves all downloaded files and directories It is recommended that the executable file for the software DADESWLv1p0exe be included in the project directory The Aerodynamicdata directory contains data files MAT format i identify ing each of the pressure taps located on the exterior building surfaces ii listing their coordinates and iii containing pressure coefficient time series from windtunnel test ing or in the future from CWE simulations corresponding to a sufficient number of directions to allow the construction of the requisite response surfaces see Sections 82 and 132 The Buildingdata directory includes the buildings geometric and structural data members properties mass matrix influence coefficients internal forces of members induced by gravity loads and modal dynamic properties The buildings structural data are calculated and prepared in advance by using finite element software following the users choice of whether secondorder effects are accounted for or disregarded see Chapter 9 The alternative option of using OpenSees to obtain the buildings structural data is available see 1 for details in which case the OpenSees directory is added Figure 181 Recommended directory structure k k k k 270 18 Tall Buildings The Climatologicaldata1 Climatologicaldata2 and Climatologicaldata3 directories contain simulated directional wind speed data of up to three distinct types of storm The Output directory contains results of calculations performed by DADESWL 1843 Software Activation Graphical User Interface To run the software the user doubleclicks the DADESWLv1p0exe file in the project directory This opens a panel Figure 182 of the Graphical User Interface GUI allow ing the user to select the type of structure steel or reinforced concrete Clicking the button Start opens the first of five pages that prompt the user to i fill in values of requisite data eg building dimensions modal periods ii choose between various options eg secondorder effects accounted for or disregarded use of input from FE analyses or OpenSees use of DAD or ESWL procedure and iii after clicking Browse buttons fill in the respective paths and names of input files used in the calculations to be performed by DADESWL At the bottom of each of the five pages there is a group of five buttons called input panel navigator Bldg modeling Wind loads Resp surface Wind effects and Results Plots see Figure 183 These are activated in succession as the calculations proceed In addition to the input panel navigator the five pages contain the following buttons Save inputs used to save input data data file paths and selected options as MAT files for future use in DADESWL Open inputs used to download the saved input data and allowing empty boxes and unselected options in the input pan els to be filled and activated and Exit which can be clicked at any time to terminate DADESWL Figure 182 Structural type selection panel k k k k 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 271 Figure 183 Page of Bldg modeling 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 1851 Building Description The structure being considered is a 47story steel building with rigid diaphragm floors outriggers and belt truss system and dimensions 40 40 160 m in depth width and height respectively Figure 184 The structure consists of approximately 2300 columns 3950 beams and 2300 diagonal bracings Columns are of three types core external core and interior columns Beams are of three types exterior internal and core beams Diagonal bracings are of two types core and outrigger bracings Each type of structural member has the same dimensions for 10 successive floors of the buildings lowest 40 floors and for the seven highest floors The columns and bracings consist of builtup hol low structural sections HSS and the beams consist of rolled Wsections selected from the AISC Steel Construction Manual 5 The steel grade is ASTM A570 steel grade 50 k k k k a 3D View b Front View c Side View d Plan view θw wind direction Width 40 m Height 160 m Building core Depth 40 m Outrigger and Belt truss system located on 15th16st 31th 32st and 47th story x y θw Corner Columns CC Core Columns COR External Columns CES External Columns CEW Internal Columns CI External Beams BESW NORTH EAST WEST External Beams BES External Beams BEW SOUTH External Beams BEWS Core Beams BOW Core Beams BOS Core Bracings XOS Internal Beams BI Core Bracings XOE Core Columns COL Wind Depth Width Front façade SOUTH ϑ Figure 184 Views and horizontal cross section of structural system k k k k 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 273 The structure is assumed to be sited in open terrain exposure in South Carolina near the shoreline of milepost 1950 for a map showing milepost locations see Figure 31 The wind speed data being used are the NIST hurricane data transformed into hourly mean speeds at the elevation of the top of the building at the empty building site The orientation angle of the building is 270 clockwise from the north that is one of the four identical façades of the building faces east The aerodynamic pressure coefficient time histories are obtained from the Tokyo Polytechnic University TPU highrise build ing aerodynamic database 6 Wind direction is defined by the clockwise angle 𝜃w with the positive xaxis heading east and the yaxis heading north see Figure 184d A total of 60 types of structural members are selected for the final design six types of column seven types of beam and two types of bracing at the 1st 17th 33rd and 45th floors Task 1 Figure 81 in Chapter 8 consists of performing the preliminary design based on for example ASCE 716 Standard provisions for buildings of all heights This task yielded the member sizes listed in Table 181 Table 181 Member sizes for the preliminary design denoted by D0 in mm and member nomenclaturea Member type Section ID Sectional type Depth Width Flange thickness Web thickness Bracing D0116 BoxTube 350 350 14 14 D1732 BoxTube 300 300 14 14 D3347 BoxTube 200 200 12 12 Column Int0116 BoxTube 700 700 30 30 Int1732 BoxTube 500 500 24 24 Int3347 BoxTube 300 300 15 15 Core0116 BoxTube 1500 1500 60 60 Core1732 BoxTube 1200 1200 50 50 Core3347 BoxTube 1000 1000 40 40 ExCore0116 BoxTube 1200 1200 50 50 ExCore1732 BoxTube 1000 1000 40 40 ExCore3347 BoxTube 800 800 30 30 Beam W10X26 IWide Flange 26162 14656 1118 660 a D0116 Diagonal bracing floors 116 and all outriggers and belt trusses D1732 Diagonal bracing floors 1732 D3347 Diagonal bracing floors 3347 Int0116 Internal columns 116 Int1732 Internal columns floors 1732 Int3347 Internal columns floors 3347 Core0116 Core columns floors 116 Core1732 Core columns floors 1732 Core3347 Core columns floors 3347 ExCore0116 External Core Columns floors 116 ExCore1732 External Core Columns floors 1732 ExCore3347 External Core Columns floors 3347 W10X26 All beams k k k k 274 18 Tall Buildings 1852 Using the DAD and the ESWL Options This section is a brief summary of salient features of the users manual in 1 which describes the software in detail DAD option Task 2 Figure 81 and Section 82 begins by clicking the button Start shown in Figure 182 and selecting the Steel Structure option This opens the page shown in Figure 183 The page activated by the button Bldg modeling contains a Building information and a Structural properties panel The user fills in the requisite data ie No of stories Building height and so forth and selects the appropriate option where a choice is offered ie for this example Secondorder analysis rather than Linear and Input analysis results from arbitrary FE software rather than Use OpenSees The user also clicks the Browse buttons and fills in the respective paths and file names containing the results obtained by FE or OpenSees depending upon the analysts choice available in the Buildingdata directory Figure 181 Task 3 starts by clicking the button Wind loads at the bottom of the GUI page shown in Figure 185 The user fills in the requisite data ie Model length scale Wind direc tions and so forth in the Wind tunnel testCWE data panel and clicks the Calculate floor wind loads from pressures measured at taps on building model facade option in Figure 185 Page Wind loads k k k k 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 275 the Floor wind loads at model scale N and Nm subpanel After clicking the Browse buttons the user selects the appropriate files from the Aerodynamicdata directory Figure 181 Input files for the pressure coefficient data CpXXXpXmat tap iden tification taplocmat and tap coordinates tapcoordmat are provided by the wind engineering laboratory as indicated in Section 183 The user then selects the interpola tion method for calculating the floor wind loads righthand side of this panel Clicking the button Calculate floor wind loads starts the automatic calculation of the floor wind loads and activates a popup window showing the progress of the computations The wind pressure information can be checked by clicking the button Display The floor wind loading data are saved for each direction in the userspecified directory in this example WLfloors as shown in Figure 185 The Wind speed range panel specifies the wind speeds used for the construction of response surfaces discussed in Sections 82 and 132 In this example wind speeds from 20 to 80 m s1 in increments of 10 m s1 were used Finally the 80 selection that pertains to ASCE 716 Standard section 3144 was made in the Lower limit requirement panel The page opened by clicking the button Resp surface contains three panels Figure 186 The Load combination cases panel specifies the gravity and wind load Figure 186 Page Resp surface k k k k 276 18 Tall Buildings combinations including the associated load factors The Calculation option panel requires the user to choose between using the DAD and the ESWL approach The user must specify the length of the initial part of the time series of inertial forces that is discarded in order to eliminate nonstationary effects Tasks 4 5 and 6 require the use of the information provided in the Response surface panel and consist of calculating the ordinates of the response surfaces that yield peak member DCIs interstory drift ratios and accelerations These are obtained by performing dynamic analyses of the structure D0 for each of the directional wind speeds with directions entered in the panel Wind tunnel testCWE data and speeds entered in the panel Wind speed range of the page Wind loads Task 4 determines for each of those directional wind speeds the effective loads consisting of the sums of the aerodynamic and inertial loads Task 5 uses the appropriate influence coefficients to determine time series of the DCIs induced by combinations of factored gravity loads and the effective wind loads obtained in Task 4 Task 6 consists of calculating the ordinates of the response surfaces for peak DCIs interstory drift ratios and accelera tions induced in structural members by each of the directional wind speeds considered in Task 4 Task 7 is executed by attending to the panels Wind climatological data and Design responses for specified MRIs of Figure 187 page Wind effects Typically the wind Figure 187 Page Wind effects k k k k 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 277 Figure 188 Page Results Plots engineering laboratory provides directional mean wind speed data at the elevation of the top of the building roof at the empty site of the building The page Results Plots allows the user to show the calculated wind effects ie DCIs interstory drift ratios and accelerations with specified MRIs as shown in Figure 188 Superscripts P M V and T of DCIs stand for axial load P and bending moment M shear force V and torsion T respectively As expected DCI values for design D0 were typically inade quate The design was modified accordingly to yield design D1 an additional iteration yielded design D2 Details are provided subsequently ESWL option Tasks 1 through 4 are identical for both ESWL and DAD How ever unlike DAD ESWL requires completing Task 4a which consists of calculating Equivalent Static Wind Loads see Figure 81 To do so in the page Resp sur face Figure 186 and the panel Calculation options the user chooses the option ESWL only applicable for DCIs and responds to the prompt No of Multiple PointsInTime see Section 32 in 1 Tasks 5 6 and 7 are performed by using the same pages as those used for the DAD option but with the input required for ESWLs DCIs for selected members are listed in Table 182 for designs D0 D1 and D2 Mem ber sizes for design D2 are shown in Table 183 The results of the ESWL calculations depend upon the number of points in time npit Calculations performed for the example k k k k 278 18 Tall Buildings Table 182 DCIs axial force and bending moments based on DAD and on ESWL for designs D0 D1 and D2 D0 D1 D2 Member IDa Method 1st 17th 33rd 45th 1st 17th 33rd 45th 1st 17th 33rd 45th CC DAD 089 114 151 042 078 076 059 013 083 088 084 087 ESWL 089 113 151 042 076 074 057 013 083 088 084 086 CEW DAD 062 097 153 037 065 076 068 015 096 080 088 087 ESWL 062 096 152 037 066 074 066 015 095 080 088 087 CI DAD 081 108 164 030 084 085 077 013 083 090 100 080 ESWL 081 107 162 029 084 084 075 013 083 090 099 079 COL DAD 199 162 102 036 079 062 052 015 084 062 090 091 ESWL 201 161 102 036 076 059 049 010 084 062 089 090 CES DAD 078 118 173 039 067 078 069 015 072 090 095 089 ESWL 078 117 169 038 067 075 066 014 072 090 095 088 COR DAD 137 144 157 061 110 092 063 023 056 067 087 078 ESWL 138 142 155 060 111 088 059 022 056 067 086 078 BESW DAD 066 100 100 066 063 096 100 066 091 098 069 097 ESWL 064 097 097 064 061 095 097 065 090 098 069 096 BES DAD 065 096 093 060 062 093 092 057 062 095 096 083 ESWL 063 093 091 058 060 092 090 056 061 094 096 083 BI DAD 086 140 162 150 085 134 156 142 085 088 095 097 ESWL 085 139 162 148 085 134 154 140 085 088 093 097 BOS DAD 077 078 081 062 077 077 080 062 077 078 081 090 ESWL 076 077 079 062 077 077 079 061 077 077 080 090 BEWS DAD 070 118 126 088 056 095 105 074 087 068 077 075 ESWL 067 113 124 087 058 095 100 071 085 067 077 075 BEW DAD 069 117 126 079 056 096 102 067 087 069 074 099 ESWL 067 113 119 077 057 096 101 067 085 068 074 099 BOW DAD 082 087 095 065 076 080 087 063 078 082 089 093 ESWL 083 086 093 064 077 080 086 063 078 082 089 092 XOS DAD 073 076 079 035 067 070 079 035 069 071 085 080 ESWL 074 070 073 033 070 064 067 031 069 071 083 080 XOE DAD 082 071 110 044 072 063 086 035 068 062 046 082 ESWL 083 071 103 042 073 060 080 035 068 060 045 082 a CC corner column CEW external column at west side of the building plan CI internal column COL core column at left side of the core CES external column at south COR core column at right side of the core BESW external beam at southern west BES external beam at south BI internal beam BOS core beam at south BEWS external beam at western south BEW external beam at west BOW core beam at west XOS core bracing at south XOE core bracing at east See Figure 184d for details k k k k 185 Steel Building Design by the DAD and the ESWL Procedures Case Studies 279 Table 183 Member sizes for design D2 in mm and member nomenclaturea Members type Section ID Depth Width Flange thickness Web thickness Bracing D0116 BoxTube 350 350 14 14 D1732 BoxTube 300 300 14 14 D3340 BoxTube 200 200 12 12 D4147 BoxTube 145 145 9 9 Column Int0116 BoxTube 600 600 35 35 Int1732 BoxTube 400 400 15 15 Int3340 BoxTube 254 254 13 13 Int4147 BoxTube 230 230 10 10 Core0110 BoxTube 1800 1800 100 100 Core1120 BoxTube 1600 1600 80 80 Core2130 BoxTube 1200 1200 50 50 Core3140 BoxTube 565 565 25 25 Core4147 BoxTube 550 550 24 24 ExCore0116 BoxTube 1300 1300 60 60 ExCore1732 BoxTube 1100 1100 45 45 ExCore3347 BoxTube 1000 1000 40 40 Beam W10X39 IWide Flange 25197 20295 1346 800 W10X26 IWide Flange 26162 14656 1118 660 W10X19 IWide Flange 25908 10211 1003 635 a D0116 Diagonal bracing from floors 116 and for all outriggers and belt trusses D1732 Diagonal bracing from floors 1732 D3340 Diagonal bracing from floors 3340 D4147 Diagonal bracing from floors 4147 Int0116 Internal column from floors 116 Int1732 Internal column from floors 1732 Int3340 Internal column from floors 3340 Int4147 Internal column from floors 4147 Core0110 Core column from floors 110 Core1120 Core column from floors 1120 Core2130 Core column from floors 2130 Core3140 Core column from floors 3140 Core4147 Core column from floors 4147 ExCore0116 External Core Column from floors 116 ExCore1732 External Core Column from floors 1732 ExCore3347 External Core Column from floors 3347 W10X39 Beam from floors 120 W10X26 Beam from floors 2135 W10X19 Beam from floors 3647 k k k k 280 18 Tall Buildings presented in this section indicated that the use of npit 10 could result in the under estimation of some peak DCIs by over 10 or more whereas for npit 10 the largest underestimation was almost constant at 3 To assess the efficiency of the ESWL procedure the ratio r between ESWL and DAD computational times required to calculate design DCIs with MRI 1700 years was obtained as a function of i the number of points npit and ii the number of members being analyzed The dependence of the ratio r upon npit was found to be almost negligible For 60 members r was approximately 04 The relative efficiency of the ESWL procedure increases when larger numbers of structural members are selected For 1000 members r was approximately 02 The computation times for the DAD calculations were found to be fully compatible with practical capabilities of structural design offices The computational times can be reduced by using parallel computing The differences between DAD and ESWLbased DCIs are sufficiently small in this case that the designs D1 and D2 obtained by the DAD procedure on the one hand and the ESWL procedure on the other are the same for all the members considered in Tables 181 and 183 As pointed out in Chapter 14 this may not be the case for wind climates where winds from an unfavorable wind direction are dominant As was also pointed out in Chapter 14 the ESWL procedure may not be practicable for buildings with irregular shapes For the number of members considered in the case study presented in this section the ESWL procedure computation time on a personal computer was in the order of hours2 The computational time would have increased had the number of distinct members and the number of storm events been larger The amount of steel required for design D1 was approximately 50 greater than for design D0 that is the capacities of the members in the preliminary design D0 were too low The iteration that followed the design D1 resulted in a design D2 for which the amount of steel was approximately 20 lower than for design D1 The evolution of the successive designs can be followed by considering Table 182 References 1 Park S Yeo D and Simiu E Databaseassisted design and equivalent static wind loads for mid and highrise structures concepts software and users manual NIST Technical Note 2000 National Institute of Standards and Technology Gaithersburg MD 2018 httpswwwnistgovwind 2 Ho T Surry D and Morrish D NISTTTU Cooperative Agreement Wind storm Mitigation Initiative Wind Tunnel Experiments on Generic Low Buildings BLWTSS202003 Boundary Layer Wind Tunnel Laboratory University of Western Ontario London Canada 2003 3 Tamura Y Aerodynamic Database of LowRise Buildings Global Center of Excel lence Program Tokyo Polytechnic University Tokyo Japan 2012 2 The system specifications were as follows Intel Xeon E5 CPU and 16 GB of RAM k k k k References 281 4 Yeo D 2014 Generation of large directional wind speed data sets for estima tion of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 5 ANSIAISC Specification for Structural Steel Buildings in ANSIAISC 36010 Chicago Illinois American Institute of Steel Construction 2010 6 TPU TPU highrise building aerodynamic database Tokyo Polytechnic University TPU Available httpwindarchtkougeiacjpsystemengcontentscodetpu k k k k 283 Part III Aeroelastic Effects Fundamentals and Applications Certain types of civil engineering structures can experience aerodynamic forces gener ated by structural motions These motions called selfexcited are in turn affected by the aerodynamic forces they generate The structural behavior associated with selfexcited motions is called aeroelastic The purpose of Part III is to provide an introduction to aeroelastic phenomena occurring in flexible civil engineering structures Chapters 19 20 and 21 consider respectively fundamental aspects of aeroelasticity phenomena asso ciated with vortex lockin galloping and torsional divergence and flutter Presented here are applications are to chimneys with circular crosssections and other slender struc tures including tall buildings Chapter 22 and to suspendedspan bridges Chapter 23 Iconic examples of aeroelastic instability are the flutter of the Brighton Chain Pier Bridge termed undulation in the 1800s Figure III1 and more than one century later the flutter of the original TacomaNarrows Bridge Figure III2 To describe the interaction between aerodynamic forces and structural motions it is in principle necessary to solve the full equations of motion describing the flow with timedependent boundary conditions imposed by the moving structure Even though progress is being made in the numerical solution of some aeroelastic problems for bluff bodies immersed in shear turbulent flow the description of aeroelastic effects still relies largely on laboratory testing and empirical modeling Owing to the violation of the Reynolds number similarity criterion the applicability to the prototype of laboratory test results and of associated empirical models needs to be assessed as thoroughly as possible However for carefully modeled structures aeroelastic test results are generally assumed to yield reasonably realistic results For additional fundamental and applied material on aeroelasticity in civil engineering see 3 The rich experience of the Japanese school of suspendedspan bridge aeroe lasticity is reflected in the abundant material contributed by Miyata in 4 Ovalling oscillations which can occur for example in certain types of silos are considered in 5 and using a Computational Wind Engineering CWE approach in 6 Aeroelastic motions of textile structures are considered in Chapter 26 see eg 7 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 284 Part III Aeroelastic Effects SKETCH showing the manner in which the 3rd span of the CHAIN PIER at BRIGHTON undulated just before it gave way in a storm on the 20th of November 1836 255 feet SKETCH showing the appearance of the 3rd span after it gave way Figure III1 Brighton chain pier failure 1836 Source From 1 Figure III2 Flutter of Tacoma Narrows suspension bridge 1940 Source From 2 k k k k Part III Aeroelastic Effects 285 A number of empirical models allow design decisions to be based on results of rel atively simple wind tunnel test results For example the designer of suspendedspan bridges can account for the possibility of flutter by using empirical data known as flut ter derivatives that can be measured in the laboratory A more thorough approach can make use of detailed observations of flow patterns associated with the aeroelastic behav ior of typically simple shapes Fundamental studies of this type are considered in 4 an example is reported in detail with exemplary rigor in 8 References 1 Russel JD On the vibration of suspension bridges and other structures and the means of preventing injury from this cause Transactions of the Royal Society of Arts 1841 reproduced in 2 2 Farquharson FB ed 19491954 Aerodynamic Stability of Suspension Bridges Part 1 Bulletin 116 Seattle WA University of Washington Engineering Experimental Sta tion 3 Scanlan RH and Simiu E 2015 Aeroelasticity in civil engineering In A modern course in aeroelasticity 5th ed ed EH Dowell Springer 4 Simiu E and Miyata T 2006 Design of Buildings and Bridges for Wind A Prac tical Guide for ASCE7 Standard Users and Designers of Special Structures 1st ed Hoboken NJ John Wiley Sons Inc 5 Paidoussis MP Price SJ and de Langre E 2012 FluidStructure Interactions Cambridge University Press 6 Hillaewaere J Degroote J Lombaert G et al 2015 Windstructure interaction simulations of ovalling vibrations in silo groups Journal of Fluids and Structures 59 328350 doi 101016jjfluidstructs201509013 7 Michalski A Kermel PD Haug E et al 2011 Validation of the computational fluidstructure interaction simulation at realscale tests of a flexible 29 m umbrella in natural wind flow Journal of Wind Engineering and Industrial Aerodynamics 99 4 400413 8 Hémon P and Santi F 2002 On the aeroelastic behaviour of rectangular cylinders in crossflow Journal of Fluids and Structures 16 7 855889 doi 101006jfls452 k k k k 287 19 VortexInduced Vibrations 191 LockIn as an Aeroelastic Phenomenon The shedding of vortices in the wake of a body gives rise to fluctuating lift forces If the body is flexible or if it has elastic supports it will experience motions due to aerody namic forces and in particular to the fluctuating lift force As long as the motions are sufficiently small they do not affect the vortexshedding frequency Ns which remains proportional to the wind speed in accordance with the relation Ns U St D 191 Section 44 where the Strouhal number St depends upon body geometry and the Reynolds number D is a characteristic body dimension and U is the mean velocity of the uniform flow or a representative mean velocity in shear flow If the vortexinduced transverse deformations are sufficiently large within an inter val NsDSt ΔU U NsDSt ΔU where ΔUU is in the order of a few percent the vortex shedding frequency no longer satisfies Eq 191 Rather because the body defor mations influence the flow the vortex shedding frequency will be constant for all wind speeds within that interval Figure 191 This is an aeroelastic effect while the flow affects the body motion the body motion in turn affects the flow insofar as it produces lockin that is a synchronization of the vortexshedding frequency with the frequency of vibration of the body 192 VortexInduced Oscillations of Circular Cylinders A variety of vortexinduced oscillation models are available in which the aeroelastic forces depend upon adjustable parameters fitted to match experimental results By con struction those models provide a reasonable description of the observed aeroelastic motions However the empirical models may not be valid as a motion predictor for conditions other than those of the experiments Consider a rigid circular cylinder in uniform smooth flow The acrosswind force act ing on the cylinder is approximately Ft 1 2𝜌U2DCLS sin 𝜔st 192 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 288 19 VortexInduced Vibrations Frequency Flow velocity Vortex shedding frequency Lockin region Natural frequency of structure Figure 191 Frequency of vortex shedding in the wake of an elastic structure as a function of wind velocity where 𝜔s 2𝜋Ns Ns satisfies the Strouhal relation Eq 191 and CLS is a lift coefficient For a circular cylinder in uniform smooth flow and Reynolds number 40 Re 3 105 CLS 06 1 p 7 For a cylinder allowed to oscillate Eq 192 is inadequate for two reasons First the acrosswind force increases with oscillation amplitude until a limiting amplitude is reached Second the spanwise correlation of the acrosswind force also increases as indicated in Figure 192 Let y denote the acrosswind displacement of a cylinder of unit length for which the imperfect spanwise force correlation is not explicitly accounted for The equation of motion of the cylinder can be written as my c y ky y y y t 193 where m is the cylinder mass c is the mechanical damping constant k is the spring stiff ness and is the fluidinduced force per unit span which may be dependent on the displacement y and its first and second derivatives as well as on time Most empirical models recognize the nearsinusoidal response of the cylinder at the Strouhal frequency and the natural frequency of vibration of the structure Unless the velocity is at the lockin values the response gives rise to a beating oscillation Figure 193 parts ac show the displacements y and their spectral densities for an elastically supported cylinder before at and after lockin respectively Scanlan 4 proposed the following simple model my 2𝜁𝜔1 y 𝜔2 1y 1 2𝜌U2D Y1K 1 𝜀 y2 D2 y U Y2K y D CLK sin𝜔t 𝜙 194 where m is the body mass per unit length 𝜁 is the damping ratio 𝜔1 is the frequency of vibration of the body D is the cylinders diameter U is the flow velocity 𝜌 is the density of the fluid K 𝜔DU is the reduced frequency and the vortexshedding k k k k 192 VortexInduced Oscillations of Circular Cylinders 289 0 0 2 4 6 8 10 2 4 6 8 10 SEPARATION rD CORRELATION 2aD 20 15 10 05 0 a 0 0 2 4 6 8 10 2 4 6 8 10 SEPARATION rD CORRELATION 2aD 20 10 0 b Figure 192 The effect of increasing the oscillation amplitude a2 of a circular cylinder of diameter D on the correlation between pressures at points separated by distance r along a generator a smooth flow b flow with 11 turbulence intensity Reynolds number 2 104 Source Reprinted from 2 with permission of Cambridge University Press frequency n 𝜔2𝜋 satisfies the Strouhal relation n U StD Y 1 Y 2 𝜀 and CL are adjustable parameters that must be fitted to experimental results As is the case for the van der Pol oscillator the amplitude y is selflimiting The first term within the brackets in the righthand side of Eq 194 is proportional to y and may therefore be viewed as a damping term of aerodynamic origin For low amplitudes y that term is positive meaning that the sum of the mechanical and aerodynamic damping forces can be negative in agreement with the physical fact that the flow promotes the cylinders motion by transferring energy to the body The reverse is true for high amplitudes where the body loses energy by transferring it to the flow At lockin 𝜔 𝜔1 and the last two terms in the righthand side of Eq 194 are rela tively small and can be neglected Then Y 1 and 𝜀 remain to be determined by experiment At steady amplitudes the average energy dissipation per cycle is zero so that T 0 4m𝜁𝜔 𝜌UDY1 1 𝜀 y2 D2 y2dt 0 195 k k k k 290 19 VortexInduced Vibrations 0 0004 0000 0004 2 4 yD t 0 00 05 10 15 20 25 30 5 10 Sf f fs fn 0 0 40 80 120 160 200 240 5 10 Sf f 0 0 1 2 3 4 5 6 5 10 Sf f fs fn 0 008 000 008 2 4 yD t 0 0004 0000 0004 2 4 yD t a b c Figure 193 Acrossflow oscillations yD of elastically supported circular cylinder a before lockin b at lockin c after lockin The Strouhal frequency and the natural frequency of vibration of the body f s and f n respectively are shown in the spectral density plots Sf 3 with permission from the American Society of Civil Engineers ASCE where T 2𝜋𝜔 Assuming that the oscillation yt is practically harmonic y y0 cos 𝜔 t 196 leads to the results T 0 y2dt 𝜔y2 0 𝜋 197 T 0 y2 y2dt 𝜔y4 0 𝜋 4 198 Then Eq 195 yields the steady amplitude solution y0 D 2 Y1 8𝜋SscSt 𝜀Y1 12 199 k k k k 192 VortexInduced Oscillations of Circular Cylinders 291 where St is the Strouhal number and Ssc 𝜁m 𝜌D2 1910 is the Scruton number If at lockin velocity the mechanical model is displaced to an initial amplitude A0 and then released it will undergo a decaying response until it reaches a steady state with amplitude y0 given by Eq 199 Figure 194 A timedependent expression for the decaying oscillation amplitude derived in 5 yields the maximum amplitudes shown in Figure 195 which are close to those yielded by an empirical formula obtained in 5 and plotted in Figure 195 y0 D 129 1 0438𝜋2St2Ssc335 1911 y Figure 194 Decaying oscillation to steady state of bluff elastically sprung model under vortex lockin excitation 10 000 005 010 15 20 25 30 35 40 yoD Scruton number Experiment Eq 1911 Figure 195 Maximum amplitudes versus Scruton number Experiment o Eq 1911 Source Reprinted from 6 with permission from Elsevier k k k k 292 19 VortexInduced Vibrations 193 AcrossWind Response of Chimneys and Towers with Circular Cross Section A model similar to Eq 194 was developed in 7 for application to the design of chimneys and towers with circular crosssection Differences between this model and Eq 194 are as follows It is noted in 7 that 𝜌U2Y 2K m𝜔2 1 The term Y 2KyD is therefore neglected and since the actual motion of the chimneys or towers is random rather than periodic the term 𝜀y2D2 of Eq 194 is replaced by the ratio y2𝜆D2 where 𝜆 is a coefficient whose physical significance is discussed subsequently The term 1 2𝜌U2DY1K 1 𝜀 y2 D2 y U 1912 of Eq 194 is written in the form 2𝜔1𝜌D2Ka0 U Ucr 1 y2 𝜆D2 y 1913 where Ka0UUcr is an aerodynamic coefficient and Ucr 𝜔1D2𝜋 St is the velocity that produces vortex shedding with frequency n1 This term is equated to the product 2m𝜁a𝜔1 where 𝜁a is defined as the aerodynamic ratio which may thus be written as 𝜁a 𝜌D2 m Ka0 U Ucr 1 y2 𝜆D2 1914 For y212 𝜆D the aerodynamic damping vanishes so the structure no longer experi ences aeroelastic effects causing the response to increase The coefficient 𝜆 may thus be interpreted as the ratio between the limiting rms value of the aeroelastic response and the diameter D The total damping ratio of the system is then 𝜁t 𝜁 𝜁a 1915 where 𝜁 is the mechanical damping ratio The aeroelastic effects are thus introduced by substituting into the equation of motion the total damping 𝜁t for the mechanical damping ratio 𝜁 This simple approach was validated in 7 against experimental results shown in Figure 196 which represents the dependence of the measured response 𝜂rms y212D upon the reduced wind speed 2𝜋U𝜔1D for various damping ratios 𝜁 Figure 197 shows calculated versus measured ratios y2 max 12 D for various values of the damp ing parameter Ks m𝜁𝜌D2 where y2 max 12 is the rms response corresponding to the most unfavorable reduced wind speed In Figure 197 i the forced vibration regime corresponds to vibrations induced quasistatically by the vorticity in the wake of the cylinder and ii the lockin regime corresponds to vibrations due to aeroelastic effects A transition regime is observed between i and ii Turbulence in the oncoming flow decreases the coherence of the vorticity shed in the wake of the body and reduces the magnitude of the acrosswind response Vibrations typical of these regimes are shown in Figure 198 The ratios of the peak to rms response are about 40 in the forced vibration regime and about 2 in the lockin regime k k k k 193 AcrossWind Response of Chimneys and Towers with Circular Cross Section 293 3 0 02 04 06 08 4 5 6 7 8 9 10 REDUCED AMPLITUDE ηrms 0002 0003 0004 0005 0009 L D 10 2m ρD2 465 ζ 2 π U ω1 D Figure 196 Response of a model stack of circular cross section and length L for different values of the mechanical damping Re subcritical Source From 8 Courtesy of National Physical Laboratory UK Based on inferences from experimental data available in the literature 7 devel oped curves representing i the dependence upon Reynolds number of the largest value of Ka0UUcr in smooth flow Ka0max and ii the dependence of the ratio Ka0UUcrKa0max upon UUcr for smooth flow and flows with various turbulence intensities u212U Figure 199 For a vertical structure experiencing random motions described by the relation y2z i 𝜉2 i y2 i z 1916 where y2 is the rms response 𝜉i and yi are the rms modal coefficient and the modal shape respectively for mode i 9 the following expression is proposed for the total damping in the ith mode 𝜁ti 𝜁i 𝜁ai 1917 𝜁ai 𝜌D2 0 mei 2K1i K2i 𝜉2 i D2 0 1918 K1i h 0 Ka0z Dz D0 2 y2 i zdz h 0 y2 i zdz 1919 k k k k 294 19 VortexInduced Vibrations 010 001 0001 01 02 04 06 08 10 20 40 Ks Transition Regime Lockin Regime Forced Vibration Regime Experimental 10 𝓡𝓮 600000 Calculated 12 y2 max D Figure 197 Measured and estimated response in smooth flow Source Reprinted from 7 with permission from Elsevier K2i h 0 Ka0zy4 i zdz 𝜆2 h 0 y2 i zdz 1920 where 𝜁i and 𝜁ai are the mechanical and the aerodynamic damping in the ith mode of vibration respectively D0 is the diameter at elevation z 0 Dz is the diameter at ele vation z h is the height of the structure mei is the equivalent mass per unit length in the ith mode of vibration defined as mei Mi h 0 y2 i zdz 1921 k k k k 193 AcrossWind Response of Chimneys and Towers with Circular Cross Section 295 05 01 003 0 003 0 01 0 05 y D y D y D KsKao 1 KsKao 1 KsKaυ 1 Figure 198 Simulated displacement histories for low moderate and high mechanical damping Source Reprinted from 7 with permission from Elsevier 08 0 02 04 06 08 10 09 10 11 12 13 14 15 16 17 UUcr Ka0 Ka0max 00 01 02 03 12 u2 U Figure 199 Dependence of ratio Ka0Ka0max upon ratio UUcr for various turbulence intensities Source Reprinted from 7 with permission from Elsevier k k k k 296 19 VortexInduced Vibrations and Mi is the generalized mass in the ith mode Equations 19171920 are based on the assumption that aeroelastic effects occurring at various elevations are linearly superposable For the relatively small values of the response that are acceptable for chimneys and stacks the estimated response depends weakly upon the assumed value of 𝜆 It is sug gested in 9 that the value 𝜆 04 is reasonable for use in estimates of the response of concrete chimneys References 1 Bishop RED and Hassan AY 1964 The lift and drag forces on a circular cylinder oscillating in a flowing fluid Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences 277 5175 2 Novak M and Tanaka H 1976 Pressure correlations on vertical cylinders In Fourth International Conference on Wind Effects on Structures ed KJ Eaton 227332 Heathrow UK 3 Goswami I Scanlan RH and Jones NP 1993 Vortexinduced vibration of circu lar cylinders I experimental data Journal of Engineering Mechanics 119 22702287 4 Simiu E and Scanlan RH 1996 Wind Effects on Structures 3rd ed Hoboken John Wiley Sons 5 Ehsan F and Scanlan RH 1990 Vortexinduced vibrations of flexible bridges Journal of Engineering Mechanics 116 13921411 6 Griffin OM Skop RA and Ramberg SE The resonant vortexexcited vibrations of structures and cable systems presented at the Offshore Technology Conference Houston Texas 1975 7 Vickery BJ and Basu RI 1983 Acrosswind vibrations of structures of circular crosssection Part I Development of a mathematical model for twodimensional conditions Journal of Wind Engineering and Industrial Aerodynamics 12 4973 8 Wooton L R and Scruton C Aerodynamic stability in Modern Design of WindSensitive Structures London UK Construction Industry Research and Infor mation Association 1971 pp 6581 9 Basu RI and Vickery BJ 1983 Acrosswind vibrations of structure of circular crosssection Part II Development of a mathematical model for fullscale applica tion Journal of Wind Engineering and Industrial Aerodynamics 12 7597 10 Wooton L R 1969 Oscillations of Large Circular Stacks in Wind Proceedings of the Institution of Civil Engineers 43 pp 573598 k k k k 297 20 Galloping and Torsional Divergence 201 Galloping Motions Galloping is a largeamplitude aeroelastic oscillation one to ten or more crosssectional dimensions of the body that can be experienced by elastically restrained cylindrical bodies with certain types of crosssection eg square section Dsection ice laden power cables For material on wake galloping of power transmission lines grouped in bundles see for example Ref 1 2011 GlauertDen Hartog Necessary Condition for Galloping Motion Consider first a fixed cylinder immersed in a flow with velocity Ur Assume the angle of attack is 𝛼 Figure 201 The positive ycoordinate in Figure 201 is downward The mean drag and lift are respectively D𝛼 1 2𝜌U2 r BCD𝛼 201 L𝛼 1 2𝜌U2 r BCL𝛼 202 The sum of the projections of these components on the direction y is Fy𝛼 D𝛼 sin 𝛼 L𝛼 cos 𝛼 203 If Fy𝛼 is written in the alternative form Fy𝛼 1 2𝜌U2BCFy𝛼 204 where U Ur cos 𝛼 it is easily verified that there follows from Eqs 201204 CFy𝛼 CL𝛼 CD𝛼 tan 𝛼 cos 𝛼 205 Consider now the case in which in a flow with velocity U the body oscillates in the acrossflow direction y Figure 202 The magnitude of the relative velocity of the flow with respect to the moving body is denoted by Ur and can be written as Ur U2 y212 206 The angle of attack denoted by 𝛼 is 𝛼 arctan y U 207 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 298 20 Galloping and Torsional Divergence L D y B Ur α Figure 201 Lift and drag on a fixed bluff object L D B Ur U α y y Figure 202 Effective angle of attack on an oscillating bluff object The equation of motion of the body in the y direction is my 2𝜁𝜔1 y 𝜔2 1y Fy 208 where m is the mass per unit length 𝜁 is the damping ratio and 𝜔1 is the natural circular frequency Fy denotes the aerodynamic force acting on the body in the direction normal to the mean flow It is assumed that the mean aerodynamic lift and drag coefficients CL𝛼 and CD𝛼 for the oscillating body and for the fixed body are the same so that Fy𝛼 is given by Eq 204 and CFy𝛼 is given by Eq 205 Consider now the case of incipient small motion that is the condition in the vicinity of y 0 wherein 𝛼 y U 0 209 For this condition Fy dFy d𝛼 𝛼0 𝛼 2010 Differentiation of Eq 205 yields dCFy d𝛼 𝛼0 dCL d𝛼 CD 𝛼0 2011 The equation of motion thus takes the form my 2𝜁𝜔1 y 𝜔2 1y 1 2𝜌U2B dCL d𝛼 CD 𝛼0 y U 2012 Considering the aerodynamic righthand side of Eq 2012 as a contribution to the overall system damping the net damping coefficient of the system is 2m𝜔1𝜁 1 2 𝜌UB dCL d𝛼 CD 𝛼0 d 2013 k k k k 201 Galloping Motions 299 The condition for the occurrence of instability is that d 0 Since 𝜁 0 for this condition to be satisfied it is necessary that dCL d𝛼 CD 𝛼0 0 2014 Equation 2014 is the GlauertDen Hartog necessary condition for incipient gallop ing motion a sufficient condition being d 0 1 It follows from Eq 2014 that circular cylinders for which dCL d𝛼 0 cannot gallop The physical interpretation of Eq 2014 is the following Let the body experience a small perturbation from its position of equilibrium that causes it to acquire a velocity y The perturbation causes an asymmetry in the aerodynamic forces that act on the body If the bodys aerodynamic properties are such that this asymmetry causes the initial velocity to increase galloping motion will occur Otherwise the body will be restored to its position of equilibrium To summarize the susceptibility of a slender prismatic body to galloping instabil ity can be assessed by evaluating its mean lift and drag coefficients and determining whether the lefthand side of Eq 2014 is negative For example plots of the drag and lift coefficients show that according to the GlauertDen Hartog criterion the octagonal cylinder of Figure 203 is susceptible to galloping for angles 5 𝛼 5 2 For a simple demonstration of the galloping motion of a square cylinder see https wwwnistgovwind CD CL D U CORNER RADIUS 005D 10 10 10 05 05 10 15 CLCD 20 30 40 50 α α Figure 203 Force coefficients for an octagonal cylinder Re 12 106 Source Courtesy of Dr R H Scanlan k k k k 300 20 Galloping and Torsional Divergence Tests have shown that the derivatives dCFy d𝛼 are not dependent upon the frequency of the body motion and can be obtained from aerodynamic force measurements on the fixed body The quantities dCFy d𝛼 are called steadystate aerodynamic lift coefficient derivatives or for short steadystate aerodynamic derivatives In the case of flutter the aeroelastic behavior is characterized by quantities of a similar nature called flutter aero dynamic derivatives that unlike the steadystate derivatives that characterize galloping motion depend upon the oscillation frequency This difference is commented upon in Chapter 23 2012 Modeling of Galloping Motion Galloping motion was described in 3 by developing the lift coefficient CFy in powers of y U CFy𝛼 A1 y U A2 y U 2 y y A3 y U 3 A5 y U 5 A7 y U 7 2015 If the dependence of CD and CL upon 𝛼 is known the coefficients A1 through A7 can be evaluated as follows First the coefficient CFy is plotted against tan 𝛼 y U using Eq 205 The coefficients in Eq 2015 can then be estimated on the basis of this plot for example by using a least squares technique Reference 3 applies the methods of Kryloff and Bogoliuboff 4 to the resulting nonlinear equation postulating as a first approximation the solution y a cos𝜔1t 𝜙 2016 where a and 𝜙 are considered to be slowly varying functions of time Depending upon whether the coefficient A1 is less than equal to or larger than zero three basic types of curves CFy are identified as functions of 𝛼 with the corresponding galloping response amplitudes as functions of the reduced velocity UD𝜔1 see Figure 204 The only pos sible oscillatory motions are those with amplitudes a traced in full lines If the speed increases from U0 to U2 Figure 204a the amplitude of the response is likely to jump from the lower to the upper branch of the solid curve If the speed decreases from U2 to U0 the jump occurs from the upper to the lower curve An elegant mathematical investigation into the nonlinear modeling of galloping motions is reported in 5 2013 Galloping of Two Elastically Coupled Square Cylinders Reference 6 describes an experiment conducted in a water tunnel on the behavior of a system of two elastically restrained and coupled aluminum square bars with sides h1 h2 635 mm and length 0215 m The spring constants were k1 56 k2 78 and k12 145 N m1 Figure 205 To prevent displacements due to drag the bar ends were attached to fixed points by thin wires with lengths r 400 mm The bars were observed to gallop in phase but except for relatively low flow speeds U this oscillatory form alter nated in unpredictable chaotic fashion with a second oscillatory form wherein the two bars galloped with higher frequency in opposite phases Figures 206a b The mean k k k k 201 Galloping Motions 301 A1 0 A1 0 A1 0 0 0 0 α α α a b c CFy CFy U0 UDω1 UDω1 UDω1 U1 U2 CFy a a a Figure 204 Three basic types of acrosswind force coefficients and the corresponding galloping response amplitudes a Source From 3 With permission from the American Society of Civil Engineers ASCE Figure 205 Schematic of double galloping oscillator k1 k2 y2 y1 V r r k12 h1 h2 k k k k 302 20 Galloping and Torsional Divergence 0030 850 920 930 940 950 900 950 1000 Time s a b Time s 1050 1100 0015 0000 0015 0030 0030 0015 0000 0015 0030 Figure 206 a Observed time history of displacement y1 b observed time history of displacements y1 solid line and y2 interrupted line Source From 6 k k k k 202 Torsional Divergence 303 exit time of the system from the region of phase space corresponding to the inphase oscillations decreased as the flow velocity increased 202 Torsional Divergence Torsional divergence also called lateral buckling can occur on airfoils or bridge decks Like galloping it can be modeled by using aerodynamic properties measured on the body at rest The parameters of the torsional divergence problem are shown in Figure 207 in which U is the horizontal wind velocity 𝛼 is the angle of rotation of the bridge deck about the elastic axis and k𝛼 is the torsional stiffness The aerodynamic moment per unit span is M𝛼 1 2𝜌U2B2CM𝛼 2017 where B is the bridge deck width and CM𝛼 is the aerodynamic moment coefficient about the elastic axis For small 𝛼 M𝛼 1 2𝜌U2B2 CM0 dCM d𝛼 𝛼0 𝛼 2018 Let 𝜆 1 2𝜌U2B2 0 Equating M𝛼 to the internal torsional moment k𝛼 𝛼 yields 𝛼 𝜆CM0 k𝛼 𝜆 dCM d𝛼 𝛼0 2019 Divergence occurs when 𝛼 goes to infinity for vanishing values of the denominator in Eq 2019 The critical divergence velocity is Ucr 2k𝛼 𝜌B2 dCM d𝛼 𝛼0 2020 In most cases of interest in civil engineering applications the critical divergence veloc ities are well beyond the range of velocities normally considered in design ELASTIC AXIS kα U α Figure 207 Parameters for the torsional divergence problem k k k k 304 20 Galloping and Torsional Divergence References 1 Den Hartog JP 1956 Mechanical Vibrations 4th ed New York McGrawHill 2 Scanlan R H and Wardlaw R L Reduction of flowinduced structural vibrations in Isolation of mechanical vibration impact and noise A colloquium presented at the ASME Design Engineering Technical Conference Cincinnati OH 1973 3 Novak M 1972 Galloping oscillations of prismatic structures Journal of the Engineering Mechanics Division 98 2746 4 Kryloff N and Bogoliuboff N 1947 Introduction to Nonlinear Mechanics Princeton Princeton University Press 5 Parkinson GV and Smith JD 1964 The square prism as an aeroelastic nonlinear oscillator The Quarterly Journal of Mechanics and Applied Mathematics 17 225239 6 Simiu E and Cook GR 1992 Empirical fluidelastic models and chaotic galloping A case study Journal of Sound and Vibration 154 4566 k k k k 305 21 Flutter Flutter is an aeroelastic phenomenon that occurs in flexible bodies with relatively flat shapes such as airplane wings and bridge decks It involves oscillations with amplitudes that grow in time and can result in catastrophic structural failure Like other aeroelastic phenomena flutter entails the solution of equations of motion involving inertial mechanical damping elastic restraint and aerodynamic forces including forces induced by selfexcited motions that depend upon the ambient flow and the shape and motion of the body Assume that the mechanical damping is negligible The motion of the body is aeroe lastically stable if following a small perturbation away from its position of equilibrium the body will revert to that position owing to stabilizing selfexcited forces associated with the perturbation As the flow velocity increases the aerodynamic forces acting on the body change and for certain elongated body shapes at a critical value of the flow velocity the selfexcited forces may cause the body to be neutrally stable For velocities larger than the critical velocity the oscillations initiated by a small perturbation from the position of equilibrium will grow in time The selfexcited forces that cause these growing oscillations can be viewed as producing a negative aerodynamic damping effect The main difficulty in solving the flutter problem for bridges is the development of expressions for the selfexcited forces For thin airfoil flutter in incompressible flow expressions for the selfexcited forces due to small oscillations have been derived by Theodorsen 1 However the airfoil solutions are in general not applicable to bridge sections Although it is accompanied at all times by vortex shedding with frequency equal to the flutter frequency flutter is a phenomenon distinct from vortexinduced oscillation The latter entails aeroelastic flowstructure interactions only for flow velocities at which the frequency of the vortex shedding is equal or close to the structures natural fre quency for velocities higher or lower than those at which lockin occurs the oscillations are much weaker than at lockin In contrast for velocities higher than those at which flutter sets in the strength of the oscillations increases monotonically with velocity To date one of the most influential contributions to solving the flutter problem for bridges is Scanlans simple conceptual framework wherein the selfexcited forces due to small bridge deck oscillations can be characterized by fundamental functions called flut ter aerodynamic derivatives 2 As shown earlier in the galloping case the selfexcited forces depend on the steadystate derivatives dCFyd𝛼 that are not significantly affected by vorticity and may therefore be obtained from measurements on the fixed body In contrast owing to the elongated shapes of bodies susceptible to flutter the aerodynamic Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 306 21 Flutter derivatives of a body susceptible to flutter must be obtained from measurements on the oscillating body This is the case because the aerodynamic pressures on the body are sig nificantly affected by vortices induced by and occurring at the frequency of the torsional oscillations of the bridge In its detail flutter in practically all cases involves nonlinear aerodynamics It has been possible in a number of instances however to treat the problem successfully by lin ear analytical approaches This is the case for two main reasons First the supporting structure is usually treatable as linearly elastic and its actions dominate the form of the response which is usually an exponentially modified sinusoidal oscillation Second it is the incipient or starting condition which may be treated as having only small amplitude that separates the stable and unstable regimes These two main features enable a flutter analysis to be based on the standard stability considerations of linear elastic systems It is characteristic of flutter as a typical selfexcited oscillation that by means of its deflections and their time derivatives a structural system taps off energy from the wind flow If the system is given an initial disturbance its motion will either decay or grow according to whether the energy of motion extracted from the flow is less than or exceeds the energy dissipated by the system through mechanical damping The theoretical divid ing line between the decaying and the sustained sinusoidal oscillation due to an initial disturbance is then recognized as the critical flutter condition Section 211 considers twodimensional 2D bridge deck behavior in smooth flow Section 212 briefly reviews the expression for the aerodynamic lift and moment act ing on airfoils Section 213 introduces the aerodynamic lift drag and moment acting on bridge decks Section 214 concerns the solution of the flutter equations for bridges Section 215 discusses the bridge response to turbulent wind in the presence of aeroe lastic effects 211 Formulation of the TwoDimensional Bridge Flutter Problem in Smooth Flow In the 2D case the bridge deformations are the same throughout the bridge span Bridge decks are typically symmetrical that is their elastic and mass centers coincide The dependence of flutter derivatives upon the oscillation frequency n of the fluttering body can be expressed in terms of the nondimensional reduced frequency K 2𝜋BnU 211 where B is the width of the deck and U is the mean wind flow velocity If the horizon tal displacement p of the deck is also taken into account the equations of motion of a twodimensional section of a symmetrical bridge deck with linear mechanical damping and elastic restoring forces in smooth flow can be written as mh ch h khh Lae 212a I 𝛼 c𝛼 𝛼 k𝛼𝛼 Mae 212b mp cp p kpp Dae 212c where h 𝛼 and p are the vertical displacement torsional angle and horizontal displace ment of the bridge deck respectively see Figure 211 for notations pertaining to h and 𝛼 k k k k 212 Aeroelastic Lift and Moment Acting on Airfoils 307 α U ab b B2 h B Figure 211 Notations similar notations are applicable to airfoils A unit span is acted upon by the aeroelastic lift Lae moment Mae and drag Dae and has mass m mass moment of inertia I vertical torsional and horizontal restoring forces with stiffness kh k𝛼 and kp respectively and mechanical damping coefficients ch c𝛼 and cp 212 Aeroelastic Lift and Moment Acting on Airfoils It is instructive at this point to briefly consider the modeling of the aeroelastic lift Lh and moment M𝛼 acting on airfoils as shown in Figure 211 For airfoils p displacements are negligible Using basic principles of potential flow theory and an elegant mathematical technique involving conformal mapping Theodorsen showed that for small airfoil motions in incompressible flow the expressions for Lh and M𝛼 are linear in h and 𝛼 and their first and second derivatives 1 The coefficients in these expressions called aero dynamic coefficients are defined in terms of the complex function CK FK iGK known as Theodorsens circulation function Figure 212 in which K b 𝜔U is the reduced frequency b is the halfchord of the airfoil U is the flow velocity and 𝜔 is the circular frequency of oscillation Theodorsens theory yields the following expressions for the harmonically oscillating lift and moment Lae 𝜋𝜌b2U 𝛼 h ab𝛼 2𝜋𝜌UCK U𝛼 h b 1 2 a 𝛼 213a Mae 𝜋𝜌b2 1 2 a Ub 𝛼 b2 1 8 a2 𝛼 abh 2𝜋𝜌U 1 2 a b2CK U𝛼 h b 1 2 a 𝛼 213b where a is the constant defining the distance ab from the midchord to the rotation point 𝜌 is the air density It was shown in Section 2011 that a galloping body experiences a singledegreeof freedom motion and that for small displacements the aeroelastic force acting on the body is linear with respect to the time rate of change of the acrosswind displacement y the proportionality factor being a function of aerodynamic origin Airfoil flutter entails k k k k 308 21 Flutter 10 08 F G G 06 04 100 F 200 300 0 02 0 2π k U nb Figure 212 Functions FK and GK motions with two degrees of freedom h and 𝛼 and the expressions for the aeroelastic forces acting on the body are therefore more elaborate than in the galloping case although conceptually they are related Indeed for small displacements the aeroelastic forces can be written as sums of terms that like their galloping counterpart are linear with respect to the rates of change of h and 𝛼 the factors of proportionality being also functions of aerodynamic origin However unlike in the case of galloping terms proportional to 𝛼 come into play as well and the factors of proportionality depend upon the reduced frequency 213 Aeroelastic Lift Drag And Moment Acting on Bridge Decks By analogy with Theodorsens results empirical expressions were proposed for the aeroelastic forces acting on bridge decks of the type 26 Lae 1 2𝜌U2B KH 1K h U KH 2KB 𝛼 U K2H 3K𝛼 K2H 4K h B KH 5K p U K2H 6K p B 214a Mae 1 2𝜌U2B2 KA 1K h U KA 2KB 𝛼 U K2A 3K𝛼 K2A 4K h B KA 5K p U K2A 6K p B 214b Dae 1 2𝜌U2B KP 1K p U KP 2KB 𝛼 U K2P 3K𝛼 K2P 4KP B KP 5K h U K2P 6K h B 214c k k k k 213 Aeroelastic Lift Drag And Moment Acting on Bridge Decks 309 where K 2𝜋nBU and n is the oscillation frequency For bridges the elastic and mass centers coincide that is a 0 Terms proportional to h 𝛼 and p ie socalled added mass terms reflecting the forces due to the body motion that result in fluid accelerations around the body are negligible in bridge engineering applications and do not appear in Eqs 214ac The role of the terms in h and p is to account for changes in the frequency of vibration of the body due to aeroelastic effects while the terms in 𝛼 reflect the role of the angle of attack The quantities hU and B 𝛼U are effective angles of attack eg the ratio hU has the same significance as in the case of galloping ie it represents the angle of attack of the relative velocity of the flow with respect to the moving body In Eqs 214 the terms containing first derivatives of the displacements are measures of aerodynamic damping If among these terms only those associated with the coefficients 12 5 4 3 2 1 0 08 04 00 0 2 4 6 8 10 12 14 0 2 4 6 8 A4 A3 A2 A1 H4 H3 H2 H1 10 UnB UnB 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 UnB UnB 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 UnB UnB 12 14 0 2 4 6 8 10 12 14 0 2 4 6 8 10 UnB UnB NORMANDY GREAT BELT TSURUMI AIRFOIL EXPERIMENTAL 12 14 04 1 20 2 10 8 6 4 2 2 0 2 00 3 2 1 0 04 08 12 16 4 0 16 12 8 4 0 4 2 0 2 4 Figure 213 Aerodynamic coefficients H i and A i for bodies shown in Figure 214 Source After 8 with permission of Professor Partha Sarkar k k k k 310 21 Flutter H1 A2 and P1 are significant the total mechanical plus aerodynamic damping can be written as ch 12𝜌UBKH1 c𝛼 12𝜌UB3KA2 cp 12𝜌UBKP1 215abc for the vertical torsional and horizontal degree of freedom respectively The nondimensional coefficients Hi Ai and Pi are known as flutter derivatives1 Unlike in the galloping case where owing to the absence of significant vortexinduced pressures on the body the derivatives can be obtained experimentally from static tests that is tests in which the body is at rest for the flutter case the coefficients of the displacements and their time rate of change must be obtained experimentally from GREAT BELT EAST BRIDGE NORMANDY BRIDGE AIRFOIL TSURUMI FAIRWAY BRIDGE 31000 27000 22300 10220 17580 3660 34000 38000 3050 3469 4400 Figure 214 Box decks for three bridges dimensions in millimeters and airfoil Source After 8 with permission of Professor Partha Sarkar 1 Equations 214ac are formulated in terms of real variables viewed by some practitioners to be best suited for structural engineering purposes An alternative wherein the aeroelastic forces and the displacements they induce in the bridge are expressed in terms of complex variables is preferred by some practitioners insofar as it may offer insights into phase relationships among various aeroelastic forces and displacements see 7 k k k k 215 TwoDimensional Bridge Deck Response to Turbulent Wind in the Presence of Aeroelastic Effects 311 measurements on the oscillating deck which owing to its elongated shape is affected by vortexinduced pressures For this reason those coefficients are called motional aerodynamic derivatives as opposed to the steadystate aerodynamic derivatives that characterize the galloping phenomenon Figure 213 shows aerodynamic coefficients H i and A i for a thin airfoil and three streamlined box decks depicted in Figure 214 The original Tacoma Narrows bridge Figure III2 had negligible H1 values for all K meaning that the total damping Eqs 215abc for motion in the h direction was positive thus precluding flutter in the vertical degree of freedom The effect of hori zontal deck motions pt was not significant However A2 was positive for values of K such that for mean velocities greater than about 20 m s1 the total damping given by Eqs 215abc was negative resulting in flutter motions involving only the torsional degree of freedom The bridges susceptibility to flutter was due to the use of an H section the horizontal line in the H representing the deck and the vertical lines repre senting the girders supporting it Owing to their inherent instability H bridge sections are no longer used 214 Solution of the Flutter Equations for Bridges The solution of the flutter equations can be obtained if plots of the flutter derivatives Hi Ai and Pi are available from measurements as functions of K It is assumed that the expressions for h 𝛼 and p are proportional to ei𝜔t These expressions are inserted into Eqs 214 and the determinant of the amplitudes of h 𝛼 and p is set to zero as the basic stability solution For each value of K a complex equation in 𝜔 𝜔1 i𝜔2 is obtained The flutter velocity is the velocity for Uc for which 𝜔2 0 that is Uc B𝜔1 Kc 216 where Kc is the value of K for which 𝜔 𝜔1 A timedomain approach to the study of suspension bridge aeroelastic behavior is presented in 9 and 10 For a simplified approach to determining the critical flutter velocity see 11 215 TwoDimensional Bridge Deck Response to Turbulent Wind in the Presence of Aeroelastic Effects The expressions for the aeroelastic forces in the turbulent flow have the same form as for the smooth flow case Eqs 214 However the aerodynamic coefficients Hi Ai Pi should be obtained from measurements in turbulent flow since turbulence may affect the aerodynamics of the bridge deck by changing the configuration of the separation layers and the position of reattachment points Through complex aerodynamic mecha nisms turbulence can affect the flutter derivatives and therefore the flutter velocity in some instances favorably but possibly also unfavorably 12 k k k k 312 21 Flutter The buffeting forces per unit span may be written as follows Lb 1 2𝜌U2B 2CL ux t U dCL d𝛼 CD wx t U 217a Mb 1 2𝜌U2B2 2CM ux t U dCM d𝛼 wx t U 217b Db 1 2𝜌U2B 2CD ux t U 217c where B is the deck width and U ux t and wt are the wind speed components in the x alongwind and vertical directions respectively For example Eq 217c is derived from the expression for the total mean plus fluctuating drag force D D D Db 1 2𝜌CDBU ut2 1 2𝜌CDBU2 2Uut 218 where U is the mean flow velocity ut is the alongwind longitudinal component of the turbulent velocity fluctuation at time t and the mean drag force is defined as D 1 2𝜌U2BCD 219 the drag coefficient CD is measured in turbulent flow and the square of the ratio ux tU is neglected For the twodimensional case the solution of the buffeting problem in the presence of aeroelastic effects is obtained from Eqs 212 in the righthand sides of which the sums Lae Lb Mae Mb Dae Db are substituted respectively for Lae Mae and Dae as defined by Eqs 214 12 The twodimensional case can provide useful insights into the behavior of a bridge However to be useful in applications to actual bridges it is necessary to obtain the solution of the threedimensional case in which the bridge displacement and the aerodynamic forces are functions of position along the span This solution is considered in Chapter 23 References 1 Theodorsen T General theory of aerodynamic instability and the mechanism of flutter NACATR496 National Advisory Committee for Aeronautics Washington DC pp 2122 1949 2 Scanlan RH and Tomko JJ 1971 Airfoil and bridge deck flutter derivatives Journal of the Engineering Mechanics Division 97 17171737 3 Singh L Jones NP Scanlan RH and Lorendeaux O 1996 Identification of lateral flutter derivatives of bridge decks Journal of Wind Engineering and Industrial Aerodynamics 60 8189 4 Scanlan RH and Simiu E 2015 Aeroelasticity in civil engineering In A Modern Course in Aeroelasticity 5the ed EH Dowell 285347 Switzerland Springer 5 Simiu E and Scanlan RH 1996 Wind Effects on Structures 3rd ed Hoboken NJ Wiley 6 Gan Chowdhury A and Sarkar PP 2003 A new technique for identification of eighteen flutter derivatives using threedegreeoffreedom section model Engineer ing Structures 25 17631772 k k k k References 313 7 Simiu E and Miyata T 2006 Design of Buildings and Bridges for Wind A Practical Guide for ASCE7 Standard Users and Designers of Special Structures 1st ed Hoboken NJ Wiley 8 Sarkar P P New identification methods applied to response of flexible bridges to wind Doctoral dissertation Civil Engineering Johns Hopkins University Baltimore MD 1992 9 Caracoglia L and Jones NP 2003 Timedomain vs frequency domain character ization of aeroelastic forces for bridge deck sections Journal of Wind Engineering and Industrial Aerodynamics 91 371402 10 Cao B and Sarkar PP 2013 Timedomain aeroelastic loads and response of flex ible bridges in gusty wind Prediction and experimental validation ASCE Journal of Engineering Mechanics 139 359366 11 Bartoli G and Mannini C 2008 A simplified approach to bridge deck flutter Journal of Wind Engineering and Industrial Aerodynamics 96 229256 12 Simiu E Buffeting and aerodynamic stability of suspension bridges in turbulent wind Doctoral dissertation Civil Engineering Princeton University Princeton NJ 1971 k k k k 315 22 Slender Chimneys and Towers This chapter presents material that complements Chapter 19 on the response of towers and chimneys with circular cross section and allows the practical calculation of that response Section 221 it briefly discusses issues related to the aeroelastic response of slender structures with square or rectangular cross section Section 222 and describes methods of alleviating windinduced oscillations of slender structures Section 223 221 Slender Chimneys with Circular Cross Section 2211 Slender Chimneys Assumed to be Rigid In turbulent flow the nominal acrosswind response 𝜎nom y of a chimney is due to a super position of two acrosswind loads The first acrosswind load due to vortex shedding in the towers wake has the expression L1z t 1 2𝜌CLz tDzU2z 221 the notations from Chapter 19 are used in this section The spectral density of the lift force L1z t is SL1z n 1 2𝜌DzU2z 2 SCLz n 222 According to 1 measurements indicate that the spectral density SCLz n can be rep resented by the bellshaped function nSCLz n C2 L 1 𝜋Bns exp 1 nns B 2 223 where n denotes frequency ns is the vortexshedding frequency given by the relation ns St Uz Dz 224 St is the Strouhal number and B is an empirical parameter that determines the spread bandwidth of the spectral curve This model is consistent with results of fullscale mea surements Figure 221 The crossspectral density of the load L1z t can be expressed as 3 SL1z1 z2 n S12 L1 z1 nS12 L1 z2 nR0z1 z2 n 225 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 316 22 Slender Chimneys and Towers 001 102 101 D 1014 m 0183 ½ 1 01 10 nSCL n nD U C2 L C2 L Figure 221 Power spectral density of lift force coefficients CL measured on the Hamburg television tower Source Reprinted from 2 with permission from Elsevier where the coherence function is assumed to be R0z1 z2 n cos2ar expar2 226 r 2 z1 z2 Dz1 Dz2 227 The parameter a is a measure of the decay of the crossspectral function SL1z1 z2 n with the distance z1 z2 Associated with the parameter a is a correlation length l a measure of the spatial separation beyond which the force fluctuations are no longer correlated The second lift force denoted by L2t is the projection on the acrosswind direction of the drag force induced by the resultant of the mean velocity Uz and the lateral turbulent velocity vzt In largescale turbulence this force has an angle of attack with respect to the alongwind direction equal to vU and its projection on that direction is L2t 1 2𝜌CDU2zvz t Uz 228 The aerodynamic parameters depend upon the Reynolds number Rez 67000 Uz Dz 229 where Uz is the wind speed at elevation z in m s1 and Dz is the outside diameter in meters the turbulence in the oncoming flow the aspect ratio hDh where h is the height of the structure and the relative surface roughness kD of the structure where k is the height of the roughness elements For steel stacks and reinforced concrete chimneys and towers 105 kD 103 4 The dependence of CD upon Reynolds number and surface roughness is represented in Figure 422 for cylinders with aspect ratios hDh 20 For cylinders with aspect k k k k 221 Slender Chimneys with Circular Cross Section 317 ratios 10 hDh 20 it may be assumed that up to the elevation h Dh the drag coefficient has the value CD Cs D 1 0015 20 h Dh 2210 where Cs D is the value of the drag coefficient taken from Figure 422 From elevation h Dh to the top of the structure the drag coefficient may be assumed to have the value CD 14Cs D regardless of aspect ratio 5 The main effect of turbulence is to decrease the Reynolds number corresponding to the onset of the critical region defined in Figure 422 The following values of the Strouhal number are suggested in 5 St 020 Re 2 105 2211a 022 St 045 2 105 Re 2 106 2211b St c 023 0007 log10 k d 5 Re 2 106 2211c For 2 105 Re 2 106 the vortex shedding is random and the Strouhal number given by Eq 2211b corresponds to the predominant frequencies of the flow fluctuations in the wake In Eq 2211c the coefficient c depends upon the aspect ratio as follows c 100 h Dh 30 0736 0012 h Dh 80 8 h Dh 30 2212ab Note that the values given by Eq 2211b differ from those obtained in a more recent study shown in Figure 415 The following values of the rms lift coefficient are suggested for design purposes 5 C2 L 12 045 Re 2 105 014 2 105 Re 2 106 d 015 0035 5 log10 k D 2 Re 2 106 2213abc where d 100 h Dh 12 08 005 h Dh 80 8 h Dh 12 2214ab No information appears to be available on the dependence of the lift coefficient upon turbulence intensity It is suggested in 3 5 that B2 0082 2u2 U2 2215 where u2 is the mean square value of the longitudinal velocity fluctuations and U is the mean speed According to 6 it may be assumed B 018 for all flows For Re 2 105 it is suggested in 3 5 that a 13 see Eq 226 to which there corresponds a correlation length l D For Re 2 105 l 25D 7 k k k k 318 22 Slender Chimneys and Towers 2212 Flexible Slender Chimneys The mechanical damping ratios 𝜁i in the ith mode of vibration depend upon the type of structure Suggested values are as follows 4 Unlined steel stacks and similar structures 00020010 Lined steel stacks 00040016 Reinforced concrete chimneys and towers 00040020 The following approximate expressions are suggested in 3 5 for the aeroelastic parameter Ka0 Ka0 U Ucr 0 U Ucr 085 ai 35 U Ucr 295 085 U Ucr 10 055ai 10 U Ucr 11 ai 275 2 U Ucr 11 U Ucr 13 ai 046 025 U Ucr 13 U Ucr 184 0 184 U Ucr 2216abcdef where ai a1 a2 a3 a4 2217 a1 10 Re 104 18 104 Re 105 10 Re 105 2218abc a2 20 U10 m 12 ms1 10 U10 m 12 ms1 2219ab a3 09 02 log10kD 5 2220 a4 10 h Dh 125 10 004 125 h Dh h Dh 125 2221ab where Ucr nDSt see Section 193 and Figure 199 2213 Approximate Expressions for the AcrossWind Response The acrosswind response in the ith mode of vibration may be estimated as 𝜎yiz 𝜉2 i 12 yiz 2222 Yiz gyi𝜎yiz 2223 gyi 2 ln3600 ni 1 2 0577 2 ln3600 ni 1 2 2224 k k k k 221 Slender Chimneys with Circular Cross Section 319 𝜉2 i 12 𝜉2 nomi 12 𝜁i 𝜁i 𝜁ai 12 2225 Siz 2𝜋ni2 h z mz1Yiz1dz1 2226 iz 2𝜋ni2 h z mz1Yiz1z1 zdz1 2227 where 𝜎yiz is the rms of the deflection at elevation z in the ith mode of vibration 𝜉2 i 12 is the rms of the corresponding generalized coordinate yiz is the ith modal shape Y iz is the peak deflection in the ith mode of vibration gyi is the peak factor ni is the natural frequency in the ith mode in Hz 𝜉2 nomi 12 is the rms nominal generalized coor dinate in the ith mode which corresponds to the response estimated by assuming that no aeroelastic effects occur and that the motion is affected only by mechanical damping in the ith mode 𝜁i is the structural damping in the ith mode 𝜁ai is the aerodynamic damping in the ith mode Siz and iz are the shear force and the bending moment at elevation z due to the acrosswind response in the ith mode and mz is the mass of the structure per unit length Note that for the ith mode the ratio of peak acceleration to peak deflection is approximately 2𝜋ni2 see Eq B16b To estimate the acrosswind response expressions are needed for the rms of the nom inal generalized coordinate in the ith mode 𝜉2 nomi 12 and the aerodynamic damping in the ith mode 𝜁ai These expressions are given next for i structures with constant cross section and ii tapered structures In both cases the expressions are valid only for relatively small ratios 𝜎yih Dh for example 3 or less to which there would correspond negligible values of the second term within the bracket of Eq 1918 In practice the design of the structure is acceptable only if the ratio 𝜎yih Dh is small Structures with Constant Cross Section The following approximate expressions based on the approach described in Section 2211 were proposed in 6 𝜉2 nomi 12 0035C2 L 12 lD12 𝜁12 i St2 𝜌D3 Mi D h 0 y2 i zdz 12 2228 𝜁ai 𝜌D2 Mi Ka01 h 0 y2 i zdz 2229 where 𝜌 125 kg m3 is the air density Mi is the generalized mass in the ith mode l is the correlation length see Section 2211 and D is the outside diameter The critical wind speed corresponding to the ith mode of vibration has the expression Ucri niD St 2230 Information on the mechanical damping ratios 𝜁i is given in Section 2212 Information on the parameters St C2 L and Ka0 is given in Sections 2211 and 2212 In Eqs 2219ab the speed U10 m corresponding to the ith mode is U10 m ln 10 z0 ln 5 6 h z0 Ucri 2231 k k k k 320 22 Slender Chimneys and Towers where h is the height of the structure in meters and z0 is the roughness length in meters for the terrain that determines the wind profile over the upper half of the chimney Table 21 Example 221 Consider a chimney with h 1936 m D 1763 m n1 0364 Hz y1zh zh167 mz 58000 kg m1 for z h2 mz 41000 kg m1 for z h2 M1 187 106 kg It is assumed 𝜁1 002 kD 105 and z0 005 m We seek the response in the first mode Assuming tentatively St 022 the critical wind speed at elevation 5 h6 1613 m is Ucr 1 292 m s1 Eq 2230 to which there corresponds a Reynolds number Re 34 107 2 106 The aspect ratio is hD 11 It can be verified that St 0178 Eqs 2211c and 2212b l D since Re 2 105 C2 L 12 0143 Eqs 2213abc and 2214ab h 0 y2 1zdz 447 m 𝜉2 nom1 12 0115 m Eq 2228 U10 191 m s1 12 m s1 Eqs 2230 and 2231 Ka0 1 0465 Eqs 22162221 𝜁a1 00043 Eq 2229 𝜉2 1 12 0130 m Eq 2225 gy1 394 Eq 2224 𝜎y1z 0130 z 1936 167 m Eq 2222 Y 1z 051 z 1936 167 m Eq 2223 10 1150 106 Nm Eq 2227 The results of the calculations depend strongly upon in particular the assumed value of the structural damping ratio 𝜁1 Had the value 𝜁1 001 been appropriate the results obtained would have been larger than those obtained in this example by a factor of 002 00043001 0004312 166 Eq 2225 Tapered Structures The following approximate expressions based on the approach described in Section 2211 were proposed in 6 𝜉2 nomizei 12 0016C2 L 12 l D 12 𝜌D4zeiyizei 𝜁12 i St2Mi𝛽12zei 2232 𝛽zei 01Dzei zei dDz dz zzei 2233 𝜁aizei 𝜌D2 0 Mi h 0 Ka0 Uz Ucrzei Dz D0 2 y2 i zdz 2234 k k k k 222 Aeroelastic Response of Slender Structures with Square and Rectangular Cross Section 321 where the notations of Eq 2228 are used D0 is the outside diameter at the base zei is the elevation corresponding to the critical velocity Ucrzei niDzei St 2235 Uz Ucrzei lnzz0 lnzeiz0 2236 and z0 is the terrain roughness that determines the wind profile over the upper half of the structure Since as in Eq 2225 𝜉2 i zei 12 𝜉2 nomizei 12 𝜁i 𝜁i 𝜁aizei 12 2237 it follows that the maximum response in the ith mode corresponds to the maximum value taken on by the function Fizei D4zeiyizei 𝛽zei𝜁i 𝜁aizei12 2238 To determine that value it is in practice necessary to calculate Fizei and in particular 𝜁aizei for a sufficient number of elevations 0 zei h As pointed out in 8 if the structure is very lightly tapered ie if dDz dz zzei and there fore 𝛽ze1 is small see Eq 2233 the chimney is assumed to behave as if it had a constant outside diameter D equal to the average diameter of its top third 6 and Eqs 22282230 are applied with the same values of the parameters St C2 L 12 and cor relation length D or for Re 105 25 D as those used in Eq 2232 In practice it is therefore necessary to calculate both the value of the response yielded by Eqs 2232 and 2234 and the value yielded by Eqs 2228 and 2229 The response to be assumed for structural design purposes is the smaller of these two values 1 222 Aeroelastic Response of Slender Structures with Square and Rectangular Cross Section AlongWind Aeroelastic Response Aerodynamic damping in tall buildings results from the interaction between the fluctuating aerodynamic forces acting on the building and the fluctuating building motions they induce Since the aerodynamic damping is due to the building motion it affects in the most general case the alongwind acrosswind and torsional motions In this section attention is restricted to the aerodynamic damping affecting the alongwind motion of an isolated building with a rectangular shape in plan The aerodynamic alongwind force depends upon the relative wind speed with respect to the moving structure If the structure is sufficiently rigid it experiences no significant motion and the relative wind speed with respect to the structure is equal in practice to the oncoming wind speed However if the structure is flexible its motions can be significant and the relative wind speed with respect to the structure is equal to the k k k k 322 22 Slender Chimneys and Towers timedependent difference between the oncoming fluctuating speed and the speed of the moving structure The procedure for estimating the aerodynamic damping presented in this section was developed in 9 The displacement at elevation z is written as xz t N i1 𝜙iz𝜉it 2239 where N number of normal modes being considered and 𝜉it and 𝜙iz generalized coordinate and modal shapes corresponding to the ith normal mode of vibration respectively In the ith modal equation of motion the generalized force is Qit L l1 𝜙izlFlzl t 2240 where L total number of taps on the windward and leeward faces mz mass distri bution and Fl zl t excitation force associated with tap l at elevation zl The force Flzl t can be written as Flzl t 1 2𝜌Uzl uzl t xzl t2ClzlAl 2241 where Uzl and uzl are the mean and the fluctuating wind speed at elevation zl xzl t is the timedependent alongwind displacement of the building at elevation zl Cl zl is the mean pressure coefficient at zl and Al is the tributary area of tap l Equation 2241 may be interpreted as follows The aerodynamic damping depends upon the degree to which the fluctuating excitation of the structure is in phase or out of phase with the windinduced velocity x If the excitation and the velocity are in phase the relative fluc tuating velocity uzl t xzl t is lower than the fluctuating velocity uzl t meaning that the fluctuating response of the structure will decrease in other terms the aerody namic damping will be positive The opposite is true if the excitation and the building velocity are in opposite phases In applying Eqs 22392241 an iterative procedure is used The force Fl is calcu lated first by neglecting the speeds xzl t The resulting equation of motion is used to calculate a first approximation to those speeds This approximation is then used in Eq 2241 and the corresponding equation of motion to obtain a second approximation to xzl t This process continues until the nth and the n 1th approximations differ insignificantly The aerodynamic damping value was obtained by a trialanderror procedure where successive total damping ratios were input in the databaseassisted design software described in Chapter 18 until the resulting rms displacements were approximately equal to the displacements calculated by the iterative procedure just described For a 60story building with dimensions 457 305 m in plan and height H 185 m and mean wind speed normal to the buildings wider face the procedure described in 9 and summarized in this section yielded values of the aerodynamic damping that were positive larger as the reduced wind increased weakly dependent upon the modal shapes and negligible for practical purposes even for mean wind speeds at the top of the building as high as 70 m s1 The results obtained for this case were comparable for practical purposes to those obtained in wind tunnel tests Figure 222a 10 k k k k 222 Aeroelastic Response of Slender Structures with Square and Rectangular Cross Section 323 In view of the uncertainties associated with the estimation of the alongwind aerodynamic damping it is prudent to neglect its favorable effect on the alongwind response AcrossWind Aeroelastic Response Unlike for alongwind response no practical analytical approach is available for the estimation of acrosswind response Based on wind tunnel test results Figure 222b shows that for sufficiently high reduced velocities the aerodynamic damping can be negative ie destabilizing although this is not the case for ratios DB sufficiently larger than unity For details see 10 which notes that the wind velocities for which the acrosswind aerodynamic damping becomes negative are much lower than the wind velocities at which galloping oscillations can occur a Alongwind response 2 DB 1 DB 05 DB 033 quasisteady theory DB 1 DB 2 DB 3 quasisteady theory DB 033CD 090 DB 05CD 102 DB 1CD 105 DB 1CD 105 DB 2CD 075 DB 3CD 070 D B D B 1 0 0 1 2 3 4 5 6 reduced wind velocity aerodynamic damping ratio ζa X aerodynamic damping ratio ζa X a DB 1 7 8 9 10 11 12 UH BD no 1 2 1 0 1 0 1 2 3 4 5 6 reduced wind velocity b DB 1 7 8 9 10 11 12 UH BD no Figure 222 Aerodynamic damping as a function of reduced wind velocity and side ratio DB 1 mechanical aerodynamic damping Source Reprinted from 10 with permission from Elsevier k k k k 324 22 Slender Chimneys and Towers b Acrosswind response DB 1 DB 2 DB 3 DB 1 DB 05 DB 033 D B D B aerodynamic damping ratio ζaY aerodynamic damping ratio ζaY 2 1 0 1 2 1 0 1 0 1 2 3 4 5 6 reduced wind velocity b DB 1 7 8 9 10 11 12 0 1 2 3 4 5 6 reduced wind velocity a DB 1 7 8 9 10 11 12 UH BD no UH BD no Figure 222 Continued A schematic of the simple experimental setup used to obtain the results reported in 10 is shown in Figure 223 For the flexible structures with square cross section tested in 10 a sufficient condition assuring adequacy of the design from an aeroelastic point of view is that the wind speeds that may be expected during the life of the structure be lower than the lowest speed denoted by Ul which induces acrosswind resonant oscillations This statement is consistent with the test results of Figure 222b which show that negative aerodynamic damping occurs at wind speeds higher than Ul The necessary condition k k k k 223 Alleviation of VortexInduced Oscillations 325 sand laser displacement transducers floor gimbal model balsa wind velocity coil spring oil damper Figure 223 Experimental setup for tests reported in 10 Source Reprinted from 10 with permission from Elsevier for galloping Eq 2014 modified to account for shear turbulent flow should also be considered A similar approach may be employed for structures with rectangular shapes in plan For additional material on aerodynamic damping see for example 11 223 Alleviation of VortexInduced Oscillations A common method of alleviating vortexinduced oscillations is the provision of spoiler devices that destroy or reduce the coherence of shed vortices 12 13 The helical strake system first proposed by Scruton 14 consists of three rectangular strakes with a pitch of one revolution in five diameters and a strake radial height of 010 m diameter to 013 m diameter for very lightly damped structures applied over the top 3340 of the stack height The effectiveness of the system is not impaired by a gap of 0005D between the strake and the cylinder surface 15 Reference 16 reports k k k k 326 22 Slender Chimneys and Towers Figure 224 Steel chimney with helical strakes Source Reprinted from 16 with permission from Elsevier the remarkable results obtained by using this system with 5mm thick strakes 06m strake height and 30m pitch in the case of a 145m all and 6m diameter steel stack Figure 224 For Reynolds numbers Re 2 105 or so in flow with about 15 turbulence intensity helical strakes were found to reduce the peak of the acrosswind resonant oscillations by a factor of about two as opposed to a factor of about 100 in smooth flow 17 It appears that the performance of strakes can be unsatisfactory in the case of stacks grouped in a row 18 Also wind tunnel tests indicate that for large vibration amplitudes eg 35 of the diameter the vortex street reestablishes itself and the aerodynamic devices become ineffective 19 It is noted that strakes increase drag as shown in Figure 225 Shrouds were also found to be effective in reducing the coherence of shed vortices A schematic view of a shroud fitted to a stack is shown in Figure 226 Results of wind tun nel experiments reported in 15 20 showed that oscillations were substantially reduced with only the 25 of the model height shrouded The most effective shrouds were found to be those with a gap width w 012D and an openarea ratio between 20 and 36 with length of square s 0052D to 0070D Improvements in the behavior of the structure under wind loads can be achieved by using tuned mass dampers see Chapter 16 and similar devices andor by increasing the mechanical damping and affecting the aerodynamic response of the structure by designing buildings with chamfered corners see 21 tapered shapes andor discon tinuous changes of shape An aerodynamic device used in the design of the New York City 85floor 425 m tall 432 Park Avenue building consists of leaving the mechanical floors open to allow air to pass through the building thus disrupting the vorticity shed in the buildings wake k k k k References 327 Plain cylinder Plain cylinder Reynolds number ℛℯ T D 15 10 05 Drag coefficient CD based on cylinder diameter 105 106 107 Cylinder with strakes TD 012 TD 006 Figure 225 Effect of strakes on drag coefficient Source From 20 Courtesy of National Physical Laboratory UK Figure 226 View of a shroud fitted to a stack Source After 15 s L l D Gap w References 1 Vickery BJ and Clark AW 1972 Lift or acrosswind response of tapered stacks Journal of the Structural Division 98 120 2 Ruscheweyh H 1976 Wind loading on the television tower Hamburg Germany Journal of Wind Engineering and Industrial Aerodynamics 1 315333 k k k k 328 22 Slender Chimneys and Towers 3 Vickery BJ and Basu RI 1983 Acrosswind vibrations of structures of circular crosssection Part I Development of a mathematical model for twodimensional conditions Journal of Wind Engineering and Industrial Aerodynamics 12 4973 4 Basu R I and Vickery B J A comparison of model and fullscale behavior in wind of towers and chimneys in Wind Tunnel Modeling for Civil Engineering Applications Proceedings of the International Workshop on Wind Tunnel Modeling Criteria and Techniques in Civil Engineering Applications Gaithersburg MD USA April 1982 T A Reinhold ed 1st ed Cambridge UK Cambridge University Press 1982 5 Basu R I AcrossWind Response of Slender Structures of Circular Cross Section to Atmospheric Turbulence Research Report BLWT23983 University of Western Ontario London Ontario Canada 1983 6 Vickery BJ and Basu R 1983 Simplified approaches to the evaluation of the acrosswind response of chimneys Journal of Wind Engineering and Industrial Aerodynamics 14 153166 7 Davenport AG and Novak M 2002 Vibrations of structures induced by Wind Chapter 29 Part II In Harris Shock and Vibration Handbook 5th ed ed CM Harris and AG Piersol 29212946 New York McGrawHill 8 Vickery B J The aeroelastic modeling of chimneys and towers in Wind Tun nel Modeling for Civil Engineering Applications Proceedings of the International Workshop on Wind Tunnel Modeling Criteria and Techniques in Civil Engineer ing Applications Gaithersburg MD USA April 1982 T A Reinhold ed 1st ed Cambridge UK Cambridge University Press 1982 9 Gabbai R and Simiu E 2010 Aerodynamic damping in the alongwind response of tall buildings Journal of Structural Engineering 136 117119 10 Marukawa H Kato N Fujii K and Tamura Y 1996 Experimental evaluation of aerodynamic damping of tall buildings Journal of Wind Engineering and Industrial Aerodynamics 59 177190 11 Kareem A and Gurley K 1996 Damping in structures its evaluation and treat ment of uncertainty Journal of Wind Engineering and Industrial Aerodynamics 59 131157 12 Zdravkovich MM 1981 Review and classification of various aerodynamic and hydrodynamic means for suppressing vortex shedding Journal of Wind Engineering and Industrial Aerodynamics 7 145189 13 Zdravkovich MM 1984 Reduction of effectiveness of means for suppressing windinduced oscillation Engineering Structures 6 344349 14 Scruton C Note on a device for the suppression of the vortexinduced oscillations of flexible structures of circular or near circular section with special reference to its applications to tall stacks NPL Aero Report No 1012 National Physical Laboratory Teddington UK 1963 15 Walshe DE and Wooton LR 1970 Preventing windinduced oscillations of structures of circular section Proceedings of the Institution of Civil Engineers 47 124 16 Hirsch G and Ruscheweyh H 1975 Fullscale measurements on steel chimney stacks Journal of Wind Engineering and Industrial Aerodynamics 1 341347 k k k k References 329 17 Gartshore I S Khanna J and Laccinole S The Effectiveness of Vortex Spoilers on a Circular Cylinder In Smooth and Turbulent Flow in Wind Engineering Pro ceedings of the Fifth International Conference Fort Collins Colorado USA July 1979 J E Cermak ed Pergamon 1980 pp 13711379 18 Ruscheweyh H 1981 Straked inline steel stacks with low massdamping parame ter Journal of Wind Engineering and Industrial Aerodynamics 8 203210 19 Ruscheweyh H 1994 Vortex excited vibrations In WindExcited Vibrations of Structures ed H Sockel 5184 WeinNew York SpringerVerlag 20 Wooton LR and Scruton C 1971 Aerodynamic stability In Modern Design of WindSensitive Structures 6581 London UK Construction Industry Research and Information Association 21 Simiu E and Miyata T 2006 Design of Buildings and Bridges for Wind A Prac tical Guide for ASCE7 Standard Users and Designers of Special Structures 1st ed Hoboken NJ Wiley k k k k 331 23 SuspendedSpan Bridges 231 Introduction Suspendedspan ie suspension and cablestayed bridges must withstand drag forces induced by the mean wind In addition they may experience aeroelastic effects which may include vortexinduced oscillations Chapter 19 flutter and buffeting in the presence of selfexcited forces Chapter 21 The study of these effects is possible only on the basis of information provided by wind tunnel tests Various types of such tests are described in Section 232 Vortexinduced vibrations of bridge decks are considered in Section 233 Section 234 is concerned with bridge buffeting in the presence of aeroelastic effects Vibrations occurring in cables of cablestayed bridges are discussed in Section 235 The action of wind must be taken into account not only for the completed bridge but also for the bridge in the construction stage In general the same methods of testing and analysis apply in both cases To decrease the vulnerability of the partially completed bridge to wind temporary ties and damping devices are used 1 2 Also to minimize the risk of strong wind loading construction usually takes place in seasons with low probabilities of occurrence of severe storms In addition to the deck and stay cables aeroelastic phenomena may affect the bridges tower and hangers on which detailed material is available in 1 232 Wind Tunnel Testing The following three types of wind tunnel tests are commonly used to obtain information on the aerodynamic behavior of suspendedspan bridges 1 Tests on models of the full bridge In addition to being geometrically similar to the full bridge such models must satisfy similarity requirements pertaining to mass dis tribution reduced frequency mechanical damping and shapes of vibration modes see Chapter 5 The construction of full bridge models is therefore elaborate and their cost is high Their usual scale is in the order of 1300 A view of a fullscale bridge model in a large specially built wind tunnel is shown in Figure 231 2 Tests on threedimensional partialbridge models In such models the main span or half of the main span is reproduced in the laboratory Typically a support struc ture consisting of taut wires or a catenary supports the simulated deck The model is typically immersed in a simulated boundarylayer flow Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 332 23 SuspendedSpan Bridges Figure 231 Model of Akashi Strait suspension bridge Source Courtesy of T Miyata of Yokohama University and M Kitagawa HonshuShikoku Bridge Authority Tokyo 3 Tests on section models Section models consist of representative spanwise sections of the deck constructed to scale with spring supports at the ends to allow both vertical and torsional motion The model is provided with end plates or other devices that reduce aerodynamic end effects Figure 232 Section models are relatively inexpensive and are built to scales in the order of 1 501 25 They are useful for performing initial assessments of a bridge decks aeroelastic stability and allow the measurement of the fundamental aerodynamic characteristics of the bridge deck on the basis of which comprehensive analytical studies can be carried out These characteristics include a The steadystate drag lift and moment coefficient defined as CD D 1 2𝜌U2B 231a CL L 1 2𝜌U2B 231b CM M 1 2𝜌U2B2 231c where D L and M are the mean drag lift and moment per unit span respectively 𝜌 is the air density B is the deck width and U is the mean wind speed in the k k k k 232 Wind Tunnel Testing 333 Figure 232 Section model of the Halifax Narrows Bridge Source Courtesy of BoundaryLayer Wind Tunnel Laboratory University of Western Ontario 0 4 α deg α deg α deg CD CL CM 4 8 12 30 20 10 10 0 20 30 40 02 08 0 06 04 02 02 04 06 08 08 16 24 32 30 20 10 10 20 B 30 D M 0 0 03 04 05 8 12 16 Figure 233 Drag lift and aerodynamic moment coefficients for replacement Tacoma Narrows Bridge 3 k k k k 334 23 SuspendedSpan Bridges oncoming flow at the deck elevation The aerodynamic coefficients are usually plotted as functions of the angle 𝛼 between the horizontal plane and the plane of the bridge deck Coefficients CD CL and CM are shown in Figure 233 for the opentruss bridge deck of the replacement Tacoma Narrows Bridge 3 and in Figure 234 for a proposed streamlined box section of the New Burrard Inlet Crossing 4 00 Handrails guardrails No handrails no guardrails 120 80 40 00 40 80 30 20 10 00 10 20 30 30 20 10 00 10 20 30 120 100 100 00 CD CL CM 100 100 00 100 100 α α α Figure 234 Drag lift and aerodynamic moment coefficients for proposed deck of New Burrard Inlet Crossing 4 Source Courtesy of National Aeronautical Establishment National Research Council of Canada k k k k 233 Response to Vortex Shedding 335 b The motional aerodynamic coefficients These coefficients characterize the selfexcited forces acting on the oscillating bridge and are discussed in Section 213 Examples of motional aerodynamic coefficients first introduced for bridge decks by Scanlan and Tomko in 1971 5 are given in Figure 213 c The Strouhal number St For details on a type of test that allows alongwind motion of the model see 6 233 Response to Vortex Shedding Open truss sections generally shred the oncoming flow to such an extent that large coherent vortices cannot occur and vortexinduced oscillations of the deck are weak However severe vortexinduced oscillations of bluff deck sections of the box type are known to have occurred A soffit plate and fairings with various dimensions were added to the original section with the results shown in Figure 235 7 The water surface was in this case close to the underside of the bridge and was modeled in the wind tunnel tests Additional shapes of streamlined bridge deck forms are shown in Figure 236 For additional material on remedial aerodynamic measures see 1 15 Velocity ms Amplitude cm 10 0 5 10 5 20 30 m fairings 24 m fairings 18 m fairings Soffit plate Basic bridge Figure 235 Vertical amplitudes of vortexinduced oscillations for various bridge deck sections of the proposed Long Creeks Bridge Source Courtesy of National Research Council Canada k k k k 336 23 SuspendedSpan Bridges Figure 236 Streamlined bridge deck forms We now present an approach to the estimation of the bridge vortexinduced response 8 Let the equation of motion of the section with unit spanwise length have the form similar to the simplified version of Eq 194 mh 2𝜁h𝜔h h 𝜔2 hh 1 2𝜌U2BKH 1 1 𝜀 h2 B2 h U 232 where K 𝜔BU and H 1 𝜀 are aerodynamic parameters obtainable experimentally Derivations similar to those of Section 192 yield the nondimensionalized steadystate amplitude h0 of the bridge deck section model h0 B 2 H 1 4Scr 𝜀H 1 12 233 where Scr 𝜁hm𝜌B2 is the Scruton number The coefficient H 1 may be viewed as the value obtained at low oscillation amplitudes by any one of the several identifica tion schemes employed to obtain flutter derivatives If the steadystate vortexinduced amplitude h0 is measured in a section model test then 𝜀 4 H 1 4Scr h0B2H 1 234 However if H 1 is not obtained from a lowamplitude model test but instead the model is allowed to oscillate down from an initial larger amplitude A0 to a steady lockedin state of measured amplitude h0 the value of H 1 may be determined from KH 1 m 2𝜌B2 𝛼 h2 0 B2 16𝜋𝜁hSt 235 where K 2𝜋 St St is the Strouhal number 𝛼 4StB2 nh2 0 ln A2 0 R2 nh2 0 A2 0 h2 0 236 and Rn is the ratio of the response amplitudes of the first to the nth cycle of amplitude decay 8 k k k k 233 Response to Vortex Shedding 337 The information given in Eq 233 is applicable to the section model only To extrapolate it to the full bridge it is necessary to consider the oscillatory structural mode involved usually a simple lowfrequency mode as well as the spanwise correlation of the lockin forces In Eq 232 it is therefore further assumed that hx t 𝜙xB𝜉t 237 where 𝜙x is the dimensionless modal shape associated with the frequency 𝜔h of the deck excited by the lockedin vorticity The corresponding generalized coordinate 𝜉t is assumed to undergo purely sinusoidal oscillations 𝜉t 𝜉0 cos 𝜔t 238 at the Strouhal frequency that is where 𝜔 2𝜋StU B 𝜔h 239 If h from Eq 237 is inserted into Eq 232 and the result is multiplied by B 𝜙x the action of the section dx of the structure associated with the spanwise point with coordinate x is described by the equation mxB2𝜙2x 𝜉t 2𝜁h𝜔h 𝜉t 𝜔2 h𝜉tdx 1 2𝜌UB3KH 11 𝜀𝜙2x𝜉2t𝜙2x 𝜉tf xdx 2310 in which f x is a function inserted to account for spanwise loss of coherence in the vortexrelated forces If integration of Eq 2310 is performed over the full bridge span there results I 𝜉t 2𝜁h𝜔h 𝜉t 𝜔2 h𝜉t 1 2𝜌UB3LKH 1C2 𝜀C3𝜉2t 𝜉t 2311 where I is the generalized fullbridge inertia of the mode in question and C2 span 𝜙2xf xdx L 2312 C4 span 𝜙4xf xdx L 2313 The strength of the vortexinduced forces is dependent upon the local oscillation amplitude of the structure There is also a loss of coherence with spanwise separation For example Figure 192 shows the correlations between local lateral pressures sepa rated spanwise along cylinders displaced sinusoidally in the vertical direction with vari ous amplitudes It is suggested that the correlation loss can be approximated by selecting f x to be the mode shape 𝜑x normalized to unit value at its highest point For example with a mode representing a halfsinusoid over a span L f x may be estimated as f x sin𝜋xL 2314 At steadystate amplitude as noted earlier the damping energy balance per cycle of oscillation will be zero a condition that defines the vortexinduced amplitude 𝜉0 2 C2H 1 4Scr 𝜀C4H 1 12 2315 k k k k 338 23 SuspendedSpan Bridges where the Scruton number is defined as Scr 𝜁hI 𝜌B4L 2316 For the case of a sinusoidal mode the values of C2 and C4 respectively are C2 L 0 sin3 𝜋x L dx L 04244 2317a C4 L 0 sin5 𝜋x L dx L 03395 2317b For a study of conditions for the occurrence of vortex shedding on a large cable stayed bridge based on fullscale data obtained by a monitoring system see 9 234 Flutter and Buffeting of the FullSpan Bridge 2341 Theory The flutter phenomenon was studied in some detail in Chapter 21 under the assumption that twodimensional geometrical conditions hold For a fullspan bridge the deforma tions of the deck are functions of position along the span so that this assumption is no longer valid This section presents a generalization of the results of Chapter 21 to the case of a fullspan bridge An example is included Let hx t px t and αx t represent respectively the vertical sway and twist deflec tions of a reference spanwise point x of the deck of a full bridge hx t N i1 hixB𝜉it 2318a 𝛼x t N i1 𝛼ix𝜉it 2318b px t N i1 pix𝜉it 2318c where hix 𝛼ix pix are respectively the values of the ith modal shape at point x of the deck and 𝜉it is the generalized coordinate of the ith mode If Ii is the generalized inertia of the full bridge in mode i the equation of motion for that mode is Ii 𝜉i 2𝜁i𝜔i 𝜉i 𝜔2 i 𝜉i Qi 2319 where 𝜁i 𝜔i are the mechanical damping ratio and the circular natural frequency in radians of the ith mode respectively and Qi deck Lae LbhiB Dae DbpiB Mae Mb𝛼idx 2320 is the generalized force in the ith mode of vibration The subscripts ae and b signify aeroelastic and buffeting respectively It is assumed that the following definitions of forces per unit span at section x hold k k k k 234 Flutter and Buffeting of the FullSpan Bridge 339 Aeroelastic selfexcited forces under sinusoidal motion Lae 1 2𝜌U2B KH 1K h U KH 2KB 𝛼 U K2H 3K𝛼 K2H 4K h B 2321a Mae 1 2𝜌U2B2 KA 1K h U KA 2KB 𝛼 U K2A 3K𝛼 K2A 4K h B 2321b Dae 1 2𝜌U2B KP 1K p U KP 2KB 𝛼 U K2P 3K𝛼 K2P 4K p B 2321c Buffeting forces Lb 1 2𝜌U2B 2CL ux t U dCL d𝛼 CD wx t U 2322a Mb 1 2𝜌U2B2 2CM ux t U dCM d𝛼 wx t U 2322b Db 1 2𝜌U2B 2CD ux t U 2322c In Eqs 2321 and 2322 it is assumed that there is no interaction between the aeroe lastic and the buffeting forces However the interaction is implicit in Eq 2322 if the aeroelastic forces are measured in turbulent flow see eg 10 12 In what follows only a singlemode approximation to the total response will be pos tulated This is justifiable by the observation that typically just one prominent mode will become unstable and dominate the flutter response of a threedimensional bridge model in the wind tunnel In this singlemode form of analysis any mode i may be con sidered in Eqs 23182322 When all but the flutter derivatives shown in Eq 2321 are considered as being of lesser importance the expression for the generalized force is Qi 1 2𝜌U2B2l KB U H 1Ghihi P 1Gpipi A 2G𝛼i𝛼i 𝜉 K2A 3G𝛼i𝛼i𝜉i deck LbhiB DbpiB Mb𝛼idx 2323 in which Gqiqi deck q2 i xdx l qi hi pi or 𝛼i 2324 and l is the span length Because of the linearity of the resulting equation of motion the conditions of system stability are independent of the buffeting forces Equation 2319 may be rewritten with a new frequency 𝜔i0 a new damping ratio 𝛾i and a buffeting force Qib as follows 𝜉i 2𝛾i𝜔i0 𝜉i 𝜔2 i0𝜉i Qibt Ii 2325 where 𝜔2 i0 𝜔2 i 𝜌B4l 2Ii 𝜔2A 3G𝛼i𝛼i 2326 2𝛾𝜔i0 2𝜁i𝜔i 𝜌B4l 2Ii 𝜔H 1Ghihi P 1Gpipi A 2G𝛼i𝛼i 2327 k k k k 340 23 SuspendedSpan Bridges Qibt 1 2𝜌U2B2l deck LbhiB DbpiB Mb𝛼idx 2328 Flutter For instability to occur it is necessary that the damping ratio 𝛾i 0 This leads to the singlemode flutter instability criterion H 1Ghihi P 1Gpipi A 2G𝛼i𝛼i 4𝜁iIi 𝜌B4l 1 𝜌B4l 2Ii A 3G𝛼i𝛼i 12 2329 in which the only significant flutter derivatives H 1 P 1 A 2 A 3 have been retained An assumption inherent in this criterion is that the flutter derivatives retain full coherence throughout the deck span The effect of the reduced coherence can be seen in a reduction of the values of the quantities Gqiqi In practice the flutter derivatives H 1 and P 1 are typically negative while A 2 may take on positive values for sufficiently large values of the reduced velocity UnB The effect of the flutter derivative A 3 an aerodynamic stiffness effect is in many practical cases negligible since the structural stiffness is typically considerably larger than the aerodynamic stiffness Buffeting The generalized force may be written as Qibt Ii 𝜌U2B2l 2Ii deck Lhi Dpi M𝛼idx l 2330 where L M and D are respectively the quantities between brackets in Eqs 2322a b and c Defining the functions 𝜑x 2CLhix CDpix CM𝛼ix 2331a 𝜓x dCL d𝛼 CD hix dCM d𝛼 𝛼ix 2331b the integrand of Eq 2330 becomes Lhi Dpi M𝛼i 𝜑xux t U 𝜓xwx t U 2332 Information on the turbulent flow fluctuations u and w is available in the form of spec tral densities Sun and Swn respectively see Chapter 2 This motivates the adoption of a frequency domain approach to the solution of Eq 2325 It is shown in 11 that the frequency domain counterpart of Eq 2325 yields the result S𝜉i𝜉i𝜔 𝜌U2B2l2Ii2 𝜔4 i01 𝜔𝜔i022 2𝛾i𝜔𝜔i02 deck 1 U2 𝜑xa𝜑xbSuu xa xb 𝜔 𝜓xa𝜓xbSww xa xb 𝜔 dxa l dxb l 2333 In Eq 2333 the effect of the crossspectra of the fluctuations u and w has been neglected The distributed cross power spectral densities are assumed to take the real forms neglecting their imaginary components Suuxa xb 𝜔 Su𝜔 exp Cu xa xb l 2334a Swwxa xb 𝜔 Sw𝜔 exp Cw xa xb l 2334b k k k k 234 Flutter and Buffeting of the FullSpan Bridge 341 Expressions for the spectra Su𝜔 and Sw𝜔 and values of Cu and Cw are suggested in Chapter 2 The standard deviation of 𝜉i is 𝜎𝜉i 0 S𝜉i𝜉indn 12 2336 where n 𝜔2𝜋 From Eqs 2318 it follows that 𝜎hix hixB𝜎𝜉i 2337a 𝜎pix pixB𝜎𝜉i 2337b 𝜎𝛼ix 𝛼ix𝜎𝜉i 2337c 2342 Example Critical Flutter Velocity and Buffeting Response of Golden Gate Bridge This section presents a set of calculations developed by Scanlan on the basis of tests performed by Ragget that illustrate the approaches developed in Section 2341 11 A 1 50 scale model section was used to obtain flutter derivatives H i and A i i 1 4 A set of those derivatives for zerodegree angle of attack in smooth flow is shown in Figures 237 and 238 This example presents calculations that illustrate the use of the approach described in this section The vibration modes and frequencies of the bridge together with their modal integrals Gqiqi were obtained for the first eight modes with the results given in Table 231 Modal forms are suggested by the notations S symmetric AS antisymmetric L lateral V vertical and T torsion Values of the modal integrals Gqiqi suggest the importance of the mode in Table 231 the largest in each category ie vertical lateral torsion is underlined The most pronounced modes are mode 6 vertical mode 1 lateral and mode 7 antisymmetric torsion Flutter The torsional aerodynamic damping coefficient A 2 exhibits a pronounced change of sign with increasing velocity indicating the possibility of singledegree of 0 4 2 6 8 10 70 00 70 140 210 U nB H1 H2 H3 H4 Figure 237 Flutter derivatives H i i 1 2 3 4 Golden Gate Bridge Source Courtesy of Dr J D Raggett West Wind Laboratory Carmel CA k k k k 342 23 SuspendedSpan Bridges 0 2 4 6 8 10 04 00 02 08 10 UnB 06 04 02 A1 A4 A2 A3 Figure 238 Flutter derivatives A i i 1 2 3 4 Golden Gate Bridge Source Courtesy of Dr J D Raggett West Wind Laboratory Carmel CA Table 231 Frequencies types of modal forms and modal integrals for Golden Gate Bridge Frequency Type Ghihi Gpipi G𝜶i𝜶i 1 0049 L 262E16 333E01 803E05 2 0087 ASV1 325E01 739E15 177E15 3 0112 L 172E14 309E01 124E02 4 0129 SV1 190E01 782E14 116E14 5 0140 V 191E01 558E14 243E14 6 0164 V 344E01 387E13 125E14 7 0192 AST1 667E12 332E02 129E 00 8 0197 ST1 250E12 247E01 255E01 freedom torsional flutter Figure 238 Mode 7 is the torsional mode with both the lowest frequency and the greatest Gqiqi value and was selected as the most vulnerable to flutter instability Figure 239 In the case of the original Tacoma Narrows Bridge the lowest antisymmetric mode was also the mostflutter prone In the Golden Gate Bridge case this mode is practically a complete sine wave along the main span with a node at the center and practically zero amplitude on the two side spans The pertinent parameters are 𝜁7 0005 assumed I7 85 109 lb ft s2 𝜌 238 106 kip ft4 s2 000238 lb ft4 s2 0002378 slugsft3 B 90 ft L 6451 ft G𝛼7𝛼7 129 k k k k 234 Flutter and Buffeting of the FullSpan Bridge 343 4 5 Reduced speed UnB 6 7 8 9 SV1 ASV1 ST1 AST1 10 3 0 5 10 15 Peak to peak displacement inches 20 25 30 35 40 2 1 0 Figure 239 Calculated peaktopeak displacements induced by buffeting in four selected modes The flutter criterion in this case reduces to A 2 4𝜁7I7 𝜌B4G𝛼7𝛼7 0131 From the graph for A 2 Figure 238 the corresponding reduced velocity value with n n7 0192 Hz is U nB 432 which corresponds to a critical laminar flow flutter velocity Ucr 4 32019290 7465 ft s1 2275 m s1 Buffeting The four modes listed in Table 232 are mainly active over the main span The following data were used lMS 4144 ft main span length z0 002 ft z 220 ft deck height u U25 lnzz0 CD 034 CL 0215 dCLd𝛼 315 CM 0 dCMd𝛼 0111 The modal shapes were assumed to have the shape of simple sinusoids hSV1 h0 sin𝜋xlMS hASV1 h0 sin2𝜋xlMS 𝛼SV1 𝛼0 sin𝜋xlMS 𝛼ASV1 𝛼0 sin2𝜋xlMS Equations 23332337 were then used to obtain the results of Figure 239 Table 232 Generalized inertia of full bridge for four modes Mode i Frequency Hz 109 Ii lb ft s2 ASV1 2 00870 1571 SV1 4 01285 615 AST1 7 01916 850 ST1 8 01972 859 k k k k 344 23 SuspendedSpan Bridges 235 Stay Cable Vibrations 2351 Cable Vibration Characteristics Stayedbridge cables have low damping small mass and low bending stiffness They can experience two types of vibration i lowamplitude highfrequency vortexinduced vibrations and ii largeamplitude lowfrequency vibrations under skewed winds that include rainwindinduced vibrations and dry galloping This section considers only vibrations of the latter type According to fullscale field observations Wind speeds at the onset of the vibrations can vary from 6 to 40 m s1 13 14 Stays are vulnerable to excitation by skewed winds with directions making either a negative or a positive angle with the direction normal to the plane containing the cable 1517 Vibrations were observed mostly in rainy weather 18 19 but also occurred in the absence of rain 13 17 Vibrations occurred in winds with both low and high turbulence intensity 19 20 Vibrations occurred in low modes of vibration mostly with frequencies of 13 Hz 15 but also in simultaneous multiple vibration modes 13 21 Stay cables susceptible to excitation eg polyethylene tubelapped cables had smooth surfaces 15 22 Reynolds numbers ranged from 6 104 to 2 105 subcritical regime 22 23 The maximum acceleration of the cables varied between 4 g and 10 g where g is the gravitational acceleration 981 m s2 16 19 24 Peaktopeak amplitudes could several times a cable diameter 13 24 The wide range of the observed data suggests that no single mechanism can explain the cable vibration phenomenon The proposed mechanisms can be roughly divided into two main categories high speed vortexinduced vibration and galloping instability According to 25 for vibrations occurring in rainy weather rain water flowing down ward owing to gravity forms a rivulet on the lower surface of the cable As the wind becomes stronger another rivulet forms on the upper surface in which the forces due to wind gravity and water surface tension are balanced Cable oscillations cause the rivulets to oscillate around their mean positions thus changing the points of separation of the wind flow and affecting the pressure distribution around the cable This results in forces that cause the cable to vibrate However to date the fundamental mechanism of these vibrations remains uncertain and could not be clarified by wind tunnel testing For references on windrain induced vibrations see 1 25 26 2352 Mitigation Approaches Common mechanical approaches to mitigating lowfrequency largeamplitude stay cable vibrations include increasing the damping by installing dampers and using crossties The damping ratios of stay cables are typically in the range of 0105 26 Most types of windinduced vibrations can be reduced to acceptable levels by increasing the Scruton number Sc m𝜁𝜌D2 Eq 1910 by increasing the cable mass and damping For rainwindinduced vibrations it is recommended that Sc 10 26 k k k k References 345 Owing to the geometrical constraints of bridge decks dampers are typically attached to stay cables near cable anchorage and are designed to mitigate the cable vibrations in the fundamental modes Transverse restrainers eg crossties between stay cables are commonly used to effectively mitigate the inplane global mode cable vibrations 27 which give rise to local modes of vibration of the interconnected stays Their excessive use may affect the aesthetics of the bridges Aerodynamic countermeasures include the modification of cable cross sections by using eg helical strakes and patternindented surfaces with a view to disturbing the formation of water rivulets on the stay cables which could cause the rainwind induced vibrations 1 According to 26 a Scruton number Sc 5 is recommended if both mechanical and aerodynamic countermeasures are used in a cable system It was reported in 28 that the effectiveness of aerodynamic countermeasures can be weaken if the Scruton number is less than 8 References 1 Simiu E and Miyata T 2006 Design of Buildings and Bridges for Wind A Practical Guide for ASCE7 Standard Users and Designers of Special Structures 1st ed Hoboken NJ Wiley 2 Davenport A G Isyumov N Fader D J and Bowen C F P A Study of Wind Action on a Suspension Bridge during Erection and Completion Report No BWLT369 with Appendix BLWT470 Faculty of Engineering Science University of Western Ontario London Canada 19691970 3 Farquharson FB ed 19491954 Aerodynamic Stability of Suspension Bridges Bulletin No 116 Seattle WA University of Washington Engineering Experiment Station 4 Wardlaw R L Static Force Measurements of Six Deck Sections for the Proposed New Burrard Inlet Crossing Report No LTRLA53 National Aeronautical Estab lishment National Research Council Ottawa Canada 1970 5 Scanlan RH and Tomko JJ 1971 Airfoil and bridge deck flutter derivatives Jour nal of the Engineering Mechanics Division 97 17171737 6 Gan Chowdhury A and Sarkar PP 2003 A new technique for identification of eighteen flutter derivatives using threedegreeoffreedom section model Engineering Structures 25 17631772 7 Wardlaw R L and Goettler L L A Wind Tunnel Study of Modifications to Improve the Aerodynamic Stability of the Long Creeks Bridge Report LTRLA8 National Aero nautical Establishment National Research Council Ottawa Canada 1968 8 Ehsan F and Scanlan RH 1990 Vortexinduced vibrations of flexible bridges Journal of Engineering Mechanics 116 13921411 9 Flamand O De Oliveira F StathopoulosVlamis A and Papanikolas P 2014 Conditions for occurrence of vortex shedding on a large cable stayed bridge Full scale data from monitoring system Journal of Wind Engineering and Industrial Aerodynamics 135 163169 10 Sarkar P P New identification methods applied to response of flexible bridges to wind Doctoral dissertation Civil Engineering Johns Hopkins University Baltimore MD 1992 k k k k 346 23 SuspendedSpan Bridges 11 Simiu E and Scanlan RH 1996 Wind Effects on Structures 3rd ed Hoboken NJ John Wiley Sons 12 Scanlan RH and Lin WH 1978 Effects of turbulence on bridge flutter deriva tives Journal of the Engineering Mechanics Division 104 719733 13 Matsumoto M Daito Y Kanamura T et al 1998 Windinduced vibration of cables of cablestayed bridges Journal of Wind Engineering and Industrial Aerody namics 7476 10151027 14 Matsumoto M Saitoh T Kitazawa M Shirato H and Nishizaki T Response characteristics of rainwind induced vibration of staycables of cablestayed bridges Journal of Wind Engineering and Industrial Aerodynamics 57 323333 1995 15 Hikami Y and Shiraishi N 1988 Rainwind induced vibrations of cables stayed bridges Journal of Wind Engineering and Industrial Aerodynamics 29 409418 16 Phelan RS Sarkar PP and Mehta KC 2006 Fullscale measurements to inves tigate rainwind induced cablestay vibration and its mitigation Journal of Bridge Engineering 11 293304 17 Zuo D and Jones N P Understanding wind and rainwindinduced stay cable vibrations from field observations and wind tunnel tests in 4th USJapan Work shop on Wind Engineering Tsukuba Japan 2006 18 Main J A and Jones N P Fullscale measurements of stay cable vibration in 10th International Conference on Wind Engineering Copenhagen Denmark 1999 pp 963970 19 Ni YQ Wang XY Chen ZQ and Ko JM 2007 Field observations of rainwindinduced cable vibration in cablestayed Dongting Lake Bridge Journal of Wind Engineering and Industrial Aerodynamics 95 303328 20 Matsumoto M Shiraishi N and Shirato H 1992 Rainwind induced vibra tion of cables of cablestayed bridges Journal of Wind Engineering and Industrial Aerodynamics 43 20112022 21 Zuo D Jones NP and Main JA 2008 Field observation of vortex and rainwindinduced staycable vibrations in a threedimensional environment Journal of Wind Engineering and Industrial Aerodynamics 96 11241133 22 Matsumoto M 1998 Observed behavior of prototype cable vibration and its gen eration mechanism In Advances in Bridge Aerodynamics ed A Larsen 189211 Rotterdam The Netherlands Balkema 23 Zuo D Understanding wind and rainwind induced stay cable vibrations Doc toral dissertation Civil Engineering Johns Hopkins University Baltimore 2005 24 Main J A Jones N P and Yamaguchi H Characterization of rainwind induced staycable vibrations from fullscale measurements in 4th International Symposium on Cable Dynamics Montreal Canada 2001 pp 235242 25 Caetano E Cable vibrations in cablestayed bridges 9 IABSE International Associ ation for Bridge and Structural Engineering 2007 26 Kumarasena S Jones N P Irwin P A and Taylor P Windinduced vibration of stay cables FHWARD05083 Federal Highway Administration McLean VA 2007 27 Yamaguchi H and Nagahawatta HD 1995 Damping effects of cable cross ties in cablestayed bridges Journal of Wind Engineering and Industrial Aerodynamics 5455 3543 28 Ruscheweyh H 1994 Vortex excited vibrations In WindExcited Vibrations of Structures ed H Sockel 5184 WienNew York SpringerVerlag k k k k 347 Part IV Other Structures and Special Topics k k k k 349 24 Trussed Frameworks and Plate Girders This chapter reviews the aerodynamic behavior of trussed frameworks and plate girders including single trusses and girders systems consisting of two or more parallel trusses or girders and square and triangular towers Test results are often presented from sev eral sources with a view to allowing an assessment of the errors that may be expected in typical wind tunnel measurements Throughout this chapter the aerodynamic coef ficients are referred to and should be used in conjunction with the effective area of the framework Af For any given wind speed the principal factors that determine the wind load acting on a trussed framework are The aspect ratio 𝜆 that is the ratio of the length of the framework to its width If end plates or abutments are provided the flow around the framework is essentially twodimensional The solidity ratio 𝜑 that is the ratio of the effective to the gross area of the frame work1 For any solidity ratio 𝜑 the wind load is for practical purposes independent of the truss configuration that is of whether a diagonal truss a Ktruss and so forth is involved The shielding of portions of the framework by other portions located upwind The degree to which shielding occurs depends upon the configuration of the spatial frame work If the framework consists of parallel trusses or girders the shielding depends on the number and spacing of the trusses or girders The shape of the members that is whether the members are rounded or have sharp edges Forces on rounded members depend on Reynolds number Re and on the roughness of the member surface see Figure 422 For trusses with sharp edges the effect of the Reynolds number and of the shape and surface roughness of the member is in practice negligible The turbulence in the oncoming flow The effect of turbulence on the drag force acting on frameworks with sharpedged members is relatively small in most cases of practi cal interest 16 A similar conclusion appears to be valid for frameworks composed of members with circular cross section in flows with subcritical Reynolds numbers 1 The effective area of a plane truss is the area of the shadow projected by its members on a plane parallel to the truss the projection being normal to that plane The gross area of a plane truss is the area contained within the outside contour of that truss The effective area and the gross area of a spatial framework are defined respectively as the effective area and the gross area of its upwind face Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 350 24 Trussed Frameworks and Plate Girders For this reason and owing to scaling difficulties in most cases wind tunnel tests for trussed frameworks are to this day conducted in smooth flow 36 The orientation of the framework with respect to mean wind direction Wind forces on ancillary parts eg ladders antenna dishes solar panels must be taken into account in design in addition to the wind forces on the trussed frameworks themselves Drag and interference effects on microwave dish antennas and their supporting towers were studied in 7 Drag coefficients for an unshrouded isolated microwave dish with depthtodiameter ratio 024 were found to be largest for angles of 030 between wind direction and the horizontal projection of the normal to the dish surface and are almost independent of the flow turbulence CD 14 For a single dish the ratio f a between the incremental total drag on the tower due to the addition of a single dish and the drag for the isolated dish depends on the wind direction and it is higher than unity as high as 13 for the most unfavorable directions This is due to the flow accelerations induced in the dish As more dishes are added at the same level of a tower interference factors are still greater than unity but tend to decrease as the number of dishes increases Various petrochemical and other industrial facilities consist of complex assemblies of pipes reservoirs vessels ladders frames trusses beams and so forth for which the determination of overall wind loads is typically difficult Estimates of wind loads for such facilities are discussed in some detail in 8 241 Single Trusses and Girders Figure 241 summarizes measurement of the drag coefficient C1 D for a single truss with infinite aspect ratio normal to the wind The data of Figure 241 were obtained in the 1930s in Göttingen by Flachsbart for trusses with sharpedged members 1 2 and in the late 1970s at the National Maritime Institute UK NMI for trusses with sharpedged and trusses with members of circular cross section all NMI measurements reported in this chapter were conducted at Reynolds numbers 104 Re 7 104 It is seen that the differences between the Göttingen and the NMI results are approximately 15 or less For single trusses normal to the wind and composed of sharpedged members Figure 242 shows ratios C1 D 𝜆C1 D 𝜆 of the drag coefficients corresponding to an aspect ratio 𝜆 and to an infinite aspect ratio Drag coefficients C1 D reported in 3 for trusses normal to the wind composed of sharpedged members and having aspect ratios 16 𝜆 6 are listed in the first line of Table 241 The second line of Table 241 lists values C1 D 𝜆 obtained from the drag coefficients of 3 through multiplication by the appropriate correction factor taken from Figure 242 Figure 243 7 summarizes results of tests on trusses with members of circular cross section 𝜆 conducted in the subsonic wind tunnel at PorzWahn Germany 9 10 and in the compressed air tunnel of the National Physical Laboratory UK 112 Note 2 Figures 243 and 24162419 are reproduced with permission of CIDECT Comité International pour le Développenent et lEtude de la de la Construction Tubulaire from HB Walker ed Wind Forces on Unclad Tubular Structures They are based in part on research work carried out by CIDECT and reported in 9 10 k k k k 241 Single Trusses and Girders 351 20 18 16 14 12 10 08 06 04 02 0 01 02 03 04 Roundsection members Anglesection members Rectangular members Solidity ratio φ 05 06 07 08 Drag coefficient CD 1 Refs 1 2 Figure 241 Drag coefficient C1 D for single truss 𝜆 wind normal to truss Source From 6 10 09 08 07 06 050 01 02 03 1λ φ 0 04 10 0975 095 075 025 01 05 1 CD λ 1 CD λ 05 0925 09 Figure 242 Ratios C1 D 𝜆C1 D 𝜆 wind normal to truss 2 k k k k 352 24 Trussed Frameworks and Plate Girders Table 241 Drag coefficients for simple trusses 𝝋 014 029 047 077 10 C1 D 1 6 𝜆 6 3 140 5 154 5 127 5 118 5 128 5 C1 D 𝜆 145 165 145 135 210 13 10 U 05 Values on curves indicate solidity ratio φ 1 CD 02 3 4 5 6 7 8 9 105 3 2 2 Re 4 5 6 7 8 9 106 λ 0 0 01 01 02 02 05 05 04 03 Figure 243 Drag coefficient C1 D for single truss with members of circular cross section 𝜆 wind normal to truss 7 Source Courtesy Comité International pour le Développement et lEtude de la Construction Tubulaire and Constructional Steel Research and Development Organization that for Re 105 the drag coefficients of Figure 243 differ by about 5 or less from the corresponding results of Figure 241 The aerodynamic force normal to a rectangular plate with aspect ratio 𝜆 510 is larger when the yaw angle ie the horizontal angle between the mean wind direction and the normal to the trusses is 𝛼 40 than if the wind is normal to the plate however for trusses with solidity ratio 𝜑 4 or so the maximum drag occurs when the wind is normal to the truss 1 242 Pairs of Trusses and of Plate Girders We consider a pair of identical parallel trusses and denote the drag coefficient corre sponding to the total aerodynamic force normal to the trusses by C2 D 𝛼 where 𝛼 is the k k k k 242 Pairs of Trusses and of Plate Girders 353 yaw angle For brevity C2 D 0 is denoted by C2 D The cases where the wind is normal to the truss 𝛼 0 and where 𝛼 0 are considered in Sections 2421 and 2422 respectively 2421 Trusses Normal to Wind Two parallel trusses normal to the wind affect each other aerodynamically so that the drag on the upwind and on the downwind truss will have drag coefficients ΨIC1 D and ΨIIC1 D respectively where C1 D is the drag coefficient for a single truss normal to the wind and in general ΨI ΨII 1 It follows that C2 D C1 D ΨI ΨII 241 Figure 244 shows values of ΨI and ΨII reported in 12 as functions of the solidity ratio 𝜑 the ratio between the truss spacing in the alongwind direction e and the truss width d Values of ΨI and ΨII also reported in 12 for four types of truss with sharpedged members and aspect ratio 𝜆 95 are shown in Figure 245 On the basis of the data in Figures 244 and 245 12 proposed the use for design purposes of the conservative values C2 D C1 D given for ed 10 in Figure 246 Measurements conducted at NMI on trusses with infinite aspect ratios are summa rized in Figure 247 The following approximate expressions based on the results of Figure 247 are suggested in 6 C2 D C1 D 2 𝜑045 e d 𝜑045 for 0 𝜑 05 242 for trusses with sharpedged members and C2 D C1 D 2 𝜑e 045 e d 𝜑e045 243 for trusses composed of members with circular cross section The nominal solidity ratio 𝜑e in Eq 243 is related to the actual solidity ratio as shown in Figure 248 Table 242 lists ratios C2 D C1 D for trusses with sharpedged members and aspect ratio 𝜆 8 4 Example 241 Consider a truss with sharpedged members solidity ratio 𝜑 018 spacing ratio ed 10 and aspect ratio 𝜆 According to both the Flachsbart and the NMI tests C1 D 170 Figure 241 and C2 D C1 D ΨI ΨII 15 Figures 244a and 247a so C2 D 170 155 265 According to the deliberately conserva tive Figure 246 C2 D C1 D 183 which exceeds by about 20 the value based on Figures 244a and 247a 2422 Trusses Skewed with Respect to Wind Direction We now consider the case in which the yaw angle is 𝛼 0 For certain values of 𝛼 the effectiveness of the shielding decreases and the drag coefficient C2 D 𝛼 characterizing the total force normal to the trusses is larger than the value C2 D k k k k 354 24 Trussed Frameworks and Plate Girders 08 06 04 ψl ψlI ψl ψlI ψl ψlI ψlI ψl ψl ψII ψl ψII φ 0427 d d φ 0404 φ 0234 φ 0404 φ 0234 φ 0545 φ 0545 I II φ 0160 φ 0178 φ 0178 φ 0160 φ 0208 φ 0232 φ 0430 φ 0511 φ 0427 φ 0430 02 0 02 10 20 a b c ed 30 40 10 20 ed 30 40 50 10 20 ed 30 40 60 10 08 06 04 02 0 02 10 08 06 04 02 0 02 10 d d e Figure 244 Factors ΨI and ΨII for three types of truss with sharpedged members and infinite aspect ratio 12 k k k k 242 Pairs of Trusses and of Plate Girders 355 02 02 10 20 30 40 ed 50 60 70 80 04 06 08 10 12 14 0 ψl ψlI ψl Model 1 φ 0627 Model 2 φ 0366 Model 4 φ 0458 Model 4 Model 1 Model 2 Model 3 Model 4 Model 3 Model 1 2 3 1 2 2 1 4 2 ψlI Model 3 φ 0435 Figure 245 Factors ΨI and ΨII for four sets of two parallel trusses with sharpedgd members 𝜆 95 wind normal to trusses 12 Figure 246 Approximate ratios C2 D C1 D proposed for design purposes in 12 22 21 20 19 18 17 16 15 14 13 12 11 0 01 02 03 04 05 06 07 08 09 10 10 20 40 ed 60 2 CD 1 CD d I II e φ Ratios max C2 D 𝛼C1 D reported in 3 for trusses with sharpedged members and aspect ratio 𝜆 8 are shown in Table 243 For example for ed 10 𝜑 0286 and 𝜆 8 the ratio C2 D 𝛼C1 D 177 Table 243 versus C2 D C1 D 159 Table 242 2423 Pairs of Solid Plates and Girders Figure 249 shows the dependence of the factors ΨI and ΨII see Eq 241 upon the spacing ratio ed for a solid disk and for three girders normal to the wind 12 13 For certain values of the horizontal angle 𝛼 between the wind direction and the normal to the k k k k 356 24 Trussed Frameworks and Plate Girders 0 01 0 01 02 03 04 05 06 07 08 02 10 I 20 30 40 ed 05 03 04 d 05 06 ψI ψII ψI ψII ψII ψI 07 08 09 10 a b φ 0 01 0 01 02 03 04 05 06 07 08 02 03 04 05 06 ψI ψII 07 08 09 10 φ 10 20 30 40 ed 05 II e I d II e ψII ψI Figure 247 Factors ΨI and ΨII for two parallel trusses with a sharpedged members and b members of circular cross section infinite aspect ratio wind normal to trusses Source From 6 k k k k 243 Multiple Frame Arrays 357 Figure 248 Equivalent solidity ratio 𝜑e for trusses with members of circular cross section and solidity ratio 𝜑 Source From 6 00 01 005 02 03 04 05 06 07 08 010 015 020 025 030 035 040 φe φ Table 242 Ratios C2 D C1 D for trusses with sharpedged members and aspect ratio 𝜆 8 wind normal to trusses ed 012 02 026 05 075 10 15 20 𝝋 0136 135 167 173 184 183 184 0286 114 147 143 156 159 159 0464 122 129 132 132 133 133 134 0773 116 115 113 110 109 108 101 101 10 101 101 101 100 101 099 095 091 Source After 5 plates the ratio C2 D 𝛼C2 D may be larger than unity For example for a plate with aspect ratio 𝜆 4 and spacing ratio ed 05 if 40 𝛼 65 then C2 D 𝛼C2 D 120 5 243 Multiple Frame Arrays The first attempts to measure aerodynamic forces on multiple frame arrays were reported in 5 6 For frames normal to the wind the drag coefficients for the first second nth frame may be written as Ψ1C1 D Ψ2C1 D ΨnC1 D where C1 D is the k k k k 358 24 Trussed Frameworks and Plate Girders Table 243 Ratios max C2 D 𝛼C1 D for trusses with sharpedged members 𝜆 8 ed 025 050 075 10 15 20 𝝋 015 185 185 186 188 193 199 03 162 166 171 177 187 197 05 140 148 154 161 176 194 08 114 119 138 148 171 184 10 101 127 136 143 161 169 Source After 5 12 10 08 06 ld 20 ld 136 ld 95 d d d 04 02 0 02 04 0 10 20 30 40 ed 50 60 70 80 ψl ψlI ψl ψlI d I II e Figure 249 Factors ΨI and ΨII for two parallel solid plates girders 12 13 drag coefficient for a single frame normal to the wind The drag coefficient for the array of frames normal to the wind is then Cn D C1 D Ψ1 Ψ2 Ψn 244 Factors Ψj j 1 2 n for arrays of three four and five parallel trusses with sharpedged members and infinite aspect ratio are given in Figure 2410 for spacings ed 05 and ed 1 6 Figure 2411 show plots of drag coefficients Cn D for the same arrays with members of circular and angle cross section 6 k k k k 243 Multiple Frame Arrays 359 Figure 2410 Factors Ψj j 1 2 n for arrays of n parallel trusses n 3 4 and 5 with sharpedged members 𝜆 wind normal to trusses a Spacing ratio ed 05 b Spacing ratio ed 10 Source From 6 10 1st Frame 2nd Frame 3rd Frame 4th Frame 5th Frame Symbol 1 2 3 4 5 Frame configuration 09 08 07 06 05 04 03 02 01 0 0 01 02 03 04 a b 05 06 07 08 φ 10 1st Frame 2nd Frame 3rd Frame 4th Frame 5th Frame Symbol 1 2 3 4 5 Frame configuration 09 08 07 06 05 04 03 02 01 0 0 01 02 03 04 05 06 07 08 φ ψj ψj k k k k 360 24 Trussed Frameworks and Plate Girders 44 Anglesection members Circularsection members n 5 n 4 n 3 n 5 n 4 n 2 n 2 n 3 n 1 n 1 40 36 32 n CD 28 24 20 16 12 08 04 00 01 02 03 04 05 06 07 08 44 48 Anglesection members Circularsection members n 5 n 4 n 3 n 5 n 4 n 2 n 2 n 3 n 1 n 1 40 36 32 n CD 28 24 20 16 12 08 0 0 01 02 03 04 05 06 07 08 a φ b φ Figure 2411 Drag coefficients Cn D for arrays of n parallel trusses 𝜆 wind normal to trusses a Spacing ratio ed 05 b Spacing ratio ed 10 Source From 6 k k k k 244 Square and Triangular Towers 361 244 Square and Triangular Towers The aerodynamic coefficients given in this chapter are in all cases referred to and should be used in conjunction with the effective area of the framework Af For square and rectangular towers Af is the effective area of one of the identical faces of the tower The dynamic response of the towers can be determined conservatively as shown in Appendix D The width of the structure used as input should be the actual width of the framework This ensures that the lateral coherence of the load fluctuations is taken into account The depth alongwind dimension of the framework should be assumed to be zero in order not to overestimate the favorable effect of the imperfect alongwind crosscorrelations of the fluctuating loads Finally the area of the framework per unit height at any given elevation used to estimate the mean and the fluctuating drag forces should be equal to the effective area per unit height at that elevation 2441 Aerodynamic Data for Square and Triangular Towers The results of wind force measurement on square towers can be expressed in terms of the aerodynamic coefficients CN𝛼 and CT𝛼 associated respectively with the wind force components N and T N T normal to the faces of the tower Figure 2412 and in terms of the aerodynamic coefficient CF𝛼 associated with the total wind force F acting at a yaw angle 𝛼 tan1TN Note that CF𝛼 C2 N𝛼 C2 T𝛼12 since as indicated earlier all aerodynamic coefficients are referenced to the effective area of one face of the framework Af For a triangular tower which has in practice and is therefore assumed here to have equal sides in plan the results of the measurements can be expressed in terms of the aerodynamic coefficients CF𝛼 Figure 2413 The aerodynamic coefficients CF0 and CF60 correspond respectively to wind forces acting in a direction normal to a side and along the direction of a median Figure 2413 Square Towers Composed of SharpEdged Members Measurements of loads on a tapered square tower model with sharpedges members aspect ratio 𝜆 and T F N 𝛼 Figure 2412 Notations solidity ratio averaged over the height of the tower 𝜑 019 ranging from 𝜑 013 at the base to 𝜑 047 at the top were reported in the 1930s 14 Until recently these mea surements have been the principal source of data on square towers The coefficients CN𝛼 CT𝛼 and CF𝛼 obtained in 4 are listed for various angles 𝛼 in Table 244 For 𝛼 45 the values of CN𝛼 and CT𝛼 should be equal as pointed out in 14 the 4 difference between these values in Table 244 is due to measurement errors Note that the value CN0 254 is close to the val ues inferred from 3 and 6 which are respectively CN0 C2 D 15 173 260 as obtained by linear interpolation for 𝜑 019 and ed 10 from Tables 241 and 242 and CN0 C2 D 17093 058 257 Eq 241 Figures 241 and 247a Note also that while the largest tension compression in the tower columns is caused by k k k k 362 24 Trussed Frameworks and Plate Girders a F0 60 60 F60 𝛼 60 b F30 𝛼 30 c Figure 2413 Notations Table 244 Aerodynamic coefficients CN𝛼 CT𝛼 and CF𝛼 for a square tower 𝜑 019 and 𝜆 14 𝜶 0 9 18 27 36 45 CN𝛼 254 275 297 301 284 260 CT𝛼 019 070 136 205 249 CF𝛼 254 276 305 330 350 360 30 25 20 CF0 15 10 0 02 04 06 08 10 𝜑 Angle membersturbulent flow Angle memberssmooth flow Square shped memberssmooth flow Figure 2414 Drag coefficients CF0 for square tower with sharpedged members measured at NMI Source From 4 winds acting in the direction 𝛼 45 the largest stresses in the bracing members occur for 𝛼 27 Measurements of forces on square towers with sharpedged members 𝜆 were more recently conducted at NMI 4 Coefficients CF0 and ratios CF𝛼CF0 based on these measurements are shown in Figures 2414 and 2415 respectively Note for k k k k 244 Square and Triangular Towers 363 Figure 2415 Ratios CF𝛼CF0 for a square tower with sharpedged members measured at NMI Source From 4 13 𝜑 0132 𝜑 0535 12 11 100 15 30 45 𝛼 CF𝛼 CF0 25 20 10 3 3 2 4 5 6 7 8 9 105 4 5 6 7 8 9 2 Re 106 Values on curves indicate solidity ratio φ U λ 0 0 01 01 02 02 CF0 03 03 04 04 05 05 055 055 Figure 2416 Drag coefficients CF0 for a square tower with members of circular cross section measured at the National Maritime Institute 7 Source Courtesy Comité International pour le Développenent et lEtude de la de la Construction Tubulaire and Constructional Steel Research and Development Organization k k k k 364 24 Trussed Frameworks and Plate Girders 25 15 10 3 3 2 4 5 6 7 8 9 105 4 5 6 7 8 9 2 106 Values on curves indicate solidity ratio φ U λ 0 0 01 0102 02 CF45 03 03055 04 05 055 Re Figure 2417 Drag coefficients CF45 for square tower with sharpedged members measured at the National Maritime Institute 7 Source Courtesy Comité International pour le Développenent et lEtude de la de la Construction Tubulaire and Constructional Steel Research and Development Organization example that for 𝜑 019 CF0 260 Figure 2414 versus CF0 254 as obtained in 14 Table 244 The agreement is less good for the ratio CF45CF0 which is about 112 according to Figure 2415 and about 140 according to data of Table 244 Square Towers Composed of Members with Circular Cross Sections Figures 2416 and 2417 9 represent respectively aerodynamic coefficients CF0 and CF45 as func tions of Reynolds number Re for towers with aspect ratio 𝜆 based on wind tunnel test results reported in 9 10 The values CF45 of Figure 2417 may be regarded as conservative envelopes that account for the loadings in the most unfavorable direc tions Results of NMI tests in both smooth and turbulent flow at Reynolds numbers Re 2 104 for solidity ratios 𝜑 017 𝜑 023 and 𝜑 031 𝜆 match the curves of Figures 2416 and 2417 to within about 5 4 Triangular Towers Composed of Members with Circular Cross Sections Figures 2418 and 2419 9 represent proposed aerodynamic coefficients CF0 CF60 and k k k k 244 Square and Triangular Towers 365 15 10 05 3 3 2 4 5 6 7 8 9 105 4 5 6 7 8 9 2 Re 106 Values on curves indicate solidity ratio φ U U λ 01 01 02 02 CF0 CF60 03 03 04 04 05 05 Figure 2418 Drag coefficients CF0 and CF60 for triangular tower with members of circular cross section 7 Source Courtesy Comité International pour le Développenent et lEtude de la de la Construction Tubulaire and Constructional Steel Research and Development Organization 15 16 10 05 3 3 2 4 5 6 7 8 9 105 4 5 6 7 8 9 2 Re 106 Values on curves indicate solidity ratio φ U λ 01 01 02 02 CF30 03 03 04 04 05 05 Figure 2419 Drag coefficients CF30 for triangular tower with members of circular cross section 8 Source Courtesy Comité International pour le Développenent et lEtude de la de la Construction Tubulaire and Constructional Steel Research and Development Organization k k k k 366 24 Trussed Frameworks and Plate Girders CF 30 as functions of Reynolds number Re for towers with aspect ratio 𝜆 based on measurements reported in 911 References 1 Flachsbart O 1934 Modellversuche über die Belastung von Gitterfachweken durch Windkräfte 1 Teil Einzelne ebene Gitterträger Der Stahlbau 7 6569 2 Flachsbart O 1934 Modellversuche über die Belastung von Gitterfachweken durch Windkräfte 1 Teil Einzelne ebene Gitterträger Der Stahlbau 7 7379 3 Georgiou P N Vickery B J and Church R Wind loading on open framed struc tures presented at the Third Canadian Workshop on Wind Engineering Vancouver Canada 1981 4 Flint A R and Smith B W The development of the British draft code of practice for the loading of lattice towers in Proceedings of the Fifth International Conference Wind Engineering Fort Collins Colorado July 1979 vol 2 J E Cermak ed NY Pergamon 1980 pp 12931304 5 Georgiou P N and Vickery B J Wind loads on building frames in Proceedings of the Fifth International Conference Wind Engineering Fort Collins Colorado July 1979 vol 1 J E Cermak ed NY Pergamon 1980 pp 421433 6 Whitbread R E The influence of shielding on the wind forces experienced by arrays of lattice frames in Proceedings of the Fifth International Conference Wind Engineering Fort Collins Colorado July 1979 vol 1 J E Cermak ed NY Perga mon 1980 pp 405420 7 Walker HB ed 1975 Wind Forces on Unclad Tubular Structures Croydon UK Constructional Steel Research and Development Organization 8 ASCE 2011 Wind Loads for Petrochemical and Other Industrial Facilities Task Committee on WindInduced Forces Petrochemical Committee of the Energy Division American Society of Civil Engineers Reston VA 9 Schulz G The Drag of Lattice Structures Constructed from Cylindrical Members Tubes and its Calculation in German CIDECT Report No 6921 Düsseldorf Germany 1969 10 Schulz G International Comparison of Standards on the Wind Loading of Struc tures CIDECT Report No 6929 Düsseldorf Germany 1969 11 Gould R W and Raymer W G Measurements over a Wide Range of Reynolds Numbers of the Wind Forces on Models and Lattice Frameworks Sc Rep No 572 National Physical Laboratory Teddington UK 1972 12 Flachsbart O 1935 Modellversuche über die Belastung von Gitterfachwerken durch Windkräfte 2 Teil Räumliche Gitterfachwerke Der Stahlbau 8 7379 13 Eiffel G 1911 La Résistance de lAir et lAviation Paris France H Dunod E Pinat 14 Katzmayr D and Seitz H 1934 Winddruck auf Fachwerkturme von quadratis chem Querschnitt Der Bauingenieur 2122 218251 k k k k 367 25 Offshore Structures Wind loads affect offshore structures during construction towing and in service They are a significant design factor especially in the case of large compliant platforms such as guyed towers and tension leg platforms Wind also affects the flight of helicopters near offshore platform landing decks 13 as potentially dangerous conditions may be created by flow separation at the edges of the platform Let the horizontal distance between the upstream edge of the platform and of the helideck be denoted by d and the depth of the upstream surface producing the separated flow be denoted by t On the basis of wind tunnel tests it has been suggested that the elevation h of the helideck with respect to the platform edge should vary from at least h 02 t if d 0 to at least h 05 t if d t 2 This chapter contains information on wind loads on offshore structures of various types Section 251 and on the treatment of dynamic effects on compliant structures Section 252 251 Wind Loading on Offshore Structures Methods for calculating wind loads on offshore platforms are recommended in 48 However laboratory and fullscale measurements indicate that these methods may in some instances have serious limitations particularly insofar as they do not account for the presence of lift forces and account insufficiently or not at all for shielding and mutual interference effects For example according to wind tunnel test results obtained for a jackup selfelevating platform 9 the methods of 4 and 5 overestimate wind loads on jackup units by at least 35 Estimates based on fullscale data for an anchored semisubmersible platform 10 suggest that the method of 5 overpredicts wind loads by as much as 100 This section briefly reviews a number of wind tunnel tests conducted for semisub mersible units and for a large guyed tower platform Wind tunnel test information on jackup units jacket structures in the towing mode and on two types of concrete plat form is available for example in 9 and 1114 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 368 25 Offshore Structures 2511 Wind Loads on Semisubmersible Units A schematic view of a semisubmersible unit used for tests reported in 15 is shown in Figure 2511 The side force and the heeling moment coefficients are defined as CY Y 12𝜌U250As 251 CK K 12𝜌U250AsHs 252 where Y is the side force K is the heeling moment 𝜌 is the air density U50 is the mean wind speed at 50 m above sea level As is the projected side area and Hs is the elevation of the center of gravity of As The coefficients CY and CK are obtained separately for the overwater and the underwater part of the unit The overwater coefficients reflect the action of wind and should be obtained in flow simulating the atmospheric boundary layer The overwater coefficients reflect account for hydrodynamic effects and should therefore be measured in uniform flow Figures 252 and 253 show values of CY and CK measured in 15 for the case of an upright draft2 TM0 1085 m corresponding for the unit being modeled to a displace ment3 of 17730 tons As noted in 15 the purpose of the tests for the underwater part is to determine the elevation of the center of reaction ie the point of application of the resultant of the underwater forces or the freefloating unit K X v ψ 10 20 Y M N Z 10 0 10 20 50 40 30 20 10 0 10 20 30 40 20 30 m 10 20 30 50 Figure 251 Schematic view of a semisubmersible unit model 15 1 Figures 251256 are excerpted from Bjeregaard E and Velschou S May 1978 Wind Overturning Effect on a Semisubmersible Paper OTC 3063 Proceedings Offshore Technology Conference Houston TX Copyright 1978 Offshore Technology Conference 2 The upright draft TM0 is the depth of immersion of the unit in even heel condition ie for an angle of heel 𝜙 0 3 The displacement is the volume of water displaced by the immersed part of the unit k k k k 251 Wind Loading on Offshore Structures 369 120 60 CK 1 05 180 240 300 5 5 5 5 15 15 15 15 25 25 25 φ 25 φ 0 0 0 0 360 deg 0 05 1 ψ 120 60 WIND ψ 90 CY 1 05 180 240 300 360 deg 0 05 1 ψ φ WIND ψ 270 Figure 252 Values CY and CK as functions of wind direction Ψ at different angles of heel 𝜙 for the overwater part 15 120 60 CK 1 05 180 240 300 5 5 5 15 15 15 25 5 15 25 25 φ 25 360 deg 0 05 1 ψ 120 60 CY 1 05 180 240 300 360 deg 0 05 1 ψ φ WL Figure 253 Values CY and CK as functions of wind direction Ψ at different angles of heel 𝜙 for the underwater part 15 Figure 254 shows estimated values of the heeling forces induced by 100knot beam winds winds blowing along the xaxis for various values of the upright draft TM0 and of the angle of heel 𝜙 The elevations of the center of action of the overwater wind force and of the center of reaction on the underwater part are shown in Figure 255 It is seen that as the angle of heel increases the elevation of the center of action of the wind force decreases This decrease is due to lift forces arising at nonzero angles of heel 𝜙 k k k k 370 25 Offshore Structures 15 5 0 50 100 150 200 250 300 Heeling Force tons TM0 643 m 900 m 1085 m 1525 m 25 φ Figure 254 Wind heeling forces corresponding to 100knot beam winds 15 15 5 0 5 10 15 5 10 15 20 25 Distance above Waterline m Above Water Under Water Water Surface 25 φ TM0 643 m 900 m 1085 m 1525 m Figure 255 Elevation of center of action of wind forces and corresponding center of reaction on the underwater part 15 The healing lever is defined as the ratio of the overturning moment to the displace ment of the vessel Values of the heeling lever for 100knot beam winds obtained from wind tests of 15 on the one hand and by using the American Bureau of Shipping method 4 on the other are shown in Figure 256 The displacements listed in 15 for the 643 900 and 1525 m drafts are 12740 16963 and 19495 tons respectively It is seen that for large angles of heel the differences between the two sets of values are significant This is largely due to the failure of 4 to account for the effects of lift It is noted in 16 that the largest overturning moments are commonly induced by quartering winds k k k k 251 Wind Loading on Offshore Structures 371 Figure 256 Wind heeling levers obtained from wind tunnel tests and from the American Bureau of Shipping ABS method 15 15 5 0 02 04 06 08 10 Heeling Lever m 0 10 20 Angle of heel 643 m A B S 900 m 1085 m 1525 m Model Tests HEELING LIMIT HEELING LIMIT 205 205 238 238 229 229 281 281 HEELING LIMIT 205 238 229 281 In the tests of 15 and 16 the water surface was modeled by the rigid horizontal surface of the wind tunnel flow Following the method described in 17 texts reported in 18 were also conducted by placing the model in a tank filled with viscoelastic material up to the level of the wind tunnel flow This facilitates the testing of models of partially submerged units Reference 18 also includes results of tests conducted in the presence of rigid obstructions aimed at representing water waves The results revealed that water waves could increase the overturning moments substantially This suggests the need for improving the simulation of the sea surface in laboratory tests The aerodynamic tests of the Ocean Ranger semisubmersible4 is reported in 19 The problem of combining hydrodynamic and wind loads was addressed by conduct ing 1 100 scale aerodynamic model tests in turbulent flow over a floor with rigid waves and using lightweight lines to apply the measured mean and fluctuating wind forces and moments to a 1 40 hydrodynamic model tested in conditions simulating those expe rienced during the storm Additional wind tunnel tests of semisubmersible units are reported in 2022 2512 Wind Loads on a Guyed Tower Platform Reference 23 presents results of wind tunnel measurements on a 1 120 scale model of the overwater part of a structure similar to Exxons Lena guyed tower platform A schematic of the platform installed in over 300 m of water in the Gulf of Mexico is shown in Figure 257 see also Figure 258 and the expression for wind speeds averaged 4 The Ocean Ranger had capsized on February 15 1982 in Hibernia Field 315 km southeast of St Johns Newfoundland in a storm with 1720 m wave heights and 120130 km h1 wind speeds It was the worlds largest submersible offshore drilling platform 46 m high from keel to operations deck and with 120 m long pontoons All of the 84 crew members were lost in the accident k k k k 372 25 Offshore Structures 2 DRILLING RIGS 3LEVEL DECK CABLE CLAMPS 250 76 m PENDANT SEGMENT 5 38 137 mm DIA BARE STRAND CONNECTOR 1700 519 m CATENARY SEGMENT 5 127 mm DIA COATED STRAND BUOYANCY TANKS 72 1829 mm DIA PERIMETER PILES 54 1372 mm DIA MAIN PILES 1000 305 m 560 171 m 3030 924 m 120 37 m 200 TON 181 mg CLUMPWEIGHT PARALEAD BENDING SHOE 1200 368m ANCHOR CABLE 5 127 mm DIA COATED STRAND 72 1829 mm DIA ANCHOR PILE Figure 257 Schematic view of the Lena guyed tower platform 24 Source With permission from ASCE over at least one minute recommended by the US Geological Survey 7 for use within the Gulf of Mexico Uz U10 z zd 10 zd 01128 253 where z is the elevation above still water in meters and zd 22 m The airwater bound ary was modeled by the rigid horizontal surface of the wind tunnel floor Force and moment coefficients were defined by relations of the type CF F 12𝜌U216AR 254 CM M 12𝜌U216ARLR 255 where F and M are the mean force and moment of interest 𝜌 is the air density U16 is the mean wind speed at 16 ft above the surface and the reference area AR and length LR were chosen as 1 ft2 and 1 ft respectively The force and moments obtained in 23 are represented in Figure 259 which also shows the notations for the respective aero dynamic coefficients The moments characterized by the coefficients CMD and CMT were taken with respect to a distance of 62 in 62 ft full scale below the still water level The measured values of the aerodynamic coefficients are represented in Figure 2510 for several platform configurations The configuration for the base case was the same k k k k 251 Wind Loading on Offshore Structures 373 3123 270 78 2249 1635 1155 580 1127 855 Drilling Derrick 2 Derrick Structure 2 Crane 2 4 4 3 3 1 Substructure Well Conductors SWL ElvOO a Deck Structure enclosed Drilling Packages 2 P Tanks Crews Quarters 2 1 2 2 Helideck 2 1605 Flare Boom 78 18 12 12 10 5 12 10 12 10 20 66 2 Flare Boom Crane Crane 30 65 28 Drilling Packages 66 132 17 Direction of Boom N CL Deck 156 156 57 Skid Base 67 32 8 P Tanks 10 120 15 1 2 Derrick Support Frame 1 2 30 30 23 Crews Quarters b Note Helidecks Rest on Top of Crews Quarters Figure 258 Guyed tower platform a side elevation b plan 25 k k k k 374 25 Offshore Structures CD CMD Y CL UP X X N U16 θ CMZ Y CT CMT Figure 259 Notations Source From 23 Copyright 1982 Offshore Technology Conference as in Figure 258 except that the deck structure was not enclosed Additional results in 23 show that the effect of enclosing the deck is negligible as is the effect of the well conductors Removing the flares boom results in torsional moment reductions but has negligible effects otherwise It is shown in 23 that drag forces and drag moments based on wind tunnel measurements are smaller by about 30 and 20 respectively than calculated values based on 7 To check the extent to which the results depend upon the laboratory facility being used the same structure was subsequently tested inde pendently in a different wind tunnel 25 In most cases of significance from a design viewpoint the results obtained in 25 were larger than those of 23 by amounts that did not exceed 2030 MOMENT OR FORCE COEFFICIENT 2 1 0 1 2 3 4 0 90 180 270 360 BASE CASE WO EAST DERRICK WO BOTH DERRICKS WO DRILLING EQUIPMENT 2 DECK CONFIGURATION DRAG FORCE CD TRANSVERSE FORCE CT WIND DIRECTION a Figure 2510 Wind tunnel test results Source From 23 Copyright 1982 Offshore Technology Conference k k k k 251 Wind Loading on Offshore Structures 375 0 15 10 05 00 05 10 15 90 180 270 360 LIFT FORCE CL TORQUE MOMENT CMZ BASE CASE WO EAST DERRICK WO BOTH DERRICKS WO DRILLING EQUIPMENT 2 DECK CONFIGURATION MOMENT OR FORCE COEFFICIENT WIND DIRECTION c MOMENT OR FORCE COEFFICIENT 2 1 0 1 2 3 4 0 90 180 270 360 BASE CASE WO EAST DERRICK WO BOTH DERRICKS WO DRILLING EQUIPMENT 2 DECK CONFIGURATION DRAG MOMENT CMD TRANSVERSE MOMENT CMT WIND DIRECTION b Figure 2510 Continued k k k k 376 25 Offshore Structures 252 Dynamic Wind Effects on Compliant Offshore Structures Compliant offshore platforms are designed to experience significant motions under load An advantage of compliance is that the forces of inertia due to the motion of the platform tend to counteract the external loads For large offshore platforms installed in deep water an additional advantage is that the natural frequencies of the platform motions in the surge sway and yaw5 degrees of freedom are very low eg from 130 to 1150 Hz depending upon type of platform and water depth Wave motions have narrow spectra centered about relatively high frequencies eg from 115 Hz for extreme events to about 1 Hz for service conditions Thus aside from possible secondorder effects compliant platforms do not exhibit dynamic amplifications of waveinduced response Unlike wave motions wind speed fluctuations in the atmospheric boundary layer are characterized by broadband spectra For this reason it has been surmised that windinduced dynamic amplification effects on compliant structures are significant 23 26 A more guarded assessment of the effects of wind gustiness was presented in 27 as a part of an evaluation of the response to environmental loads of the North Sea Hutton tension leg platform Figure 2511 see also 28 According to 27 Wind gusts are typically broadbanded and may contain energy which could excite surge motions at the natural period These would be controlled by surge damp ing Theoretical and experimental research is required to clarify the importance of this matter Investigations into the behavior of tension leg platforms under wind loads reported in 29 and 30 were based on the assumption that the response to wind is described by a system with proportional damping with damping ratio in the order of 5 However it was shown in 31 that for structures comparable to the Hutton platform the effective hydrodynamic damping is considerably stronger and that the windinduced dynamic amplification for lowfrequency motions is for this reason negligible Section 2521 describes the approach used in 31 to estimate the response of a tension leg platform to wind in the presence of current and waves and a simple method for estimating the order of magnitude of the damping inherent in the hydrodynamic loads 2521 Turbulent Wind Effects on Tension Leg Platform Surge Under the assumption that the external loads are parallel to one of the sides of the plat form shown in Figure 2511 the equation of surge motion can be written as Mx Fxt 256 where Fxt Fut Fht Rt 257 and Fut Fht and Rt denote the wind force the hydrodynamic force and the restor ing force respectively Not included in Eq 257 is the damping force due to internal 5 Displacements in the longitudinal transverse and vertical direction are called surge sway and heave respectively Rotations in a transverse longitudinal and horizontal plane are called roll pitch and yaw respectively k k k k 252 Dynamic Wind Effects on Compliant Offshore Structures 377 Figure 2511 Schematic view of the Hutton tension leg platform Source From 28 Copyright 1982 Offshore Technology Conference friction within the structure which is negligible compared to the damping forces due to hydrodynamic effects Wind Loads To estimate the windinduced drag force it is assumed that the elemental drag force per unit of area projected on a plane P normal to the mean wind speed is py z t 1 2𝜌aCpy zuy z t xt2 258 where 𝜌a is the air density Cpy z is the pressure coefficient at elevation z and horizon tal coordinate y in the plane P t is the time x is the surge displacement the dot denotes differentiation with respect to time and uz y t is the wind speed upwind of the struc ture in the direction of the mean wind The speed uz y t can be expressed as a sum of the mean speed Uz and the fluctuating speed uy z t uz y t Uz uz y t 259 The total windinduced drag force is Fut Aa py z tdydz 2510 where Aa is the projection of the abovewater part of the platform on a plane normal to the mean wind speed k k k k 378 25 Offshore Structures The mean wind speeds and the turbulence spectrum and cospectrum can be modeled as in Chapter 2 Neglecting secondorder terms it follows from Eqs 2582510 that the mean windinduced drag is Fut 1 2𝜌aCaAaU2za 2511 where the overall aerodynamic drag coefficient is Ca 1 AaU2zaAa Cpy zU2zdydz 2512 and za is the elevation of the aerodynamic center of the abovewater part of the platform The fluctuating part of the windinduced drag is F urt 𝜌aAa Cpy zUzuz tdydz 2513 where the subscript r refers to the fact that the platform is at rest As shown in 31 the spectral density of the fluctuating part of the windinduced drag is SFurn 𝜌aCaAaUza2Sueqn 2514 where for typical drilling and production platform geometries the equivalent wind fluc tuation spectrum can be defined as Sueqn Suza nJn 2515 Jn is a reduction factor that accounts for the imperfect spatial coherence if the fluctu ations u with the expression Jn 2 E expE 1 1 E expE 1 2516 E Cyb n Uza 2517 where b is the width of main deck and Cy is the horizontal exponential decay coefficient in Eq 294 Hydrodynamic Loads The total hydrodynamic load Fh can be written Fh Fv Fe Ax B x 2518 where Fv is the total hydrodynamic viscous force Fe is the total waveinduced exciting force A is the surgeadded mass and B is the surge waveradiation damping coefficient It was assumed for convenience in 31 that the wave motion is monochromatic hence the absence of secondorder drift forces in Eq 2518 It was also assumed that B 0 since the radiation damping at low frequencies is negligible 32 33 The total waveinduced exciting force and the surgeadded mass can be estimated numerically on the basis of potential theory Alternatively they may be assumed to be given by the inertia component of the Morison equation A 𝜌w i j ijCmij 1 2519 k k k k 252 Dynamic Wind Effects on Compliant Offshore Structures 379 Fe 𝜌w i j ijCmij vij t vi vij x vij X 2520 34 p 31 where 𝜌w is the water density ij is the elemental volume of the submerged structure Cmij is the surge inertia coefficient corresponding to ij X is the horizontal distance from some arbitrary origin to the center of ij along the direction parallel to surge motion vi and vij are the current velocity and horizontal particle velocity due to wave motion respectively at the center of ij Equations 2519 and 2520 may be employed if for the component being considered the ratio of diameter to wave length DL 02 34 p 283 Since for Tw 15 s L gTw 22𝜋 it follows that for members of typical tension leg platform structures for which D 20 m or so the use of Eqs 2519 and 2520 is acceptable if threedimensional flow effects are not taken into account The wave motion can be described by deep water linear theory so vij 𝜋H Tw ekwzi cos kwXj 2𝜋t Tw 2521 where H is the wave height and kw is the wave number given by kw 1 g 2𝜋 Tw 2 2522 34 p 157 The total hydrodynamic viscous load may be described by the viscous com ponent of Morisons equation Fv 05𝜌w i j CdijApij vi vij x vi vij x 2523 where Apij is the area of elemental volume ij projected on a plane normal to the direction of the current and Cdij is the drag coefficient corresponding to Apij If the relative motion of the body with respect to the fluid is harmonic the drag and inertia coefficients in the Morison equation can be determined on the basis of exper imental results as functions of local oscillatory Reynolds number Re 2𝜋 D2𝜈 Tf KeuleganCarpenter number K VTf D and relative body roughness where D is the diameter of the body 𝜈 is the kinematic viscosity and V and Tj are the amplitude and period of the relative fluidbody velocity However actual relative fluidbody motions are not harmonic This introduces uncertainties in the determination of the drag and inertia coefficients even if experimental information for harmonic relative motions were available in terms of Re and K Unfortunately such information is not available for the small numbers K in the order of 2 and the large Reynolds numbers in the order of 106 of interest in tension leg platform design For this reason calculations should be carried out for various sets of values Cd Cm and investigations should be conducted into the sensitivity of the results to changes in these values Restoring Force The surgerestoring force in a tension leg platform is supplied by the horizontal projection of the total force in the tethers Most of this force is the result of pretensioning which is achieved by ballasting the floating platform tying it by means of the tethers to the foundations at the sea floor then deballasting it The tension forces k k k k 380 25 Offshore Structures ln Δln T ΔT ln x Figure 2512 Notations in the tethers should exceed the compression forces induced by pitching and rolling moments due to extreme loads Under the assumption that the tethers are straight at all times the restoring force can be written as Rt T ΔT x ln Δln 2524 where T is the initial pretensioning force ΔT is the incre mental tension due to surge motion ln is the nominal length of the tethers at x 0 Δln is the incremental length and T ΔT ln Δln T ln CNL1 1 xln2 2525 where CNL is the downdraw coefficient equal to the weight of water displaced as the draft is increased by a unit length 32 Figure 2512 In reality hydrodynamic and inertia forces cause the teth ers to deform transversely The angle between the horizon tal and the tangent to the tether axis at the platform heel can therefore differ significantly for the values correspond ing to the case of a straight tether Nevertheless owing to the relatively small role of the restoring force in the dynamics of typical tension leg plat forms the effect of such differences on the motion of the platforms appears to be negli gible for practical purposes 3537 Surge Response The surge response is obtained by solving Eq 256 This equation is nonlinear the strongest contribution to the nonlinearity being due to the hydrodynamic viscous load Fv Its solution is sought in the time domain The nominal natural period in surge is Tn 2𝜋 Meff k 12 2526 where Meff is the coefficient of the term in x and k is the coefficient of the term in x in Eq 256 From Eqs 256 2518 and 2524 it follows that Tn 2𝜋 M Aln T 12 2527 A calculated time history of the surge response is represented in Figure 2513 as a function of time for a platform with the geometrical configuration of Figure 2514 under the following assumptions platform mass M 343 106 kg total initial tension in legs T 156 105 kN it follows from these assumptions and Eqs 2519 and 2527 that for the platform of Figure 2514 the nominal natural frequency is Tn 100 s Mori son equation coefficients Cmij 18 Cdij 06 wave height and period H 25 m and Tw 15 s respectively current speed varying from 14 m s1 at the mean water level to 015 m s1 at 550 m depth aerodynamic parameter CaAa 4320 m2 elevation of aero dynamic center za 50 m atmospheric boundary layer parameters 𝜅 0002 𝛽 60 Lx u 180 m Cy 16 see Chapter 2 and mean wind speed Uza 45 m s1 It is shown k k k k 252 Dynamic Wind Effects on Compliant Offshore Structures 381 0 25 30 35 40 45 50 500 1000 TIME s 1500 2000 SURGE m Figure 2513 Calculated time history of a surge response 31 32 m 67 m Diam 18 m Diam 9 m Depth 600 m Figure 2514 Geometry of a platform 31 k k k k 382 25 Offshore Structures in 31 that the contributions of the mean wind speed and the wind speed fluctuations about the mean are about 40 and 12 respectively It can be verified that this conclusion is equivalent to stating that windinduced resonant amplification effects are negligible in the cases investigated in 31 Sensitivity studies showed that the results were affected insignificantly by uncertain ties with respect to the actual values of the atmospheric boundarylayer parameters It is shown in 31 that the damping ratio in a linear system equivalent to the nonlin ear system studies in this section is in the order of 𝜁 05 and 𝜁 02 for the platforms with ln 600 m and ln 150 m respectively The coefficients Cdij 06 and Cmij 18 on which these results were based may not be realistic for members with large diameters such as those depicted in Figure 2514 The use of alternative values for those coefficients showed that the damping ratios were in all cases sufficiently large to prevent the occur rence of significant windinduced dynamic amplification effects However for some values of Cdij calculations in which the assumed currents would be lower than those of 31 could result in reduced nominal damping rations for certain wind climatological conditions Because windwave tests violate Reynolds number and KeuleganCarpenter number similarity they cannot provide a reliable indication of the equivalent damping ratio for the prototype This is a continuing cause of uncertainty in the assessment of dynamic effects induced by wind acting alone or in the case of a nonlinear analysis in conjunction with waveinduced slow drift References 1 Davies M E Cole L R and ONeill P G G The Nature of Air Flows Over Off shore Platforms NMI R14 OTR7726 National Maritime Institute Feltham UK June 1977 2 Davies M E Wind Tunnel Modeling of the Local Atmospheric Environment of Off shore Platforms NMI R58 OTR7935 National Maritime Institute Feltham UK May 1979 3 Littleburg K H Wind tunnel testing techniques for offshore gasoil production platforms Paper OTC 4125 Proceedings Offshore Technology Conference Houston TX 1981 4 American Bureau of Shipping 1980 Rules for Building and Classing Mobile Off shore Drilling Units New York American Bureau of Shipping 5 Det Norske Veritas 1981 Rules for the Construction and Classification of Mobile Offshore Units Oslo Det Norske Veritas 6 Det Norske Veritas Rules for the Design Construction and Inspection of Offshore Structures Appendix B Loads Det Norske Veritas Oslo 1977 Reprint with correc tions 1979 7 USGS Requirements for Verifying the Structural Integrity of OCS Platforms Appen dices United States Geological Survey OCS Platform Verification Division Reston VA 1979 8 API API Recommended Practice for Planning Designing and Constructing Fixed Offshore Platforms API RP 2A American Petroleum Institute Washington DC 1981 k k k k References 383 9 Norton D J and Wolff C V Mobile offshore platform wind loads Paper OTC 4126 Proceedings Offshore Technology Conference Houston TX 1981 10 Boonstra H 1980 Analysis of fullscale wind forces on a semisubmersible platform using operators data Journal of Petroleum Technology 32 771776 11 Ponsford P J Measurements of the Wind Forces and Measurements of an Oil Production Jacket Structure in TowOut Mode NMI R30 OTR7801 National Maritime Institute Feltham UK January 1978 12 Cowdrey C F TimeAveraged Aerodynamic Forces and Moments on a Model of a ThreeLegged Concrete Production Platform NMI R36 OTR7808 National Maritime Institute Feltham UK June 1982 13 Miller B L and Davies M E Wind Loading on Offshore Structures A Summary of Wind Tunnel Studies NMI R36 OTR7808 National Maritime Institute Feltham UK September 1982 14 Davenport A G and Hambly E C Turbulent wind loading and dynamic response of of jackup platform OTC Paper 4824 Proceedings Offshore Technology Conference Houston TX May 1984 15 Bjerregaard E and Velschou S Wind overturning effects on a semisubmersible OTC Paper 3063 Proceedings Offshore Technology Conference Houston TX May 1978 16 Bjerregaard E and Sorensen E Wind overturning effects obtained from wind tun nel tests with various submersible models OTC Paper 4124 Proceedings Offshore Technology Conference Houston TX May 1981 17 Ribbe J H and Brusse J C Simulation of the airwater interface for wind tunnel testing of floating structures Proceedings Fourth US National Conference Wind Engineering Research B J Hartz Ed Department of Civil Engineering University of Washington Seattle July 1981 18 Macha J M and Reid D F Semisubmersible wind loads and wind effects Paper no 3 Annual Meeting New York November 1984 New York The Society of Naval Architects and Marine Engineers 1984 19 Wardlaw R L Laurich P H and Mogridge G R Modeling of dynamic loads in wave basin tests of the semisubmersible drilling platform ocean ranger Proceedings International Conference on FlowInduced Vibrations BownessonWindermere UK May 1214 1987 20 Cowdrey C F and Gould R F TimeAveraged Aerodynamic Forces and Moments on a National Model of a Submersible Offshore Rig NMI R25 OTR7748 National Maritime Institute Feltham UK September 1982 21 Ponsword P J Wind Tunnel Measurements of Aerodynamic Forces and Moments on a Model of a Semisubmersible Offshore Rig NMI R34 OTR7807 National Maritime Institute Feltham UK June 1982 22 Troesch AW Van Gunst RW and Lee S 1983 Wind loads on a 1115 model of a semisubmersible Marine Technology 20 283289 23 Pike P J and Vickery B J A wind tunnel investigation of loads and pressure on a typical guyed tower offshore platform OTC Paper 4288 Proceedings Offshore Technology Conference Houston TX May 1982 24 Glasscock MS and Finn LD 1984 Design of a guyed tower for 1000 ft of water in the Gulf of Mexico Journal of Structural Engineering 110 10831098 k k k k 384 25 Offshore Structures 25 Morreale T A Gergely P and Grigoriu M Wind Tunnel Study of Wind Loading on a Compliant Offshore Platform NBSGCR84465 National Bureau of Standards Washington DC December 1983 26 Smith J R and Taylor R S The development of articulated buoyant column sys tems as an aid to economic offshore production Proceedings European Offshore Petroleum Conference Exhibition London October 1980 pp 545557 27 Mercier J A Leverette S J and Bliault A L Evaluation of Hutton TLP response to environmental loads OTC Paper 4429 Proceedings Offshore Technology Confer ence Houston TX May 1982 28 Ellis N Tetlow J H Anderson F and Woodhead A L Hutton TLP ves sel Structural configuration and design features OTC Paper 4427 Proceedings Offshore Technology Conference Houston TX May 1982 29 Kareem A and Dalton C Dynamic effects of wind on tension leg platforms OTC Paper 4229 Proceedings Offshore Technology Conference Houston TX May 1982 30 Vickery B J Wind loads on compliant offshore structures Proceedings Ocean Structural Dynamics Symposium Department of Civil Engineering Corvallis OR September 1982 pp 632648 31 Simiu E and Leigh SD 1984 Turbulent wind and tension leg platform surge Journal of Structural Engineering 110 785802 httpswwwnistgovwind 32 Salvesen N von Kerczek C H Vue D K et al Computations of nonlinear surge motions of tension leg platforms OTC Paper 4394 Proceedings Offshore Technology Conference Houston TX May 1982 33 Pinkster J A and Van Oortmerssen G Computation of first and secondorder forces on oscillating bodies in regular waves Proceedings Second International Conference on Ship Hydrodynamics University of California Berkeley 1977 34 Sarpkaya T and Isaacson M 1981 Mechanics of Wave Forces on Offshore Struc tures New York Van Nostrand Reinhold 35 Jefferys ER and Patel MH 1982 On the dynamics of taut mooring systems Engineering Structures 4 3743 36 Simiu E Carasso A and Smith CE 1984 Tether deformation and tension leg platform surge Journal of Structural Engineering 110 14191422 37 Simiu E and Carasso A Interdependence between dynamic surge motions of platform and tethers for a deep water TLP Proceedings Fourth International Conference on Behavior of Offshore Structures BOSS 15 July 1985 Delft The Netherlands pp 557562 k k k k 385 26 Tensile Membrane Structures Tensile membrane structures owe their capacity to resist loads to tension stresses in membranes supported by cables columns other members such as beams or arches andor pressurized air 1 2 For a number of small structures with commonly used simple shapes cones ridgeandvalley shapes hyperbolic paraboloids also known as saddle shapes can tilevered canopies external and internal constant pressure coefficients specified for welldefined zones on the membrane surfaces are available in the literature eg 3 4 in formats similar to those used in codes and standards for ordinary structures Tentative aerodynamic information is also provided in 3 for the preliminary design of certain types of open stadium roofs For tensile membrane structures with unusual shapes andor with longspans eg exceeding 100 m say it is necessary to resort to wind tunnel testing Commonly per formed on rigid models such testing can provide time histories of pressures at large numbers of points on the structures surfaces The deformations induced by the time and spacedependent aerodynamic pressures can be calculated by accounting for geo metric and material nonlinearities and for dynamic effects Because these deformations are typically large and can therefore significantly affect the structures shape the rigid model that reproduces the original surface needs to be modified accordingly The mod ified rigid model is used to measure a new set of pressure time histories The stresses and deformations induced by those pressures can then be determined with improved accuracy 5 Deformations measured in aeroelastic tests are reported in 6 which notes that the prototype Froude number was reproduced in the laboratory No other information on the aeroelastic testing technique is provided in 6 Computational Wind Engineering CWE simulations are increasingly being per formed with a view to modeling aerodynamic or aeroelastic response 7 Their results have been validated in some cases see eg 8 In the absence of appropriate validation CWE results are generally not accepted for design purposes The form of tensile membrane structures must be consistent with specified i geo metric boundary conditions support geometry and cable or fixed edges and ii cable and fabric prestress Form finding is an intricate process that requires the use of special ized software see eg wwwformfinderat Prestressing and anticlastic shapes shapes with double curvature ie saddle forms are designed to prevent the occurrence of membrane flutter and of compression in the membrane and cables For structures with common shapes classified as small ie with dimensions in the order of 10 m to less Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 386 26 Tensile Membrane Structures than 100 m it is suggested in 3 that the sum of the ratios of prestress in the warp and weft directions in kN m1 to the respective radii of curvature in m is a useful indica tor of structural behavior if the sum exceeds 03 kN m2 the performance was typically found to be sound whereas if it is less than 02 kN m2 a detailed investigation of wind effects is in order Based on results of a carefully designed round robin exercise it is noted in 9 that different formfinding procedures can yield significantly different forms It is strongly emphasized in 9 that geometric and material nonlinearities render the structural anal ysis far more complex than is the case for typical structures For this reason and in the absence of a clear and consistent basis for ensuring structural safety by accounting for the various uncertainties inherent in the analysis it was found in 9 that estimated design stress factors varied among the round robin participants between 28 and 71 In addition to aerodynamic information applicable to the design of small membrane structures with simple shapes 3 provides tentative information that may be used for the preliminary design of a few types of open stadium roofs Measurements of pressures performed on rigid models by using pneumatic averaging are described extensively in 6 Similar though much more complete and accurate pressure measurements can currently be performed by using the pressure scanner technique Such measurements performed iteratively following the approach described in 5 can be employed in nonlinear finiteelement static and dynamic analyses to obtain the requisite design information While analyses of this type can in principle follow the databaseassisted design approaches described in Part II of this book it is shown in 10 that they present formidable difficulties that can result in incorrect response predictions This can be the case even if the use of follower wind forces see eg 11 ie wind forces that change direction by remaining normal to the moving membrane surface is included in the analyses see also 12 However according to 13 for a lowprofile cablereinforced airsupported structure fullscale measurements in strong winds showed that windtunnel pressure measurements on a rigid model used in conjunction with a straightforward linear model of the dynamic response provided a reasonable representation of the structures behavior under wind loads References 1 ASCE Tensile Membrane Structures in ASCE Standard ASCESEI 5516 Reston VA American Society of Civil Engineers 2016 2 Beccarelli P 2015 Biaxial Testing for Fabrics and Foils Milan Springer 3 Forster B and Mollaert M 2015 European Design Guide for Tensile Surface Struc tures Brussels Tensinet Publications 4 CEN Eurocode 1 Actions on structures Part 14 General actions Wind actions in EN 199114 European Committee for Standardization CEN 2005 5 Hincz K and GamboaMarrufo M 2016 Deformed shape wind analysis of tensile membrane structures Journal of Structural Engineering 142 04015153 6 Vickery BJ and Majowiecki M 1992 Wind induced response of a cable sup ported stadium roof Journal of Wind Engineering and Industrial Aerodynamics 4144 14471458 k k k k References 387 7 Heil M Andrew LH and Jonathan B 2008 Solvers for largedisplacement fluidstructure interaction problems segregated versus monolithic approaches Computational Mechanics 43 1 91101 8 Michalski A Kermel PD Haug E et al 2011 Validation of the computational fluidstructure interaction simulation at realscale tests of a flexible 29 m umbrella in natural wind flow Journal of Wind Engineering and Industrial Aerodynamics 99 4 400413 9 Gosling PD Bridgens BN Albrecht A et al 2013 Analysis and design of membrane structures results of a round robin exercise Engineering Structures 48 313328 doi 101016jengstruct201210008 10 Lazzari M Masowiecki M Vitaliani RV and Saetta AV 2009 Nonlinear FE analysis of Montreal Olympic Stadium roof under natural loading conditions Engineering Structures 31 1631 11 Lazzari M Vitaliani RV Majowiecki M and Saetta AV 2003 Dynamic behavior of a transgrity system subjected to follower wind loading Computers and Structures 81 21992217 12 Gil Pérez M Kang THK Sin I and Kim SD 2016 Nonlinear analysis and design of membrane fabric structures modeling procedure and case studies Journal of Structural Engineering 142 05016001 Nov 13 Mataki Y Iwasa Y Fukao Y and Okada A 1988 Windinduced response of lowprofile cable reinforced airsupported structures Journal of Wind Engineering and Industrial Aerodynamics 29 253262 k k k k 389 27 Tornado Wind and Atmospheric Pressure Change Effects 271 Introduction Tornadoes are storms containing the most powerful of all winds Their probabilities of occurrence at any one location are low compared to those of other extreme winds It has therefore been generally considered that the cost of designing structures to withstand tornado effects is significantly higher than the expected loss associated with the risk of a tornado strike risk being defined as the product of the loss by its probability of occurrence For this reason tornadoresistant design requirements are not included in current building codes or standards This is changing however as efforts are underway to develop standard requirements for the design of such facilities as fire stations police stations hospitals and power plants whose survival of a tornado strike is considered essential from a community resilience point of view The consequences of failure would be especially grave for nuclear power plants In the United States construction permits or operating licenses for nuclear power plants are issued or continued only if their design is consistent with Regulatory Guides issued by the US Nuclear Regulatory Commission or is otherwise acceptable to the Regulatory staff of that agency 1 2 Tornado effects may be divided into three groups 1 Wind pressures caused by the direct aerodynamic action of the air flow on the struc ture 2 Atmospheric pressure change effects 3 Impactive forces caused by tornadoborne missiles This chapter and Chapter 28 present design criteria and procedures developed to ensure an adequate representation of tornado effects on nuclear power plants Reference 1 uses a model of the tornado wind flow characterized by the following parameters i maximum rotational wind speed ii translational wind speed of the tornado vortex V tr iii radius of maximum rotational wind speed Rm iv pressure drop pa and v rate of pressure drop dpadt Values of these parameters specified by the US Nuclear Regulatory Commission 1 as a design basis for nuclear power plants are listed in Section 343 The use of this model for the estimation of wind pressures on structures is discussed in Section 272 Section 273 is concerned with atmospheric pressure change loading Recent experimental work on the modeling of tornadoes and of the pressures they induce on buildings is briefly reviewed in Section 274 Tornadoborne missile speeds are discussed in Chapter 28 which also discusses hurricaneborne missile speeds Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 390 27 Tornado Wind and Atmospheric Pressure Change Effects 272 Wind Pressures A procedure for calculating wind pressures proposed in 3 assumes the following 1 The wind velocities and therefore the wind pressures do not vary with height above ground 2 The rotational velocity component Figure 271 is given by the expressions VR r Rm VRm 0 r Rm 271 VR Rm r VRm r Rm 272 where VRm is the maximum rotational wind speed and Rm is the radius of maximum rotational wind speed 3 The wind flow model described by Eqs 271 and 272 moves horizontally with a translation velocity V tr The corresponding maximum wind speed is Vmax VRm Vtr 273 The flow described by Eqs 271273 is called the combined Rankine vortex Figure 271 The wind pressure pw used in designing structures or parts thereof may be written as pw qFCp qMCpi 274 where Cp is the external pressure coefficient Cpi is the internal pressure coefficient qF is the basic external pressure and qM is the basic internal pressure The quantities qF and qM may be calculated as follows qF CF s pmax 275 qM CM s pmax 276 pmax 1 2𝜌V 2 max 277 VR Vtr Figure 271 Rankine combined vortex model Source From 1 where 𝜌 is the air density and V max is the maximum horizon tal wind speed Table 32 If V max is expressed in mph and pmax in lb ft2 12𝜌 000256 lb ft2 mph2 The quantities CF s and CM s are reduction or size coefficients that account for the nonuniformity in space of the tornado wind field CF s can be determined from Figure 272 as a function of the ratio LRm where L is the horizontal dimension normal to the wind direction of the tributary area of the structural ele ment concerned if the wind load is distributed among several structural elements eg by a horizontal diaphragm L is the horizontal dimension normal to the wind direction of the total area tributary to those elements If the size and dis tribution of the openings are relatively uniform around the periphery of the structure CM s is determined in the same way as CF s using a value of L equal to the horizontal dimension of k k k k 272 Wind Pressures 391 02 04 05 06 07 08 09 10 04 06 08 10 12 14 16 18 20 Cs F L Rm Figure 272 Size coefficient CF s 3 the structure normal to the wind direction If the sizes and distribution of the openings are not uniform the following weighted averaging procedure is used 1 Determine the quantity r1Rm such that r1 Rm Rm r1 L 278 2 Locate the plan of the structure drawn at appropriate scale within the non dimensional pressure profile of Figure 273 with the left end of the structure at the coordinate r1Rm 3 Determine Cq from Figure 273 for each exposed opening 4 Determine CM s from Eq 279 CM s N i A0iCqi N i A0i 279 where A0i is the area of the opening at location i Cqi is the factor Cq at location i and N is the number of openings The coefficient Cq in Figure 273 represents nondimensionalized wind pressures and was calculated using Eqs 271273 and 277 To obtain Figure 272 the nondimensionalized pressures of Figure 273 were integrated between the limits r1 and r1 L where r1 is given by Eq 278 and the results of the integration were normalized the coefficient CF s is thus an approximate measure of the average pressure coefficient over the interval L 3 Numerical Example The sizes and distribution of the openings not represented in Figure 274 are assumed to be uniform around the periphery of the structure The ratio between area of openings and the total wall area is A0Aw 025 It is assumed that V max 200 mph 894 m s1 Rm 150 ft 46 m The pressures on the 100 ft 305 m side walls induced by wind blowing in the direction shown in Figure 274 are calculated as follows pmax 000256 2002 1024 lb ft2 4900 N m2 Eq277 k k k k 392 27 Tornado Wind and Atmospheric Pressure Change Effects 02 04 06 08 10 12 14 16 18 20 22 24 26 28 30 01 02 03 04 05 06 07 08 09 10 Cq r Rm Figure 273 Coefficient Cq 3 200 ft 61 m Wind direction 50 ft 152 m 100 ft 305 m Figure 274 Schematic view of building Basic external pressures L 200 ft 61 m L Rm 133 CF s 057 Figure 272 qF 057 1024 58 lb ft2 2800 N m2 Eq 275 Basic internal pressures CF s 057 Figure 272 qM 057 1024 58 lb ft2 2800 N m2 Eq 276 k k k k 273 Atmospheric Pressure Change Loading 393 The pressure coefficients are assumed in this example to have the following values Cp 07 Cpi 03 Wind pressure pw 07 58 03 58 58 lb ft2 2800 N m2 Eq 274 273 Atmospheric Pressure Change Loading Consider the cyclostrophic equation Eq 14 in which the term affected by the Coriolis acceleration may be neglected written as dpa dr 𝜌 V 2 R r 2710 where dpadr is the atmospheric pressure gradient at radius r from the center of the tornado vortex To obtain the pressure drop pa Eq 2710 is integrated from infinity to r Using the expression for V t given by Eqs 271 and 272 par 𝜌 V 2 Rm 2 2 r2 R2 m 0 r Rm 2711a par 𝜌 V 2 Rm 2 R2 m r2 Rm r 2711b In structures with no openings ie unvented structures the internal pressure remains equal to the atmospheric pressure before the passage of the tornado Therefore during the passage of the tornado the difference between the internal pressure and the atmo spheric pressure is equal to pa It follows from Eqs 2711 that the maximum value of pa which occurs at r 0 is pmax a 𝜌V 2 Rm 2712 If the structures are completely open the internal and external pressures are equal ized for practical purposes instantaneously so the loading due to atmospheric pres sure changes approaches zero In structure with openings ie vented structures the internal pressures change during the tornado passage by an amount pit Denoting the atmospheric pressure change by pat the atmospheric differential pressure that acts on the external walls is pat pit A useful model for pat can be obtained by assuming in Eqs 2711 that r V tr t where V tr is the tornado translation speed and t is the time A simpler model in which the variation of pat with time is given by the graph of Figure 275 may also be used The time varying internal pressures pit may be estimated by iteration as follows Assume that the building consists of a number n of compartments The air mass in compartment N where N n at time tj 1 is denoted by W Ntj 1 and may be written as WNtj1 WNtj GNintj GNouttjΔt 2713 where GNin and GNout denote the mass of air flowing into and out of compartment N per unit of time respectively and Δt is the time increment The air mass flow rates GN can be calculated as functions of the pressures outside and within the compartment N k k k k 394 27 Tornado Wind and Atmospheric Pressure Change Effects Time Atmospheric pressure change Rm Vtr 3 Rm Vtr Rm Vtr pa Figure 275 Idealized atmospheric pressure change versus time function 3 and of relevant geometrical parameters including opening sizes as shown subsequently The internal pressure in compartment N at time tj 1 piNtj 1 is then written as piNtj1 WNtj1 WNtj k piNtj 2714 where k 14 is the ratio of specific heat of air at constant pressure to specific heat of air at constant volume The air mass flow rate can be modeled as follows G 06CcA22𝛾1p1 p212 2715 where the nondimensional compressibility coefficient Cc has the expression Cc p2 p1 2k k k 1 1 p2p1k1k 1 p2p1 1 A2A12 1 A2A12p2p12k 12 2716 and A1 is the area on the side of compartment 1 of the wall between compartments 1 and 2 A2 is the area connecting compartments 1 and 2 k 14 p1 is the pressure in compartment 1 p2 is the pressure in compartment 2 p2 p1 and 𝛾1 is the mass per unit volume of air in compartment 1 If in compartments provided with a blowout panel the differential pressure exceeds the design pressure for a panel the blowout area is transformed into a wall opening To account for threedimensional effects disregarded in Eq 2715 the atmospheric differential pressures on external walls obtained by the procedure just described are multiplied by a factor of 12 3 Figure 276 is an illustration of the pressure distribution and of the flow pattern in a building during depressurization An illustration of a structure depressurization model with values of geometric parameters required as input in the calculations and an example of a calculated corresponding differential pressuretime history are shown in Figures 277 and 278 respectively k k k k 273 Atmospheric Pressure Change Loading 395 Port open to atmosphere Connectivity Compartment Air flow pattern Figure 276 Pressure distribution and flow pattern during building depressurization 3 Port 3 Port 1 Port 4 Port 2 A 34 ft2 Comp 8 V 15 100 ft3 Comp 6 V 121 700 ft3 Comp 5 V 320 200 ft3 Comp 4 V 295 800 ft3 Comp 7 V 15 840 ft3 Comp 3 V 194 700 ft3 Comp 2 V 41 800 ft3 Comp 1 V 1559 000 ft3 A 2250 ft2 A 294 ft2 A 358 ft2 A 390 ft2 A 44 ft2 A 66 ft2 A 66 ft2 A 66 ft2 A 42 ft2 A 21 ft2 1 ft2 0093 m2 1 ft3 0028 m3 A 63 ft2 A 24 ft2 A 21 ft2 A 22 ft2 Figure 277 Illustration of a structure depressurization model 3 k k k k 396 27 Tornado Wind and Atmospheric Pressure Change Effects 120 100 80 60 40 20 0 20 40 60 80 100 120 Between compartment 3 and outside atmosphere Between compartments 1 and 3 2 4 6 8 12 1 lbft2 48 Nm2 Time s Differential Pressure lbft2 10 Figure 278 Differential pressuretime history for compartments 1 and 3 Input time history based on Figure 275 using 3RmVtr 9 s and pa 432 lb ft2 3 274 Experimental Modeling of TornadoLike Wind Flows Facilities aimed to simulate tornado flows have been developed since the early in the early 1970s eg 47 see Figures 512 and 513 for two examples Their objective is to produce vortex flows with a strong rotation combined with a radial sink flow 8 From the nondimensional NavierStokes equations in cylindrical coordinates Lewellen 9 established that such flows depend upon three nondimensional parame ters the aspect ratio a hr0 where h and r0 are the axial inflow height and the updraft radius respectively the swirl ratio S and a radial Reynolds number Rer Q2𝜋𝜈 where 𝜈 is the kinematic viscosity and Q is the volumetric flow rate per unit axial length A commonly used expression for the swirl ratio is S r0Γ 2Qh 2717 where the circulation Γ Γ 2𝜋r0Vtmax 2718 and V tmax is the maximum tangential velocity Experimental and numerical results showed that flows with approximately swirl ratios S 05 and S 10 produced onecell k k k k References 397 and multiplecell vortices respectively 10 11 In addition it is established that the wind flows depend upon terrain roughness eg 12 Descriptions of flow fields associated with various values of S and Rer are presented in 8 and with the added benefit of modern measurement and flow visualization techniques in 7 A transition from laminar axisymmetric core to a turbulent core with greatly expanded radius termed vortex breakdown is noted in 13 the transition is due to development of an adverse pressure gradient as the laminar core spreads out in radius with increasing downstream distance 8 13 14 Measurements of pressures induced by tornadoes on structures are reported in 6 11 1517 It appears that aerodynamic pressures induced by tornadoes on lateral walls of lowrise buildings may differ in some cases than those induced by straight winds this is especially the case for suctions on roofs owing to suctions induced by atmospheric pressure defects on fully unvented structures In situ observations reported in 18 are a first promising attempt to document the structure of tornado wind flows near the ground References 1 US Nuclear Regulatory Commission Regulatory Guide 176 DesignBasis Tornado and Tornado Missiles for Nuclear Power Plants Revision 1 2007 2 US Nuclear Regulatory Commission NUREG0800 Standard Review Plan 332 Tornado Loads p 3326 Revision 3 March 2007 3 Rotz J V Yeh G C K and Bertwell W Tornado and Extreme Wind Criteria for Nuclear Power Plants Topical Report No BCTOP3A Revision 3 Bechtel Power Corporation San Francisco 1974 4 Ward NB 1962 The exploration of certain features of tornado dynamics using a laboratory model Journal of the Atmospheric Sciences 29 11941204 5 Haan FL Sarkar PP and Gallus WA 2008 Design construction and perfor mance of a large tornado simulator for wind engineering applications Engineering Structures 30 11461159 6 Tang Z Feng C Wu L et al 2017 Characteristics of tornadolike vortices simulated in a largescale wardtype simulator BoundaryLayer Meteorology doi 101007s1054601703057 7 Refan M and Hangan HM 2014 Characterization of tornadolike flow fields in a new model scale wind testing chamber Journal of Wind Engineering and Industrial Aerodynamics 151 107121 8 Church CR Snow JT Baker GL and Agee EM 1979 Characteristics of tornadolike vortices as a function of swirl ratio a laboratory investigation Journal of the Atmospheric Sciences 36 17551776 9 Lewellen WS 1962 A solution for threedimensional vortex flows with strong cir culation Journal of Fluid Mechanics 14 420432 10 DaviesJones RP 1973 The dependence of core radius on swirl ratio in a tornado simulator Journal of the Atmospheric Sciences 30 14271430 11 Haan FL Jr Kumar Balaramudu V and Sarkar PP 2010 Tornadoinduced wind loads on a lowrise building Journal of Structural Engineering 136 106116 k k k k 398 27 Tornado Wind and Atmospheric Pressure Change Effects 12 Natarajan D and Hangan H 2012 Large eddy simulations of translation and surface roughness effects on tornadolike vortices Journal of Wind Engineering and Industrial Aerodynamics 104106 577584 13 Hall MG 1972 Vortex breakdown Annual Review of Fluid Mechanics 4 195218 14 Tari PH Gurka R and Hangan H 2010 Experimental investigation of tornadolike vortex dynamics with swirl ratio the mean and turbulent flow fields Journal of Wind Engineering and Industrial Aerodynamics 98 936944 15 Kikitsu H Sarkar P P and Haan F L Jr Experimental study on tornadoinduced loads of lowrise buildings using a large tornado simulator Proceedings of the 13th International Conference on Wind Engineering Amsterdam Netherlands July 1015 2011 16 Thampi H Dayal V and Sarkar PP 2011 Finite element analysis of interac tion of tornadoes with a lowrise timber building Journal of Wind Engineering and Industrial Aerodynamics 99 369377 17 Mishra AR James DL and Letchford CW 2008 Physical simulation of a singlecelled tornadolike vortex flow field characterization Journal of Wind Engi neering and Industrial Aerodynamics 96 12431257 doi 101016jjweia200802063 18 Wurman J Kosiba K and Robinson P 2013 In situ Doppler radar and video observations of the interior structure of a tornado and the winddamage relation ship Bulletin of the American Meteorological Society 94 835846 k k k k 399 28 Tornado and HurricaneBorne Missile Speeds 281 Introduction Debris produced by windinduced damage to structures and various other objects that may be carried by strong winds can acquire sufficiently high speeds to cause serious damage to the structures or building components they impact during their flight Dam age that may be produced by certain types of objects for example roof gravel and light fences can be avoided by appropriately regulating their use in high wind zones objects such as roof pavers can be prevented from becoming windborne by adequately attach ing them to their supporting structure and openings can be protected from damage through the use of shutters However for the design of nuclear power plants or other facilities whose failure to perform adequately could be catastrophic specific allowance must be made in design for the impacts produced by windborne missiles in tornadoes or hurricanes The purpose of this chapter is to review approaches to determining tornado and hurricaneborne missile speeds for structural design purposes Sections 282 and 283 concern tornadoborne and hurricaneborne missiles respectively For additional information on windborne debris hazards see 17 282 TornadoBorne Missile Speeds To estimate speeds attained by an object under the action of aerodynamic forces induced by tornado winds a set of assumptions is needed concerning The aerodynamic characteristics of the object The detailed features of the wind flow field The initial position of the object with respect to the ground and to the tornado center and the translation velocity vector For the design of nuclear power plants objects commonly considered as potential mis siles include bluff bodies such as planks steel rods steel pipes utility poles and auto mobiles1 This section reviews approaches to the tornadoborne missile problem based on i deterministic and ii probabilistic modeling 1 Information on the behavior of automobiles in strong winds is presented in 1 2 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 400 28 Tornado and HurricaneBorne Missile Speeds 2821 Deterministic Modeling of DesignBasis Missile Speeds Equations of Motion and Aerodynamic Modeling The motion of an object can be described by solving a system of three equations of balance of momenta and three equations of balance of moments of momenta For bluff bodies in motion a major difficulty in writing these six equations is that the aerodynamic forcing functions are not known In the absence of a satisfactory model for the aerodynamic description of the missile as a rigid body it is customary to resort to the alternative of describing the missile as a material point acted upon by a drag force D 12𝜌 CDAVw VMVw VM 281 where 𝜌 is the air density Vw is the wind velocity VM is the missile velocity A is a suit ably chosen area and CD is the corresponding drag coefficient This model is reasonable if during its motion the missile either maintains a constant or almost constant attitude with respect to the relative velocity vector Vw VM or has a tumbling motion such that with no significant errors a mean value of the quantity CD A can be used in the expres sion for the drag D The assumption of a constant body attitude with respect to the flow would be credible if the aerodynamic force were applied at all times exactly at the center of mass of the body which is highly unlikely or if the body rotation induced by a nonzero aerodynamic moment with respect to the center of mass were prevented by aerodynamic forces intrinsic in the bodyfluid system There is no evidence to this effect so the assumption that windborne missiles will tumble during their flight is reasonable Assuming then that Eq 281 is valid and that the average lift force vanishes under tumbling conditions the motion of the missile viewed as a threedegreeof freedom system is governed by the relation dVM dt 1 2𝜌CDA m Vw VMVw VM gk 282 where g is the acceleration of gravity k is the unit vector along the vertical axis and m is the mass of the missile It follows form Eq 282 that for a given flow field and given initial conditions the motion depends only upon the value of the parameter a CDAm For a tumbling body this value can in principle be determined experimentally Unfortu nately little information on this topic appears to be available Information on tumbling motions under flow conditions corresponding to Mach numbers 0535 is available in 3 Those data were extrapolated in 4 to lower subsonic speeds according to this extrapolation for a randomly tumbling cube the quantity CDAm equals approximately the average of the projected areas corresponding to all positions statistically possible times the respective static drag coefficients 4 pp 1317 and 1416 In the absence of more experimental information it appears reasonable to assume that the effective product CDA is given by the expression CDA cCD1 CD2 CD3 283 where CDiAii 1 2 3 are products of the projected areas corresponding to the cases in which the principal axes of the body are parallel to the vector Vw VM times the respective static drag coefficients and c is a coefficient assumed to be 050 for planks k k k k 282 TornadoBorne Missile Speeds 401 rods pipes and poles and 033 for automobiles In the case of circular cylindrical bodies rods pipes poles the assumption c 05 is conservative Computation of Missile Speeds A computer program for calculating and plotting tra jectories and velocities of tornadoborne missiles is listed in 5 The program includes specialized subroutines incorporating the assumed model for the tornado wind field and the assumed drag coefficients which may vary as functions of Reynolds number Input statements include values of relevant parameters and the initial conditions of the missile motion In Eq 282 both Vw and VM are referenced with respect to an absolute frame The velocity Vw is usually specified as a sum of two parts The first part represents the wind velocity of a stationary tornado vortex and is referenced with respect to a cylindrical coordinate system The second part represents a translation velocity of the tornado with respect to an absolute frame of reference Transformations required to represent Vw in an absolute frame are derived in 5 and are incorporated in the computer program Maximum calculated horizontal missile speed V max Mh are reported in 5 as functions of the parameter CDAm under the following assumptions The rotational velocity of the tornado vortex V R is described by Eq 281 The radial velocity component V r and the vertical velocity component V z are given by the expressions suggested in 6 Vr 050VR 284 Vz 067VR 285 The radial component is directed toward the center of the vortex the vertical com ponent is directed upward The translation velocity of the tornado vortex V tr is directed along the xaxis The initial conditions at time t 0 are x0 Rm y0 0 z0 40 m VMx 0 VMy 0 VMz 0 where x y z are the coordinates of the center of mass of the missile and VMx V y VMz are the missile velocity components along the x y and zaxes Also at t 0 the center of the tornado vortex coincides with the origin O of the coordinate axes Similar calculations were performed independently by the US Nuclear Regulatory Commission for a set of potential missiles listed in Table 281 assuming the validity of the tornado model with the characteristics listed in Table 32 for Regions I II and III corresponding to Regions 1 2 and 3 in Figure 35 For details see 1 The ANSIANS232011 R2016 Standard 7 contains a number of differences in the specification of missile speeds with respect to the values of 1 A critique of various models of the wind field in tornadoes was recently pre sented in 8 and a novel improved modeling of tornadoborne missile flight was proposed in 9 2822 Probabilistic Modeling of DesignBasis Missile Speeds Reference 10 proposed a procedure for estimating speeds with 107year mean recur rence intervals of postulated missiles that strike a given set of targets within a nuclear k k k k 402 28 Tornado and HurricaneBorne Missile Speeds Table 281 Designbasis tornado missile spectrum and maximum horizontal speeds Vmax Mh Missile Type Schedule 40 Pipe Automobile Solid Steel Sphere Regions I and II 5 m 2 m 13 m Dimensions 0168 m dia 458 m long Region III 1 in dia 45 m 17 m 15 m 254 cm dia Regions I and II 1810 kg Mass 130 kg Region III 1178 kg 00669 kg Regions I and II 00070 m2 kg1 CDAm 00043 m2 kg1 Region III 00095 m2 kg1 00034 m2 kg1 Region I 41 m s1 41 m s1 8 m s1 VMh max Region II 34 m s1 34 m s1 7 m s1 Region III 24 m s1 24 m s1 6 m s1 power plant or similar installation The procedure is based on assumptions concerning the number and location of potential missiles the magnitude of the force opposing mis sile takeoff the direction of the tornado axis of translation and the size of the target area The results of the calculations depend upon the parameter CDAm and the ratio k between the minimum aerodynamic force required to cause missile takeoff and the weight of the missile A listing of the computer program used in the procedure is avail able in 11 A more elaborate approach to the development of a riskinformed approach is pro posed in 12 which defines a missile impact probability MIP as the number of hits per missile per unit of target area In this approach the hit frequency given a target structure is proportional to the tornado frequency the number of missiles the target area and the MIP and can be used for probabilistic risk assessments of core damage and radioactive release In 12 the MIP was computed using data from 13 The MIP depends on tornado characteristics height of target shielding inherent in the configu ration of buildings in a plant and area of spread of the missiles initial location and is independent of tornado frequency An innovative approach that does not require the use of Monte Carlo simulations is described in 14 which uses a threedegreeoffreedom model of the missile motion rather than a sixdegreeof freedom model The translating tornado wind velocity field can be described either by using the Rankine vortex or the Fujita model Also included in this approach is a model for the lifting of potential missiles initially located on the ground in the tornado path k k k k 283 HurricaneBorne Missile Speeds 403 283 HurricaneBorne Missile Speeds Calculated hurricaneborne missile speeds for the design of nuclear power plants are listed in 15 for the missiles considered in 1 and in addition for a platelike and a planklike missile that arise from metallic siding dislodged during a tornado event The assumptions on the basis of which the calculations were performed and the properties of the missiles being considered are considered in Section 2831 A sample of results of the numerical calculations is presented in Section 2832 Closed form as opposed to numerical solutions can be obtained for the case of wind speeds independent of height above ground and are presented in Section 2833 The closed form equations provide useful insights into the missiles dynamic behavior as a function of the various parameters of the motion initial conditions hurricane wind speeds parameters defining missile properties A summary of the numerical results of interest for regulatory purposes is presented in 16 2831 Basic Assumptions This section considers the assumptions on the basis of which the calculations were per formed 1 Unlike for tornadoes for hurricanes winds updraft speeds may be neglected It fol lows that forces tending to increase the elevation of the missile with respect to the ground level may be assumed to be negligible as well In particular no updraft forces are available to lift automobiles 2 The missiles start their motion with zero initial velocity from an elevation h above ground As was the case for the tornado missile analyses performed for Regulatory Guide 176 it was assumed h 40 m In addition the assumptions h 30 20 and 10 m were used These assumptions imply that the change in the hurricane wind field through which the missile travel during its flight time is small Indeed for h 40 m the flight time tmax that is the time it takes the missile to reach the ground from its initial eleva tion is tmax 2 40 g 12 286 seconds where g 981 m s2 is the acceleration of gravity Therefore for all the elevations h assumed in the calculations tmax 3 s Let the hurricane speed be 100 m s1 say and the radius of maximum wind speed be 15 km the vast majority of hurricanes have radii of maximum wind speeds one order of magnitude larger Assume conservatively that the horizontal distance traveled by the missile is in the order of 100 m s1 3 s 300 m and that the missiles horizontal trajectory is tangent to the circle with radius 15 km assumed conservatively to represent the hurricanes radius of maximum wind speeds At the end of the trajectory the distance from the center of the circle to the missile will then be r 1500 cos tan1 3001500 1530 m 286 k k k k 404 28 Tornado and HurricaneBorne Missile Speeds For practical purposes the wind flows at 1500 and 1530 m from the center can be assumed to be the same The differences between wind fields at the beginning and end of the missile trajectory ie over a time interval in the order of 3 s may similarly be assumed to be small 3 Suburban terrain exposure and open terrain exposure represent respectively Expo sure B and C as defined in the ASCE 7 Standard For open terrain exposure the wind speed vh considered in the calculations represents the peak 3second gust speed and varies with height above ground z in accordance with the power law vopen h z vopen h 10 z 10 195 287a where 10 is the peak 3second gust speed at 10 m above ground in open terrain A simplified model of the wind field adopted in the ASCE 705 Standard 2006 is based on the assumption that the retardation of the wind flow by friction at the ground surface becomes negligible at an elevation referred to conventionally as the gradient height z 274 m At the gradient height the wind speed is in accordance with Eq 287a 274 m 142 10 m In that simplified model it is further assumed that for suburban terrain exposure the retardation of the wind flow by friction at the ground surface becomes negligible at a gradient height z 366 m The retar dation of the wind flow by surface friction is effective up to higher elevations than over open exposure because the friction is stronger over suburban than over open terrain For suburban terrain exposure the wind speed considered in the calculations repre sents the peak 3second gust speed and varies with height above ground z in accor dance with the power law vsub h z vsub h 366 m z 366 m 17 287b z in meters Since 366 m 142 10 m Eq 287b can be written as vsub h z vopen h 10 142 z 366 17 287c For example if vopen h 10 m 40 and 150 m s1 𝛼 195 then vsub h 10 m 34 and 1275 m s1 𝛼 17 respectively The equations of motion of the missiles used in conjunction with Eqs 287a and 287b can only be solved numerically Results of numerical calculations are presented in Section 2832 For simplified representations of the hurricane flow field it is possible to solve the equations of motion in closed form Such closed form solutions are presented in Section 2833 4 As in the case of tornadoborne missiles the aerodynamic force acting on a missile at any point of its trajectory was assumed to be proportional to the square of the velocity at that point times the parameter a 1 2 𝜌 CD A m 288 k k k k 283 HurricaneBorne Missile Speeds 405 where 𝜌 is the air density 12 kg m3 CD is the drag coefficient characterizing the average aerodynamic pressure acting on the missile A is the effective area of the missile that is the area by which pressures must be multiplied to yield the aero dynamic force and m is the mass of the missile For a plank with length and width 305 m 0305 m A 093 m2 mass m 38 kg steel board batten siding coated in PVC for a slab with length and width 305 m 153 m A 467 m2 mass m 38 kg The assumptions concerning the areas A are conservative For these two missiles it is assumed CD 12 Therefore a 0176 and a 00885 m1 respectively For the other missiles being considered the parameters a have the same values as in Table 281 Software for the calculation of hurricaneborne missile speeds based on the assump tions listed in this section is available at httpswwwnistgovwind 2832 Numerical Solutions Reference 15 lists Terminal horizontal missile speeds ie horizontal speeds at the time the missile reaches the ground Terminal total missile speeds ie resultants of the horizontal and vertical missile speeds at the time the missile reaches the ground Maximum horizontal wind speeds ie largest horizontal wind speeds reached during the missile flight Maximum total missile speeds for the following conditions Wind flows corresponding to 3second wind speeds 10 m 40150 m s1 in incre ments of 10 m s1 at 10 m above terrain with open exposure i over open terrain and ii over suburban terrain Missiles starting from rest from elevations 40 30 20 and 10 m For values of the parameter a 0006 in particular for the four missiles covered by Regulatory Guide 176 the differences between the maximum missile speeds and the speeds at the time the missiles reach the ground level are not significant However for values of the parameter a 0006 m1 those differences can be large The explanation for the decrease of the missile speeds from their maximum values is the following After reaching those maximum speeds the difference vh vmh between the hurricane wind speed and the horizontal missile speed can become negative as the missile moves at lower elevations where owing to friction at the ground level hurricane speeds are low The missile motion is then decelerated Figure 281 shows an example of results obtained by numerical calculations For example for hurricanes and tornadoes with 230 mph 103 m s1 maximum 3second wind speeds at 10 m above terrain with open exposure calculated maximum horizontal speeds of missiles listed in Table 281 are shown in Table 282 Results obtained in 15 were used to develop Regulatory Guide 1221 16 k k k k 406 28 Tornado and HurricaneBorne Missile Speeds 40 50 60 70 80 90 100 110 120 130 140 150 005 010 015 020 0 50 100 150 Missile Characteristic Parameter a m1 ms1 Figure 281 Maximum total missile speeds in m s1 for parameters 0005 m1 a 0200 m1 and wind speeds over terrain with open exposure 10 m 40 50150 m s1 Missiles start at 40 m above ground level Table 282 Calculated maximum horizontal missile speeds in hurricanes and tornadoes in m s1 Hurricanes Tornadoes Region I Solid steel sphere 48 8 Schedule 40 pipe 54 41 5 m automobile 68 41 2833 Simplified Flow Field Closed Form Solutions It is now shown that a closed form solution can be obtained under the assumption that the wind speed vh does not depend on height above ground To check the validity of the algorithm by which they were obtained numerical solutions corresponding to that assumption were compared to their closed form counterparts It was assumed that the vertical drag force is negligible and that the parameter a is given by Eq 288 k k k k 283 HurricaneBorne Missile Speeds 407 The equation of horizontal motion of the missile can be written as dvmh dt avh vmh2 289 where vmh is the horizontal missile velocity Equation 289 can be written as follows dvh vmh dt avh vmh2 2810 Let vh vmh y Eq 2810 becomes dy dt ay2 2811 It follows that dy y2 adt 2812 1 y at C 2813 vh vmh 1 at C 2814 vmh vh 1 at C 2815 For t 0 vmh 0 so C 1vh Therefore vmh vh vh avht 1 2816 For example for vh 100 m s1 a 00042 m1 a 40 m initial elevation of the mis sile and therefore it takes the missile a time t 2 4098112 286 s to reach the ground level under the action of gravity and the horizontal missile speed at that time is vmh 100100120 1 5455 m s1 The horizontal distance traveled by the missile in 286 s is a small fraction of the hurri canes radius of maximum wind speeds assumed conservatively to be 15 km Denoting the horizontal position of the missile by xmh with the change of variable t 1 avh 𝜏 2817 integration of Eq 2818 in which vmh dxmhdx yields xmh vh𝜏 1a log 𝜏 𝜏0 B 2818 where the integration constant C was written in the form C B 1aln 𝜏0 and 𝜏0 is the value taken on by 𝜏 for t 0 After some algebra since for t 0 xmh 0 x vht 1 a log1 avht 2819 For vh 100 m s1 a 00042 m1 t 286 seconds xmh 286 100042 log 1 286 00042 100 98 m It is shown in 15 that this result differs negligibly from its counterparts obtained numerically thus verifying the numerical procedure being used A similar verification was performed for tornadoborne missile speeds k k k k 408 28 Tornado and HurricaneBorne Missile Speeds References 1 Paulikas MJ Schmidlin TW and Marshall TP 2016 The stability of passenger vehicles at tornado wind intensities of the enhanced Fujita scale Weather Climate and Society 8 8591 2 Haan FL Sarkar PP Kop GA and Stedman DA 2017 Critical wind speeds or tornadoinduced vehicle movements Journal of Wind Engineering and Industrial Aerodynamics 168 18 3 Hansche E and Rinehart JS 1952 Air drag on cubes at Mach numbers 05 to 35 Journal of the Atmospheric Sciences 19 8384 4 Hoerner S F FluidDynamic Drag published by the author 1958 5 Simiu E and Cordes MR TornadoBorne Missile Speeds NBSIR 761050 National Bureau of Standards Washington DC 1976 httpswwwnistgovwind 6 McDonald JR Mehta KC and Minor JE 1974 Tornadoresistant design of nuclear powerplant structures Nuclear Safety 15 432439 7 American Nuclear Society ANSIANS232011 Estimating tornado hurricane and extreme straight wind characteristics at nuclear facility sites La Grange Park Illinois reaffirmed Jun 29 2016 8 Gillmeier S Sterling M Hemida H and Baker CJ 2018 A reflection on analytical tornadolike vortex flow field models Journal of Wind Engineering and Industrial Aerodynamics 174 1027 9 Baker CJ and Sterling M 2017 Modelling wind fields and debris flight in torna does Journal of Wind Engineering and Industrial Aerodynamics 168 312321 10 Simiu E and Cordes MR 1983 Tornadoborne missile speed probabilities Jour nal of Structural Engineering 109 154168 Online publication date January 1 1983 httpswwwnistgovwind 11 Cordes M R and Simiu E Probabilistic Assessment of TornadoBorne Missile Speeds NBSIR 802117 National Bureau of Standards Washington DC 1980 httpswwwnistgovwind 12 Pensado O Analysis of Missile Impact Probability for Generic Tornado Hazard Assessments Prepared for US Nuclear Regulatory Commission Division of Risk Assessment of the Office of Nuclear Reactor Regulation Southwest Research Insti tute Center for Nuclear Waste Regulatory Analyses 2016 13 EPRI Tornado Missile Risk Analysis Report NP768 Electric Power Research Institute Washington DC 1978 14 Eguchi Y Murakami T Hirakuchi H et al 2017 An evaluation method for Tor nado missile strike probability with stochastic correlation Nuclear Engineering and Technology 49 395403 15 Simiu E and Potra F Technical Basis for Regulatory Guidance on DesignBasis Hurricane HurricaneBorne Missile Speeds for Nuclear Power Plants NUREGCR7004 S Sancaktar NRC Project Manager National Institute of Stan dards and Technology Gaithersburg MD NRC Code 6726 Nov 2011 httpswww nistgovwind 16 US Nuclear Regulatory Commission NRC DesignBasis Hurricane and Hurricane Missiles for Nuclear Power Plants Regulatory Guide 1221 2011 17 ASCE WindBorne Debris Hazards NB Kaye ed Environmental Wind Engineer ing Committee Wind Engineering Division American Society of Civil Engineers 2018 Appendices k k k k 411 Appendix A Elements of Probability and Statistics A1 Introduction A11 Definition and Purpose of Probability Theory Following Cramér 1 probability theory will be defined as a mathematical model for the description and interpretation of phenomena showing statistical regularity Examples are phenomena such as the wind intensity at a given location the turbulent wind speed fluctuations at a point the pressure fluctuations on the surface of a building or the fluc tuating response of a structure to wind loads Probabilistic models arising in connection with the wind loading of structures are discussed in Sections A1A6 Consider an experiment that can be repeated an indefinite number of times and whose outcome can be the occurrence or nonoccurrence of an event A If for large values of trials n the ratio mn called the relative frequency of the event A differs little from some unique limiting value PA the number PA is defined as the probability of occurrence of event A For example if a coin is tossed the ratio of the number of heads observed in a very large recorded sequence of Hs heads and Ts tails should be close to 12 so that in any one toss the probability of occurrence of a head would be 12 Consider however the recorded sequence H T H T H T H T H T H T consisting of alternating Hs and Ts If in this sequence the observed outcome of a toss is a head the probability of a head in the next toss will not be 12 2 Indeed for the definition of probability just advanced to be meaningful it is required that the sequence S previously referred to satisfy the condition of randomness This condition states that the relative frequency of event A must have the same limiting value in the sequence S as in any partial sequence that might be selected from it in any arbi trary way the number of terms in any sequence being sufficiently large and the selection being made in the absence of any information on the outcomes of the experiment 3 The hypothesis that limiting values of the relative frequencies exist is confirmed for a wide variety of random phenomena by a large body of empirical evidence A12 Statistical Estimation Data obtained from observations must be fitted to mathematical models provided by probability theory by using statistical methods Such methods fall into two broad Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 412 Appendix A Elements of Probability and Statistics categories parametric and nonparametric Parametric models aim to estimate parameters of the probabilistic models Nonparametric parameterfree models are typically applied to large samples of rankordered data obtained in some applications by numerical simulation Like probability theory statistics is a vast field Basic statistical notions and meth ods used in applications connected with the wind loading of structures are discussed in Chapter 3 and Appendices C and E References that complement the material covered in this Appendix include 413 A2 Fundamental Relations A21 Addition of Probabilities Consider two events A1 and A2 associated with an experiment Assume that these events are mutually exclusive ie cannot occur at the same time The event that either A1 or A2 will occur is denoted by A1 A2 The probability of this event is PA1 A2 PA1 PA2 A1 The empirical basis of the addition rule Eq A1 is that if the relative frequency of event A1 is m1n and that of event A2 is m2n the frequency of either A1 or A2 is m1 m2n Equation A1 then follows from the relation between frequencies and probabilities and can obviously be extended to any number of mutually exclusive events A1 A2 An Example A1 For a fair die the probability of throwing a five is 16 and the probability of throwing a six is 16 The probability of throwing either a five or a six is then 16 16 13 Let the nonoccurrence of event A be denoted by A Events A and A are mutually exclusive Also the event that A either occurs or does not occur is certain that is its probability is unity PA A 1 A2a Equation A2a follows immediately from the addition rule Eq A1 applied to the events A and A the probabilities of which are the limiting values of the relative fre quencies mn and n mn respectively The probability that A does not occur can be written as PA 1 PA A2b Two events for which Eq A2b holds are said to be complementary A22 Compound and Conditional Probabilities The Multiplication Rule Consider events A and B that may occur at the same time The probability of the event that A and B will occur simultaneously is called the compound probability of events A and B and is denoted by PA1 A2 The probability of event A given that event B has k k k k Appendix A Elements of Probability and Statistics 413 already occurred is denoted by PAB and is known as the conditional probability of event A under the condition that B has already occurred Formally PAB is defined as follows PA B PA B PB A3a In Eq A3a it is assumed that PB 0 Similarly if PA 0 PB A PA B PA A3b Example A2 In a certain region records show that in an average year 60 days are windy 200 days are cold and 50 days are both windy and cold Let the probability that a day will be windy and the probability that a day will be cold be denoted by PW and PC respectively If it is known that condition C ie cold weather prevails the probability that a day is windy PWC is PW C PC 50365 200365 50 200 From Eqs A3a and A3b it follows that PA B PBPA B PAPB A A4 Equation A4 is referred to as the multiplication rule of probability theory A23 Total Probabilities If the events B1 B2 Bn are mutually exclusive and PB1 PB2 PBn 1 the probability of event A is PA PA B1PB1 PA B2PB2 PA BnPBn A5 Equation A5 is referred to as the theorem of total probability Example A3 With reference to the previous example we denote the probability of occurrence of winds as PW the probability of occurrence of winds given that a day is cold as PWC the probability that a day is not cold as PW C the probability that a day is cold as PC and the probability that a day is not cold as PC From Eq A5 it follows that PW PW CPC PW CPC 50 200 200 365 10 165 165 365 60 365 A24 Bayes Rule If B1 B2 Bn are n simultaneously exclusive events the conditional probability of occurrence of Bi given that the event A has occurred is PBi A PA BiPBi PA B1PB1 PA BnPBn A6 k k k k 414 Appendix A Elements of Probability and Statistics Equation A6 follows immediately from Eqs A3b and A4 in which B is replaced by Bi and Eq A5 Equation A6 allows the calculation of the posterior probabilities PBiA in terms of the prior probabilities PB1 PB2 PBn and the conditional probabilities PAB1 PAB2 PABn Example A4 On the basis of experience with destructive effects of previous tor nadoes it was estimated subjectively that the maximum wind speeds in a tornado were 5070 m s1 It was further estimated also subjectively that the likelihood of the speeds being about 50 60 and 70 m s1 is P50 03 P60 05 and P70 02 These values are prior probabilities According to a subsequent failure investigation the speed was 50 m s1 However associated with the investigation were uncertainties that were estimated subjectively in terms of conditional probabilities P 50Vtrue that is of probabilities that the speed estimated on the basis of the investigation is 50 m s1 given that the actual speed of the tornado was Vtrue The estimated values of P 50Vtrue were P 5050 06 P 5060 03 P 5070 01 It follows from Eq A6 that the posterior probabilities that is the probabilities calcu lated by taking into account the information due to the failure investigation are P50 50 P 50 50P50 P 50 50P50 P 50 60P60 P 50 70P70 051 P60 50 043 P70 50 006 Whereas the prior probabilities favored the assumption that the speed was 60 m s1 according to the calculated posterior probabilities it is more likely that the speed was only 50 m s1 This result is of course useful only to the extent that the various subjective estimates assumed in the calculations are reasonably correct A25 Independence In the example following Eq A3b the occurrence of winds and the occurrence of low temperatures were not independent events Indeed in the region in question if the weather is cold the probability of windiness increases Assume now that event A consists of the occurrence of a rainy day in Pensacola Florida and event B consists of an increase in the world market price of gold It is reasonable to state that the probability of rain in Pensacola is in no way dependent upon whether such an increase has occurred or not In this case it is then natural to state that PA B PA A7 k k k k Appendix A Elements of Probability and Statistics 415 Two events A and B for which Eq A7 holds are called stochastically1 independent By virtue of Eqs A1 and A7 an alternative definition of independence is PA B PAPB A8 Example A5 The probability that one part of a mechanism will be defective is 001 for another part independent of the first this probability is 002 The probability that both parts will be defective is 001 002 00002 Three events A B and C are stochastically independent only if in addition to Eq A8 the following relations hold PA C PAPC PB C PBPC PA B C PAPBPC A9 In general n events are said to be independent if relations similar to Eq A9 hold for all combinations of two or more events A3 Random Variables and Probability Distributions A31 Random Variables Definition Let a numerical value be assigned to each of the events that may occur as a result of an experiment The resulting set of possible numbers is defined as a random variable Example A6 1 A coin is tossed The numbers zero and one are assigned to the outcome heads and the outcome tails respectively The set of numbers zero and one constitutes a ran dom variable 2 To each measurement of a quantity a number is assigned equal to the result of that measurement The set of all possible results of the measurements constitutes a ran dom variable Random variables are called discrete or continuous according to whether they may take on values restricted to a set of integers as in Example A6 1 or any value on a seg ment of the real axis as in Example A6 2 It is customary to denote random variables by capital letters eg X Y Z Specific values that may be taken on by these random numbers are then denoted by the corresponding lower case letters x y or z A32 Histograms Probability Density Functions Cumulative Distribution Functions Let the range of the continuous random variable X associated with an experiment be divided into equal intervals ΔX Assume that if the experiment is carried out n times 1 The word stochastic means connected with random experiments and reliability and is derived from the Greek 𝜎𝜏o𝜒𝛼𝜁o𝜇𝛼𝜄 meaning to aim at seek after guess surmise k k k k 416 Appendix A Elements of Probability and Statistics n1 n2 n3 n4 n5 n6 n7 n8 n9 X0 X1 X2 X3 X4 X5 X6 X7 X8 X9 X Figure A1 Histogram the number of times that X has taken on values in the given intervals X1 X0 X2 X1 Xi Xi1 is n1 n2 ni respectively A graph in which the numbers ni are plotted as in Figure A1 is called a histogram similar graphs may be plotted for discrete variables Let the ordinates of the histogram in Figure A1 be divided by nΔX The resulting diagram is called the frequency density distribution The relative frequency of the event Xi 1 X Xi is then equal to the product of the ordinate of the frequency distribution ninΔX by the interval ΔX Since the area under the histogram is n1 n2 ni ΔX nΔX the total area under the frequency density diagram is unity As ΔX becomes very small so that ΔX dx and as n becomes very large the ordinates of the frequency density distribution approach in the limit values denoted by f x where x denotes a value that may be taken on by the random variable X The function f x is known as the probability density function PDF of the random variable X Figure A2a It follows from this definition that the probability of the event x X x dx is equal to f xdx and that f xdx 1 A10a In the experiment reflected in Figure A1 the number of times that X has assumed values smaller than Xi is equal to the sum n1 n2 ni Similarly the probability that X x called the cumulative distribution function CDF of the random variable X and denoted by Fx can be written as Fx x f xdx A10b that is the ordinate at X in Figure A2b is equal to the shaded area of Figure A2a It follows from Eq A10b that f x dFx dx A11 k k k k Appendix A Elements of Probability and Statistics 417 Figure A2 a Probability density function b Cumulative distribution function X x fx X Fx a x b A33 Changes of Variable We consider here only the change of variable y x ab where a and b are constants We assume the CDF FXx is known and we seek the CDF FYy and the PDF f Yy We can write FXx PX x P X a b x a b FYy A12abc Since Eq A12c implies that dFXx dFYy or f Xxdx f Yydy it follows that fXx 1 bfYy A13 A34 Joint Probability Distributions Let X and Y be two continuous random variables and let f x ydxdy be the probability that x X x dx and y Y y dy The quantity f x y is called the joint PDF of the random variables X and Y Figure A3 The probability that X x and Y y is called the joint cumulative probability distribution of X and Y and is denoted by Fx y k k k k 418 Appendix A Elements of Probability and Statistics fx y X Y Figure A3 Probability density function fx y From the definition of f x ydxdy it follows that Fx y x y f x ydx dy A14a and f x ydx dy 1 A14b It follows from Eq A14a that f x y 2Fx y xy A15 If f x y is known the probability that x X x dx denoted by f Xxdx is obtained by applying the addition rule to the probabilities f x y dx dy over the entire Y domain fXx f x y dy A16 The function f Xx is called the marginal PDF of X Finally the probability that y Y y dy under the condition that x X x dx is denoted by f yxdy The function f yx is known as the conditional probability function of Y given that X x If Eq A3a is used it follows that f y x f x y fXx A17 If X and Y are independent f yx f Y y and f x y fXxfYy A18 Similar definitions hold for any number of discrete or continuous random variables k k k k Appendix A Elements of Probability and Statistics 419 A4 Descriptors of Random Variable Behavior A41 Mean Value Median Mode Standard Deviation Coefficient of Variation and Correlation Coefficient The complete description of the behavior of a random variable is provided by its proba bility distribution in the case of several variables by their joint probability distribution Useful if less detailed information is provided by such descriptors as the mean value the median the mode the standard deviation and in the case of two variables their correlation coefficient The mean value also known as the expected value or the expectation of the discrete random variable X is defined as EX m i1 xi fi A19 where m is the number of values taken on by x The counterpart of Eq A19 in terms of relative frequencies of the quantity EX is EX m i1 xi ni n A20 If the random variable X is continuous the expected value of X is written in complete analogy with Eq A18 as EX x f x dx A21 The median of a continuous random variable X is the value x that corresponds to the value 12 of the CDF The mode of X corresponds to the maximum value of the PDF Since Prob x X x dx fx dx the mode may be interpreted as the value of the variable that has the largest probability of occurrence in any given trial The mean value the median and the mode are measures of location The expected value of the quantity x EX2 is the variance of the variable X By virtue of the definition of the expected value Eq A21 the variance can be written as Varx EX EX2 x EX2f xdx A22 The quantity SDX VarX12 is the standard deviation of the random variable X The ratio SDXEX is the coefficient of variation CoV of X The variance the standard deviation and the CoV are useful measures of the scatter or dispersion of the random variable about its mean The correlation coefficient of two continuous random variables X and Y is defined as CorrX Y x EX y EY f x y dx dy SDXSDY A23 The correlation coefficient is similarly defined if the variables are discrete It can be shown that 1 CorrX Y 1 A24 k k k k 420 Appendix A Elements of Probability and Statistics It follows from Eq A23 that if two random variables are linearly related Y a bX A25 then CorrX Y 1 A26 The sign in the righthand side of Eq A26 is the same as that of the coefficient b in Eq A25 It can be proved that conversely Eq A26 implies Eq A25 The correla tion coefficient may thus be viewed as an index of the extent to which two variables are linearly related If X and Y are independent then CorrX Y 0 This follows from Eqs A23 A18 and A21 However the relation CorrX Y 0 does not necessarily imply the inde pendence of X and Y 4 A5 Geometric Poisson Normal and Lognormal Distributions A51 The Geometric Distribution Consider an experiment of the type known as Bernoulli trials in which i the only possible outcomes are the occurrence and the nonoccurrence of an event A ii the probability s of the event A is the same for all trials and iii the outcomes of the trials are independent of each other Let the random variable N be equal to the number of the trial in which the event A occurs for the first time The probability pn that event A will occur on the nth trial is equal to the probability that event A will not occur on each of the first n 1 trials and will occur on the nth trial Since the probability of nonoccurrence of event A in one trial is 1 s Eq A2 and since the n trials are independent it follows from the multiplication rule Eq A8 pn 1 sn1s n 1 2 3 A27 This probability distribution is known as the geometric distribution with parameter s The probability Pn that event A will occur at least once in n trials can be found as follows The probability that event A will not occur in n trials is 1 sn The probability that it will occur at least once is therefore Pn 1 1 sn A28 The expected value of N is by virtue of Eqs A19 and A27 N n1 n1 sn1s A29 The sum of this series can be shown to be N 1 s A30 The quantity N is called the mean return period or the mean recurrence interval MRI k k k k Appendix A Elements of Probability and Statistics 421 Example A7 For a die the probability that a four occurs in a trial is s 16 If the total number of trials is large it may be expected that in the long run a four will appear on average once in N 116 6 trials The extension of the Mean Recurrence Inter vals MRI concept to extreme wind speeds is discussed in Section 311 A52 The Poisson Distribution Consider a class of events each of which occurs independently of the other and with equal likelihood at any time 0 t T A random variable is defined consisting of the number N of events that will occur during an arbitrary time interval 𝜏 t2 t1t1 0 t1 t2 T Let pn 𝜏 denote the probability that n events will occur during the inter val 𝜏 If it is assumed that pn 𝜏 is not influenced by the occurrence of any number of events at times outside this interval it can be shown that pn 𝜏 𝜆𝜏n n e𝜆𝜏 n 0 1 2 3 A31 If Eqs A21 and A22 are used it is found that the expected value and the variance of n are both equal to 𝜆𝜏 Since 𝜆𝜏 is the expected number of events occurring during time 𝜏 the parameter 𝜆 is called the average rate of arrival of the process and represents the expected number of events per unit of time The applicability of Poissons distribution may be illustrated in connection with the incidence of telephone calls in a telephone exchange Consider an interval of say 15 minutes during which the average rate of arrival of calls is constant During any subinterval of those 15 minutes the incidence of a number n of calls is as likely as during any other equal subinterval In addition it may be assumed that individual calls are independent of each other Therefore Eq A31 applies to any time subinterval 𝜏 lying within the 15minute interval Example A8 The estimated mean annual rate of arrival of hurricanes in Miami is λ 056year Consider a period 𝜏 3 years Therefore λτ 168 What is the proba bility that there will be two hurricane occurrences in Miami during a period τ 3 years From Eq A31 pn 2 τ 3 0263 A53 Normal and Lognormal Distributions Consider a random variable X that consists of a sum of small independent contributions X1 X2 Xn It can be proved that under very general conditions if n is large the PDF of X is f x 1 2𝜋𝜎x exp x 𝜇x2 2𝜎2 x A32 where 𝜇x EX and 𝜎2 x VarX are the mean value and the variance of X respec tively This statement is known as the central limit theorem The distribution represented by Eq A32 is called normal or Gaussian It can be shown that the distribution of a linear function of a normally distributed variable is also normal as is the sum of inde pendent normally distributed variables If the distribution of the variable Z ln X is normal the distribution of X is called lognormal Lognormal distributions are heavytailed meaning that the ordinates of its k k k k 422 Appendix A Elements of Probability and Statistics PDF are still significant for values X for which the ordinates of the Gaussian PDF are negligibly small A6 Extreme Value Distributions A61 Extreme Value Distribution Types Let the variable X be the largest of n independent random variables Y 1 Y 2 Y n The inequality X x implies Y1 x Y2 x Yn x Therefore FX x ProbY1 x Y2 x Yn x FY1xFY2x FYnx A33ab where to obtain Eq A33b from Eq A33a the generalized form of Eq A8 is used In the particular case in which the variables Y i are identically distributed ie have the same distribution FYxFYx Eq A33b becomes FXx FYxn A34 The distribution FYy is called the underlying or the initial distribution of the vari able Y which constitutes the parent population from which the largest values X have been extracted It has been shown that depending upon the properties of the initial distribution there exist three types of extreme value distributions the FisherTippett Type I Type II and Type III distributions of the largest values also known as the Gum bel Fréchet and reverse Weibull distributions In extreme wind climatology the initial distributions can be tentatively determined only for a few types of storm that do not include for example tropical storms For this reason in practice the choice among the three distributions can only be made on an empirical basis see Section A7 A611 Extreme Value Type I Distribution FIx exp exp x 𝜇 𝜎 x 𝜇 0 𝜎 A35 where 𝜇 and 𝜎 are the location and scale parameter respectively Equations A35 A21 and A22 yield the mean value and the standard deviation of the variate X EX 𝜇 05722𝜎 A36ab SDX 𝜋 6 𝜎 The percentage point function defined as the inverse of the CDF is xFI 𝜇 𝜎 ln ln FI A37 The estimated extreme value with MRI N years can be determined from Eqs A35A37 vIN EX 078SDXln N 0577 A38 where N 11 Fx k k k k Appendix A Elements of Probability and Statistics 423 A612 Extreme Value Type II Distribution FIIx exp x 𝜇 𝜎 𝛾 𝜇 x 𝜇 0 𝜎 𝛾 0 A39 where 𝜇 𝜎 and 𝛾 are the location scale and shape or tail length parameters For 𝛾 2 both the mean value and the standard deviation of the variate X diverge A613 Extreme Value Type III Distribution FIIIx exp x 𝜇 𝜎 𝛾 x 𝜇 A40 xFIII 𝜇 𝜎 lnFIII1𝛾 A41 The mean value and the standard deviation of the variate X are related to the parameters 𝜇 𝜎 and 𝛾 as follows SDX 𝜎 Γ 1 2 𝛾 Γ 1 1 𝛾 212 A42 EX 𝜇 𝜎 Γ 1 1 𝛾 A43 where Γ is the gamma function In wind engineering practice it is typically assumed that the Extreme Value Type I Gumbel distribution is an appropriate distributional model The rationale for this assumption is discussed in Section 332 A62 Generalized Extreme Value GEV Distribution The GEV distribution is applied to independent extreme data eg extreme wind speeds peak wind effects that exceed an optimal threshold Its CDF is FGEVx 𝜇 𝜎 k exp 1 k x 𝜇 𝜎 1k A44 where 1 kx 𝜇𝜎 0 𝜇 and 0 𝜎 For the shape parameter k 0 and k 0 Eq A44 corresponds to the EV II and EV III distribution respectively In the limit k 0 the GEV CDF is FGEVx 𝜇 𝜎 0 exp exp x 𝜇 𝜎 A45 and corresponds to the EV I distribution Equation A45 is the conditional CDF of the variate X given that X u where u is a sufficiently large optimal threshold The GEV is used with a different notation in Section C2 A63 Generalized Pareto Distribution GPD The GPD is applied to differences between independent extreme data and an optimal threshold Its expression is for c 0 FGPDy a c 1 1 c y a 1c A46 k k k k 424 Appendix A Elements of Probability and Statistics for c 0 FGPDy a 0 1 exp y a A47 where a 0 y 0 when c 0 and 0 y ac when c 0 Equation A46 is the conditional CDF of the excess of the variate X over the optimal threshold u Y X u given X u for u sufficiently large The tail length parameters c 0 c 0 and c 0 correspond respectively to EV II EV I and EV III distribution tails For c 0 Eq A47 the expression between braces is understood in a limiting sense as the exponential expya The relations between the parameters a and c and the mean value EY and standard deviation SDY of the variate Y are 14 a 1 2EY 1 EY SDY 2 A48a c 1 2 1 EY SDY 2 A48b A64 Mean Recurrence Intervals MRIs for Epochal and PeaksoverThreshold POT Approaches Epochal Approach Consider the largest value of the variate X within each of number of fixed epochs each assumed to be one year Given the CDF Fx of the variate X the probability of exceedance of x is 1 Fx and the MRI in years is N 11 Fx POT Approach We first consider the GEV distribution Let 𝜆 denote the average number per unit time ie the mean rate of arrival of exceedances of the thresh old u by the variate X and let the unit of time be 1 year The average number of exceedances in N years is then 𝜆 N An average epoch the average length of time between successive exceedances is then equal to 1𝜆 years For example if 𝜆 2 exceedancesyear the average epoch is 12 years if 𝜆 05 exceedancesyear the average epoch is 2 years The MRI in terms of the number of average epochs between exceedances of the value x is 1FX x 𝜆N Therefore the MRI of the event X x in years is N 1 𝜆1 FX x A49 FX x 1 1 𝜆N A50 A similar equation in which Y X u and y x u are substituted in Eq A50 for X and x applies to the Generalized Pareto Distribution that is FY y 1 1 𝜆N A51 1 1 c y a 1c 1 1 𝜆N A52 Therefore y a1 𝜆Nc c A53 k k k k Appendix A Elements of Probability and Statistics 425 and the value being sought is xN y u A54 where N is the MRI of x in years A7 Statistical Estimates A71 Goodness of Fit Confidence Intervals Estimator Efficiency Data obtained from observations may be viewed as observed values of random variables The behavior of the data may then be assumed to be described by models governing the behavior of random variables that is by mathematical models used in probability theory In practical applications from the nature of the phenomenon being investigated and on the basis of observations one must infer the probability distribution that will ade quately describe the behavior of the data and unless a nonparametric approach is used the parameters of that distribution or at least some characteristics of that distribution for example the mean and the standard deviation In practice given a set of observed data or a data sample it is hypothesized in the parametric approach that its behavior can be modeled by means of some probability dis tribution believed to be appropriate This hypothesis must then be tested Techniques are available that incorporate some measure of the degree of agreement or goodness of fit between the model including hypothesized values of its parameters and the data or conversely of the degree to which the data deviate from the model Techniques that allow the selection of the most appropriate distributional model and the estima tion of its best fitting parameters include among others the method of moments least squares the probability plot correlation coefficient and DATAPLOT and maximum likelihood For details on such techniques see also the publicly available NIST SEMAT ECH eHandbook of Statistical Methods 13 and R A Language and Environment for Statistical Computing 12 For details on Wstatistics see Appendix C An estimator is defined as a function 𝛼X1 X2 Xn of the sample data such that 𝛼 is a reasonable approximation of the unknown value 𝛼 of the distribution parameter or characteristic being sought As a function of random variables Xi i 1 2 n 𝛼 is itself a random variable This is illustrated by the following example Consider the observed sequence of 14 outcomes of an experiment consisting of the tossing of a coin H T T T H T H H T H H H T H A55a The random numbers associated with this experiment are the numbers zero and one which are assigned to the outcome heads and the outcome tails respectively The data sample corresponding to the observed outcome is then 0 1 1 1 0 1 0 0 1 0 0 0 1 0 A55b This sample is assumed to be extracted from an infinite population that in the case of an ideally fair coin will have a mean value denoted in this case by 𝛼 equal to 12 A reasonable estimator for the mean 𝛼 is the sample mean a 1 n n i1 Xi A56 k k k k 426 Appendix A Elements of Probability and Statistics where n is the sample size and Xi are the observed data For the sample of size 14 in Eq A55b 𝛼 37 If the samples consisting of the first seven and the last seven obser vations in Eq A55b are used 𝛼 47 and 𝛼 27 respectively As a random variable an estimator 𝛼 will have a certain probability distribution with nonzero dispersion about the true value 𝛼 Thus given a sample of statistical data it is not possible to calculate the true value 𝛼 being sought Rather confidence inter vals can be estimated of which it can be stated with a specified confidence level q that they contain the unknown value 𝛼 Typically a nominal 95 confidence interval is considered which corresponds for the Gaussian distribution to EX 2 SDX where EX and SDX denote the estimated mean value and standard deviation of the vari ate X In order for the confidence interval corresponding to a given confidence level q to be as narrow as possible it is desirable that the estimator being used be efficient Of two different estimators 𝛼1 and 𝛼2 of the same quantity being estimated the estimator 𝛼1 is said to be more efficient if E𝛼1 𝛼2 E𝛼2 𝛼2 A72 Parameter Estimation for Extreme Wind Speed Distributions Among the numerous methods for estimation of extreme wind distribution parameters by the epochal approach we mention the method of moments as applied to the EV I distribution and the Lieblein method which was developed specifically for the EV I distribution Both are covered in Section 333 For the POT approach wind speed data separated by intervals of five days or more may be regarded as independent although more rigorous methods for declustering data are available see Appendix C in which Poisson processes are applied to the estimation of extremes Let the wind speed data be denoted by xi Generalized Pareto Distribution GPD The analysis is performed on data xi u where u denotes the threshold If the threshold u is too large the size of the data sample will be small and the estimated values will be affected by large sampling errors If the threshold is too low the estimates biased by the presence in the sample of nonextreme wind data The analysis is carried out for a sufficiently large set of thresholds u For a subset of those thresholds the analysis will yield approximately the same estimated values of the parameters being sought A threshold within that subset referred to as optimal yields the estimates being sought The determination of the subset is performed visually and is subjective and slow An objective approach is presented in Appendix C Two methods for the estimation of the GPD are now presented In the method of moments the estimated GPD parameters are obtained by applying Eq A48 to the sample mean value and standard deviation of the data yi From Eq A54 it follows that the estimated wind speed with an Nyear MRI is xN yN u A57 where u is an optimal threshold In the de Haan method 15 the number of data above the threshold is denoted by k so that the threshold u represents the k 1th k k k k Appendix A Elements of Probability and Statistics 427 highest data point We have 𝜆 knyears where nyears is the length of the record in years The highest second highest kth highest k 1th highest data points are denoted by Xnn Xn1n Xnk 1n Xnkn u respectively Compute the quantities Mr n 1 k k1 i0 lnXn1n lnXnknr r 1 2 A58 The estimators of c and a are c M1 n 1 1 21 M1 n 2M2 n a uM1 n 𝜌1 A59ab 𝜌1 1 c 0 𝜌1 11 c c 0 A60ab Figure 33 is a POT plot of the estimated wind speeds obtained by Eqs A59 and A60 as functions of threshold u in mph and of sample size corresponding to the threshold u Generalized Extreme Value Distribution GEV The GEV distribution is applied to data that exceed a threshold u Unlike in the GPD the statistical analysis is performed on the data themselves rather than on the differences between the data and the threshold see Appendix C A8 Monte Carlo Methods Monte Carlo methods are a branch of mathematics pertaining to experiments on ran dom numbers The simulation of the statistics of interest is achieved by appropriate transformations of sequences of random numbers The new sequences thus obtained may be viewed as data the sample statistics of which are representative of the statistical properties of interest The following example illustrates the application of Monte Carlo techniques We con sider a sequence of uniformly distributed random numbers 0 yi 1 i 1 2 n The numbers yi are viewed as values of the CDF FI xi of a variate X with EV I distribution that is yi FI xi From Eq A37 it the follows that xyi 𝜇 𝜎 ln ln yi A61 From the sample xyi i 1 2 n of the variate X it is possible to obtain estimates of 𝜇 𝜎 and percentage points xFI for any specified FI The procedure is repeated a large number m of times A number m of sets of values 𝜇 𝜎 and xFI and corresponding histograms can then be obtained From the m sets statistics of those estimates can be produced For example large directional wind speed datasets of synoptic windstorms can be generated from relatively short measured wind datasets by using Monte Carlo simulations 16 k k k k 428 Appendix A Elements of Probability and Statistics A9 NonParametric Statistical Estimates A91 Single Hazards Consider a data sample of size n at a location where the mean arrival rate of the variate of interest 𝜆year If the rate were 𝜆 1year the estimated probability that the highest value of the variate in the set would be exceeded is 1n 1 and the corresponding estimated MRI would be N n 1 years on average n 1 trials would be required for a storm to exceed that highest valued Section 3112 Example 32 The estimated probability that the qth highest value of the variate in the set is exceeded is qn 1 the corresponding estimated MRI in years is N n 1q and the rank of the variate with MRI N is q n 1N In general 𝜆 1 and the estimated MRI is therefore N n 1q𝜆 years For example if n 999 hurricane wind speed data and 𝜆 05year the esti mated MRI of the event that the highest wind speed in the sample will occur is N n 1q𝜆 100005 2000 years the estimated MRI of the second highest speed is 1000 years and so forth The rank of the speed with a specified MRI N is q n 1N𝜆2 Example A9 Nonparametric MRI estimates for hurricane wind speeds from a spec ified directional sector at a specified coastal location The use of nonparametric esti mates of MRIs is illustrated for quantities forming a vector vk k 1 2 n where n is the number of trials The methodology is the same regardless of the nature of the vari ate which can represent wind effects or as in this example hurricane wind speeds We consider speeds blowing from the 225 sector centered on the SW ie 225 direction at milestone 2250 near New York City where 𝜆 0305year The data being used were obtained from the site httpswwwnistgovwind as indicated in Section 31 They are rankordered in Table A1 It is sufficient to consider the first 55 rankordered data since higherrank data are small The qth largest speed in the set of 999 speeds corresponds to a MRI N n 1 q𝜆 10000305q For the first highest and second highest speeds listed in Table A1 N 10000305 3279 years and N 10000305 2 1639 years respectively The peak 3second gust speed with a 100year MRI has rank q 10000305100 3278 that is 33 and is seen from Table A1 to be 17 m s1 Note that the precision of the estimates is poorer for higherranking speeds owing to the relatively large differences between successive higherranking speeds in Table A1 eg 54 vs 39 m s1 for the highest vs the second highest speed For this reason it is appropriate to develop datasets covering periods longer by a factor of 3 say than the specified design MRI A92 Multiple Hazards We now consider the case of multiple hazards for example synoptic wind speeds and thunderstorm wind speeds or hurricanes and earthquakes 2 A formula that takes into account the possibility that two or more hurricanes may occur at a site in any one year and is more exact for short MRIs eg 5 years is N 11 exp𝜆qn 1 For example for n 999 𝝀 05 and q 2 N 10005 years k k k k Appendix A Elements of Probability and Statistics 429 Table A1 Rankordered peak 3second gust speeds in m s1 from SW direction at 10 m above open terrain for 225 sector at milepost 2550 1minute speed in knots 0625 3second speed in m s1 Rank q SW 225 Rank q SW 225 Rank q SW 225 1 54 19 19 39 14 2 39 20 19 40 14 3 33 21 18 41 14 4 30 22 18 42 13 5 27 23 18 43 13 6 26 24 17 44 13 7 26 25 17 45 13 8 23 26 17 46 13 9 23 27 17 47 13 10 22 28 17 48 12 11 22 29 17 49 12 12 21 30 17 50 12 13 20 31 17 51 11 14 20 32 17 52 10 15 20 33 17 53 10 16 19 34 16 54 9 17 19 35 16 55 2 18 19 36 16 Example A10 Assume that the mean annual rates of synoptic storm and thun derstorm arrival at the location of interest are 𝜆s 4year and 𝜆t 35year The rankordered DCIs induced in a structural member by 10 000 synthetic synoptic storms and 10 000 thunderstorms are listed in Table A2 The MRI of DCIs 100 induced by synoptic storms is Ns ns 1qs𝜆s 100015 4 500 years so the probability that the DCI induced by synoptic winds is greater than 100 is 1500 in any one year Similarly the probability that the DCI induced by thunderstorm winds is greater than 100 is 110 0019 35 1317 in any one year The probability that the DCI induced by synoptic winds or by thunder storms is greater than 100 is 1500 1317 in any one year hint see Section 312 Eq 33 This corresponds to an MRI of the occurrence of the event DCI 100 equal to N 194 years A similar approach can be used for regions such as South Carolina and Hawaii sub jected to both hurricane and earthquake hazards see 17 For the approach to be appli cable in this case it is necessary to provide in addition to a probabilistic model of the extreme wind speeds at the location of interest and a procedure for determining the DCI demandtocapacity indexes induced in the structure by those speeds a prob abilistic model of the strength of the seismic events at that location and a procedure for determining the DCIs induced in the structure by those events k k k k 430 Appendix A Elements of Probability and Statistics Table A2 Rankordered DCIs Induced by synoptic storm and thunderstorm winds DCIs Induced by Synoptic Storms DCIs Induced by Thunderstorms Rank DCI Rank DCI 1 134 2 130 3 126 4 123 1 122 5 121 2 116 6 118 3 110 7 110 4 104 8 102 5 101 9 101 6 099 10 098 References 1 Cramér H 1955 The Elements of Probability Theory New York Wiley 2 Mihram AG 1972 Simulation New York Academic Press 3 von Mises R 1957 Probability Statistics and Truth London G Allen Unwin 4 Benjamin JR and Cornell CA 1970 Probability Statistics and Decision for Civil Engineers New York McGrawHill 5 Montgomery DC and Runger GC 2013 Applied Statistics and Probability for Engineers 6th ed Hoboken Wiley 6 Montgomery DC Runger GC and Hobele NF 2011 Student Solutions Manual Engineering Statistics 5the Hoboken Wiley 7 Kay S 2006 Intuitive Probability and Random Processes Using MATLAB New York Springer 8 Gumbel EJ 1958 Statistics of Extremes New York Columbia University Press 9 Coles S 2001 An Introduction to Statistical Modeling of Extreme Values London Springer 10 Castillo E Hadi AS Balakrishnan N and Sarabia JM 2004 Extreme Value and Related Models with Applications in Engineering and Science 1st ed Hoboken New Jersey Wiley 11 Beirlant J Goegebeur Y Segers J and Teugels J 2004 Statistics of Extremes Theory and Applications Chichester Wiley k k k k Appendix A Elements of Probability and Statistics 431 12 RDevelopmentCoreTeam R A Language and Environment for Statistical Com puting R Foundation for Statistical Computing Available httpwwwRprojectorg 2011 13 NISTSEMATECH eHandbook of Statistical Methods Available httpswwwitlnist govdiv898handbook 2012 14 Hosking JRM and Wallis JR 1987 Parameter and quantile estimation for the generalized Pareto distribution Technometrics 29 339349 15 de Haan L 1994 Extreme value statistics In Extreme Value Theory and Applica tions vol 1 ed J Galambos J Lechner and E Simiu 93122 Boston MA Kluwer Academic Publishers 16 Yeo D 2014 Generation of large directional wind speed data sets for estimation of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 17 Duthinh D and Simiu E 2010 Safety of structures in strong winds and earth quakes multihazard considerations Journal of Structural Engineering 136 330333 httpswwwnistgovwind k k k k 433 Appendix B Random Processes Consider a process the possible outcomes of which form a collection or an ensemble of functions of time yt A member of the ensemble is called a sample function or a ran dom signal The process is called a random process if the values of the sample functions at any particular time constitute a random variable Let a numerical value be assigned to each of the events that may occur as a result of an experiment The resulting set of possible numbers is defined as a random variable Examples i If a coin is tossed the numbers zero and one assigned to the outcome heads and to the outcome tails constitute a discrete random variable ii To each mea surement of a quantity a number is assigned to the result of that measurement The set of all possible results of the measurements constitutes a continuous random variable A timedependent random process is stationary if its statistical properties eg the mean and the mean square value do not depend upon the choice of the time origin and do not vary with time A stationary random signal is thus assumed to extend over the entire time domain The ensemble average or expectation of a random process is the average of the values of the member functions at any particular time A stationary random process is ergodic if its time averages equal its ensemble averages Ergodicity requires that every sample function be typical of the entire ensemble A stationary random signal may be viewed as a superposition of harmonic oscilla tions over a continuous range of frequencies Some basic results of harmonic analysis are reviewed in Sections B1 and B2 The spectral density function Section B3 the autocovariance function Section B4 the crosscovariance function the cospectrum the quadrature spectrum and the coherence function Section B5 are defined next Mean upcrossing and outcrossing rates are introduced in Section B6 The estimation of peaks of Gaussian random signals is considered in Section B7 B1 Fourier Series and Fourier Integrals Consider a periodic function xt with zero mean and period T It can be easily shown that xt C0 k1 Ck cos2𝜋kn1t 𝜙k B1 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 434 Appendix B Random Processes where n1 1T is the fundamental frequency and C0 1 T T2 T2 xtdt B1a Ck A2 k B2 k12 B1b 𝜙k tan1 Bk Ak B1c Ak 2 T T2 T2 xt cos2𝜋kn1tdt B1d Bk 2 T T2 T2 xt sin2𝜋kn1tdt B1e Equation B1 is the Fourier series expansion of the periodic function xt If a function yt is nonperiodic it is still possible to regard it as periodic with infinite period It can be shown that if yt is piecewise differentiable in every finite interval and if the integral ytdt B2 exists the following relation holds yt Cn cos2𝜋nt 𝜙ndn B3 In Eq B3 called the Fourier integral of yt in real form n is a continuously varying frequency and Cn A2n B2n12 B3a 𝜙n tan1 Bn An B3b An yt cos2𝜋ntdt B3c Bn yt sin2𝜋ntdt B3d From Eqs B3a through B3d and the identities sin 𝜙 tan 𝜙 1 tan2𝜙12 B4a cos 𝜙 1 1 tan2𝜙12 B4b it follows that yt cos2𝜋nt 𝜙ndt Cn B5 The functions yt and Cn which satisfy the symmetrical relations Eqs B3 and B5 form a Fourier transform pair k k k k Appendix B Random Processes 435 Successive differentiation of Eq B3 yields yt 2𝜋nCn sin2𝜋nt 𝜙ndn B6a yt 4𝜋2n2Cn cos2𝜋nt 𝜙ndn B6b B2 Parsevals Equality The mean square value of the periodic function xt with period T Eq B1 is 𝜎2 x 1 T T2 T2 x2tdt B7 Substitution of Eq B1 into Eq B7 yields 𝜎2 x k0 Sk B8 where S0 C2 0 and Sk 12 C2 k k 1 2 The quantity Sk is the contribution to the mean square value of xt of the harmonic component with frequency kn1 Equation B8 is a form of Parsevals equality For a nonperiodic function for which an integral Fourier expression exists Eqs B3 and B5 yield y2tdt yt Cn cos2𝜋nt 𝜙ndn dt Cn yt cos2𝜋nt 𝜙ndt dn C2ndn 2 0 C2ndn B9 Equation B9 is the form taken by Parsevals equality in the case of a nonperiodic function B3 Spectral Density Function of a Random Stationary Signal A relation similar to Eq B8 is now sought for functions generated by stationary pro cesses The spectral density of such functions is defined as the counterpart of the quan tities Sk Let zt be a stationary random signal with zero mean Because it does not satisfy the condition B2 zt does not have a Fourier transform An auxiliary function yt is therefore defined as follows Figure B1 yt zt T 2 t T 2 B10a k k k k 436 Appendix B Random Processes zt yt T Figure B1 Definition of function yt yt 0 elsewhere B10b The function yt so defined is nonperiodic satisfies condition B2 and thus has a Fourier integral From the definition of yt it follows that lim T yt zt B11 By virtue of Eqs B9 and B10 the mean square value of yt is 𝜎2 y lim T 1 T T2 T2 y2tdt 1 T y2tdt 2 T 0 C2ndn B12 The mean square of the function zt is then 𝜎2 z lim T 𝜎2 y lim T 2 T 0 C2ndn B13 With the notation Szn lim T 2 T C2n B14 Equation B13 becomes 𝜎2 z 0 Szndn B15 The function Szn is defined as the spectral density function of zt To each frequency n 0 n there corresponds an elemental contribution Sn dn to the mean square value 𝜎2 z 𝜎2 z is equal to the area under the spectral density curve Szn Because in k k k k Appendix B Random Processes 437 Eq B15 the spectrum is defined for 0 n only Szn is called the onesided spec tral density function of zt This definition of the spectrum is used throughout this text A different convention may be used where the spectrum is defined for n and the integration limits in Eq B15 are to This convention yields the twosided spectral density function of zt From Eqs B6ab following the same steps that led from Eq B3 to Eq B14 there result the expressions for the spectral density of the first and second derivative of a ran dom process S zn 4𝜋2n2Szn B16a Szn 16𝜋4n4Szn B16b B4 Autocorrelation Function of a Random Stationary Signal From Eqs B3a B3c and B3d it follows that 2 T C2n 2 T A2n B2n 2 T AnAn BnBn 2 T yt1 cos2𝜋nt1dt1 yt2 cos2𝜋nt2dt2 yt1 sin2𝜋nt1dt1 yt2 sin2𝜋nt2dt2 2 T yt1yt2 cos2𝜋nt2 t1dt1dt2 B17 Using the notations 𝜏 t2 t1 and R𝜏 1 T yt1yt1 𝜏dt1 B18 Equation B17 can be written as 2 T C2n 2 R𝜏 cos2𝜋n𝜏d𝜏 B19 Equations B19 B11 and B14 thus yield Szn 2Rz𝜏 cos2𝜋n𝜏d𝜏 B20 where Rz𝜏 lim T 1 T T2 T2 ztzt 𝜏dt B21 The function Rz𝜏 is defined as the autocovariance function of zt and provides a mea sure of the interdependence of the variable z at times t and t 𝜏 From the stationarity of zt it follows that Rz𝜏 Rz𝜏 B22 k k k k 438 Appendix B Random Processes Since Rz𝜏 is an even function of 𝜏 2Rz𝜏 sin2𝜋n𝜏d𝜏 0 B23 A comparison of Eqs B5 and B20 shows that Szn and 2Rz𝜏 form a Fourier transform pair Therefore Rz𝜏 1 2 Szn cos2𝜋n𝜏dn B24a Since as follows from Eq B20 Szn is an even function of n Eq B24a may be written as Rz𝜏 0 Szn cos2𝜋n𝜏dn B24b Similarly by virtue of Eqs B20 and B22 Szn 4 0 Rz𝜏 cos2𝜋n𝜏d𝜏 B25 The definition of the autocovariance function Eq B21 yields Rz0 𝜎2 z B26 For 𝜏 0 the products ztzt 𝜏 are not always positive as is the case for 𝜏 0 so Rz𝜏 𝜎2 z B27 For large values of 𝜏 the values zt and zt 𝜏 bear no relationship to each other so lim 𝜏 Rz𝜏 0 B28 The nondimensional quantity Rz𝜏𝜎2 z called the autocorrelation function of the function zt is equal to unity for 𝜏 0 and vanishes for 𝜏 B5 CrossCovariance Function CoSpectrum Quadrature Spectrum Coherence Consider two stationary signals z1t and z2t with zero means The function Rz1z2𝜏 lim T 1 T T2 T2 z1tz2t 𝜏 dt B29 is defined as the crosscovariance function of the signals z1t and z2t From this defi nition and the stationarity of the signals it follows that Rz1z2𝜏 Rz2z1𝜏 B30 However in general Rz1z2𝜏 Rz1z2𝜏 For example if z2t z1t 𝜏0 it can imme diately be seen from Figure B2 that Rz1z2𝜏0 Rz10 B31 Rz1z2𝜏0 Rz12𝜏0 B32 k k k k Appendix B Random Processes 439 z1t z2t a a b b τ0 τ0 2τ0 Figure B2 Functions z1t and z2t z1t𝜏0 The cospectrum and the quadrature spectrum of the signals z1t and z2t are defined respectively as SC z1z2n 2Rz1z2𝜏 cos2𝜋n𝜏d𝜏 B33 SQ z1z2n 2Rz1z2𝜏 sin2𝜋n𝜏d𝜏 B34 It follows from Eq B30 that SC z1z2n SC z2z1n B35a SQ z1z2n SQ z2z1n B35b The coherence function is a measure of the correlation between components with fre quency n of two signals z1t and z2t and is defined as Cohz1z2n SC z1z2n2 SQ z1z2n2 Sz1nSz2n 12 B36 Example B1 The animation in Figure 427 shows pressures on the exterior surface of a building induced by wind blowing in the direction shown by the arrow If pres sures at any two points were perfectly coherent spatially at any given time the shades representing their intensity would be the same regardless of the distance between the points B6 Mean Upcrossing and Outcrossing Rate for a Gaussian Process Let zt be a stationary differentiable process with mean zero The process crosses a level k at least once in a time interval t t Δt if zt k and zt Δt k If zt has smooth samples and Δt is sufficiently small zt will have a single kcrossing with pos itive slope ie a single kupcrossing The probability of occurrence of the event k k k k 440 Appendix B Random Processes Cluster Cluster Figure B3 Upcrossings of a random process indicated by rectangles Clusters are groups of two or more local peaks within an interval defined by two successive upcrossings zt k zt Δt k can be approximated by the probability of the event zt k zt zt Δt The mean rate of kupcrossings of zt is vk 0 zfz zb zd z B37a where fz zdenotes the joint probability density function of zt zt For a stationary pro cess the variables z and z are independent1 so 𝜈k E zt zt k fzk B37b E zt fzk B37c where fz denotes the probability density function of zt and E zt zt k denotes the expectation of the positive part of zt conditional on zt k A plot showing zero upcrossings of a random process is shown in Figure B3 If zt is a stationary Gaussian process with mean zero f zzz z 1 2𝜋𝜎 z𝜎z exp 1 2 z2 𝜎2 z z2 𝜎2 z B38 and the mean kupcrossing rate is 𝜈k E zt f k 𝜎 z 2𝜋 1 2𝜋𝜎z exp k2 2𝜎2 z B39 where 𝜎z and 𝜎 z denote the standard deviations of zt and zt Equation B37a can be extended to the case in which the random process is a vector x Let 𝜈D denote the mean rate at which the random process ie the tip of the vector with specified origin O crosses in an outward direction the boundary FD of a region containing the point O The rate 𝜈D has the expression 𝜈D FD dx 0 xnfx xnx xnd xn B40 1 For a stationary process Ez2t const so dEz2tdt 2Eztdztdt 0 for a fixed arbitrary time t meaning that zt and dztdt are uncorrelated If zt is Gaussian so is dztdt It then follows from the expression for the joint Gaussian distribution of two correlated variables that if their correlation vanishes the two variates are independent k k k k Appendix B Random Processes 441 where xn is the projection of the vector x on the normal to FD and fx xnx xn is the joint probability distribution of x and xn Eq B40 can be written as 𝜈D FD 0 xnf xn xnX xd xn fXxdx FD E 0 Xn X x fXxdx B41 where f X X probability density of the vector X and E 0 XnX x is the aver age of the positive values of Xn given that X x If Xn and X are independent E 0 XnX x E 0 Xn Equation B41 has been used in an attempt to estimate mean recurrence intervals of directional wind effects that exceed outcross a limit state defined by a boundary FD Objections to this approach include the perception by structural engineers that it lacks transparency see Appendix F the fact that the vector x which represents a struc tural response to wind eg a demandtocapacity index may be nonGaussian the fact that the limit state boundary cannot be defined unless the structural design is finalized which is in practice not the case at the time the outcrossing calculations are performed and the fact that if the size of the available directional wind speeds data is small rather than creating a larger data set by Monte Carlo simulation some practitioners make use of what are purported to be parent population data that is nonextreme wind speeds that may include morning breezes and other types of wind that differ from a meteoro logical point of view from the extremes and cannot therefore constitute a reliable basis for estimating extreme values B7 Probability Distribution of the Peak Value of a Random Signal with Gaussian Marginal Distribution The probability distribution of the set of values zt of the random process is called the marginal distribution of that process Since 𝜎2 z 0 Szn dn B42 𝜎2 z 4𝜋2 0 n2Szn dn B43 Eq B16a denoting 𝜈 12𝜋𝜎 x𝜎x B44 𝜅 k𝜎x B45 it follows from Eq B39 that the upcrossing rate of the level 𝜅 in units of standards deviations of the process is E𝜅 𝜈 exp 𝜅2 2 B46 k k k k 442 Appendix B Random Processes where 𝜈 0 n2Szndn 0 Szndn 12 B47 is the mean zero upcrossing rate that is 𝜈 E0 B48 Peaks greater than k𝜎z may be regarded as rare events Their probability distribution may therefore be assumed to be of the Poisson type The probability that in the time interval T there will be no peaks equal to or larger than k𝜎z can therefore be written as p0 T expEkT B49 The probability p0 T can be viewed as the probability that given the interval T the ratio K of the largest peak to the rms value of zt is less than 𝜅 that is PK 𝜅T expE𝜅T B50 The probability density function of K that is the probability pK𝜅T that 𝜅 K 𝜅 d𝜅 is obtained from Eq B50 by differentiation P𝜅T 𝜅T E𝜅 expE𝜅T B51 The expected value of the largest peak occurring in the interval T may then be calcu lated as K 0 𝜅pK𝜅Td𝜅 B52 The integral of Eq B52 is approximately K 2 ln 𝜈T12 0577 2 ln 𝜈T12 B53 1 where 𝜈 is given by Eq B47 The estimation of statistics of peaks of random signals with arbitrary marginal prob ability distributions is discussed in detail in Appendix C Reference 1 Davenport AG 1964 Note on the distribution of the largest value of a random function with application to gust loading Journal of the Institution of Civil Engineers 24 187196 k k k k 443 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks C1 Introduction The estimation of the distribution of the peak of a random process yt with specified duration T from a single finite time series of length T1 T and of the corresponding uncertainties has applications in Extreme wind climatology where the time series consists of a record of extreme wind speeds over a time interval T1 N1 years and the statistics of the largest wind speed during a longer time interval T N years are of interest Aerodynamics and structural engineering where a time series of length T1 of wind effects eg measured pressure coefficients or calculated internal forces demandtocapacity indexes interstory drift accelerations is available and the statistics of the peak wind effect for a time series with length T T1 are of interest For the particular case in which the marginal distribution of a process yt is Gaussian a closedform expression for the distribution of the peak is available see Section B7 If the distribution is not Gaussian a nonlinear mapping procedure referred to as transla tion has been developed by which those statistics can be obtained 1 The translation procedure depends heavily on the users ability to choose an appropriate marginal prob ability distribution In practice because of the difficulty of this task the performance of the translation method can be unsatisfactory A simple procedure in which the time history of length T1 is divided into n equal segments epochs was proposed in 2 A data sample is created consisting of the peak of each of those segments and a Gumbel Cumulative Distribution Function CDF is fitted to that sample The length T1n of the segments must be sufficient for the peaks of different segments to be mutually independent To obtain the largest peak for a time history of length T rT1n r n the Gumbel CDF describing the probabilistic behav ior of the segment peaks is raised to the rth power Because that CDF is an exponential function this operation results in an alternative Gumbel distribution that describes the probabilistic behavior of the peak of the time history of length T 3 This procedure is most efficiently implemented by using the BLUE Best Linear Unbiased Estimator method to estimate the parameters of the Gumbel distribution of the segment peaks see Section 333 and 4 httpswwwnistgovwind However as shown in Section C3 a Dr A L Pintars leading role in the development and application of the procedure described in this Appendix is acknowledged with thanks Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 444 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 40 30 20 10 00 60 80 100 20 Seconds Pressure 0 Figure C1 Time history of pressure coefficients drawback of the BLUE method is that the estimates can depend significantly on n with no criterion for an optimal choice of n being available in the literature The purposes of this Appendix are i to describe a PeaksoverThreshold POT Poissonprocess procedure for the estimation of the distribution of the peak of a sta tionary random process of specified duration Section A2 and of the corresponding uncertainties and ii to assess the performance of the procedure through compari son of its results with observed data and with results obtained by the BLUE method A software implementation of the procedure applicable to time histories of pressures or pressure effects eg internal forces in structural members that leverages the R environ ment for statistical computing and graphics 5 is available in 6 httpsgithubcom usnistgovpotMax which also contains detailed instructions for installation and use The procedure is described and illustrated in what follows with reference to the time history of pressure coefficients of Figure C1 To allow the reader to replicate the cal culations described herein we note that the data were obtained from the NISTUWO Aerodynamic Database for Rigid Buildings 7 httpswwwnistgovwind dataset jp1 Building 7 open terrain tap 1715 at middle of eave sampling rate 500 Hz wind direc tion 270 For similar software applied to the estimation of extreme wind speeds see Section 335 C2 Peak Estimation by PeaksOverThreshold PoissonProcess Procedure Description of Procedure The POT approach is applied to observations yt within a time series that exceed a threshold u The POT approach is chosen over the epochal approach for two reasons First the POT approach generally allows the use of more observations than does the epochal approach potentially leading to less uncertainty Second and more important a procedure is available for an optimal selection of the threshold u 8 The steps of the procedure are as follows 1 Reverse the signs of the time series if necessary The procedure is developed for pos itive peaks The peaks of interest in Figure C1 being negative the signs of this time series were reversed If analysts are interested in both positive and negative peaks the procedure is applied twice first with the original signs and second with reversed signs k k k k Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 445 2 Choose a model For the reasons indicated in Section 332 it is assumed for wind climatological purposes that the peaks of the variate yt are described probabilis tically by an Extreme Value Type I EV I distribution For the same reasons the restriction to the EV I distribution also holds for the peak of time series considered in aerodynamics and structural engineering applications However if interested in considering EV II or EV III distributional models the analyst can choose to do so as is indicated subsequently Supposing that the variables t y follow a Poisson pro cess with intensity function 𝜆t y the random number of peak values of y events that occur in a time interval t2 t1 and have magnitude between y1 and y2 can be described by the Poisson distribution pn t2 t1 y2 y1 𝜆t ydt dy n n exp t2 t1 y2 y1 𝜆t ydt dy C1 Let us consider the particular case in which the intensity function 𝜆t y const and y2 is the largest possible value of y under the assumptions that the peaks y have an EV I or EV II distribution y2 is infinitely large if y has an EV III distribution it has a finite upper bound In that case the expected number of events is y1 y2t2 t1 𝜆 where the constant intensity function 𝜆 is the rate of arrival of those events How ever Eq C1 allows for more complex cases In one such case the random process is not stationary For example if y represents wind speeds in either synoptic storms or thunderstorms the process y should have two different constant intensity func tions rates of arrival 𝜆syn and 𝜆th applicable to the time intervals in which there occur synoptic storms and thunderstorms respectively In the case of a stationary process for peak values y that cross a high threshold asymptotic arguments lead to the expressions 𝜆t y 1 𝜎 1 ky 𝜇 𝜎 11k C2 𝜆t y 1 𝜎 exp y 𝜇 𝜎 C3 9 In Eq C2 the subscript means that negative values of the quantity 1 ky𝜇 𝜎 are raised to zero Depending upon whether k 0 or k 0 Eq C2 is the POT equivalent of a Type II Fréchet or Type III reverse Weibull extreme value dis tribution respectively Equation C3 is the POT equivalent to the Type I Gum bel extreme value distribution it is the limit as k approaches zero of Eq C2 The POT Poissonprocess procedure is designated as FpotMax if used with Eq C2 and GpotMax if used with Eq C3 the letters F and G stand for full and Gumbel respectively The parameters 𝜇 and 𝜎 are respectively the location and scale param eters of the distribution of the peak value of yt The volume 𝜆t ydt dy is equal to the expected number of peaks per elemental area dt dy 3 Decluster Figure C2a depicts the same raw time series as Figure C1 Thresholded variants with the threshold u 18 and u 20 are depicted in Figure C2b and Figure C2c respectively In raw time series successive peaks can be separated by time intervals smaller than the time between an upcrossing of the mean and the subsequent downcrossing of the mean see Appendix B Figure B3 Such successive k k k k 446 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 30 15 00 30 00 15 30 15 00 Pressure a b c Seconds 0 25 50 75 100 Figure C2 a Raw time series observations in raw time series b above threshold u 18 and c above threshold u 20 peaks are typically strongly correlated as shown by Figure C3 where it is seen that the autocorrelation function remains strong and positive for observations separated by more than 40 increments of time in this case 40500 Hz 008 seconds Poisson processes are not appropriate for highly autocorrelated data without further processing because of the independence assumption that underlies them Clusters are data blocks within time intervals defined by an upcrossing of the mean and the subsequent downcrossing of the mean see Figure B4 Declustering is an operation that is effective in removing the high autocorrelation from the data It proceeds by discarding in each cluster all data other than the cluster maximum Figure C4 displays the counterparts of Figure C2 after declustering The estimated autocorrelation function of the data analysis of the time series in Figure C4a shows that declustering is highly effective After removing the autocorrelation in the series or declustering the use of Poisson processes as models for crossings of a high threshold is justified They are used for such purposes in many papers for example 8 1012 k k k k Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 447 20 10 00 02 04 06 ACF 08 10 0 30 Lag Samples 40 Figure C3 Estimated autocorrelation function for the time series in Figure C2a 4 Select optimal threshold Historically a hurdle to the use of the POT models has been the appropriate choice of a threshold Since the threshold dictates the data that are included in or omitted from the sample used to fit the model its impact on the results can be large The extreme value model becomes more appropriate as the threshold increases as more nonextreme values of the variate are excluded from the sample being analyzed but the threshold cannot be too high because too few obser vations will remain for fitting the model since observations are taken over a finite period of time Any approach to choosing a threshold must balance these competing aspects A common and easy to implement approach though not necessarily optimal is to pick a high quantile of the series eg 95 13 p 489 The approach of 8 is superior insofar as it uses an optimal threshold based on the fit of the model to the data as judged by the statistics called Wstatistics defined in 11 Eq 130 The Wstatistic is unitless and defines a transformation of the data such that if the Poissonprocess model were perfectly correct the transformed data would follow exactly an exponential distribution with mean one Figure C5 shows a plot of the ordered Wstatistics versus quantiles of the standard exponential distribution using the optimal threshold for the series in Figure C4a If the data fitted perfectly to the model the points would fall exactly on the diagonal line The threshold is chosen by creating such a plot for a sequence of potential thresholds and selecting the threshold that minimizes the maximum absolute vertical distance to the diagonal line This method for selecting the threshold is comparable to the method used in 14 5 Estimate model parameters The model parameters 𝜂 𝜇 𝜎 for the intensity func tion in Eq C1 are estimated by maximum likelihood from the set of declustered k k k k 448 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 30 15 00 30 00 15 30 15 00 Pressure a b c Seconds 0 20 40 60 80 100 Figure C4 a Declustered time series resulting observations b above u 18 and c above u 20 4 3 2 exp1 quantiles Ordered WStatistics WStatistic Plot 1 0 0 1 2 3 4 Figure C5 Plot of the Wstatistics versus their corresponding standard exponential quantiles for the declustered series depicted in Figure C4a using the optimal threshold Best fit of data using GpotMax with a 21 threshold 45 data points k k k k Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 449 Distribution of the Peak Value Mean 40 Peak Value Density 45 50 35 30 25 12 10 08 06 04 02 00 Figure C6 Histogram of the estimated distribution of the peak value starting with the time series depicted in Figure C4a The triangle shows the mean of the distribution data corresponding to the chosen threshold The likelihood is given in Equation 2 of 8 and maximum likelihood is discussed in for example 15 Section 722 6 Empirically build the distribution of the peak by Monte Carlo simulation A series of desired length T is generated from the fitted model and the peak of the generated series is recorded This is repeated nmc times The recorded peaks form an empirical approximation to the distribution of the peak A histogram of the simulated peaks over 100 seconds with nmc 1000 for the example data set is shown in Figure C6 in which the mean value is marked by the triangle 7 Quantify uncertainty The objective of the computations is to estimate the distribu tion of the peak of the time series under study Thus the uncertainty in the estimate of the entire distribution of the peak is being quantified not just for example the uncer tainty in the mean of that distribution To accomplish this a second layer of Monte Carlo sampling is performed The input to step 6 was the maximum likelihood esti mate of the vector 𝜂 denoted by 𝜂 However because only a finite sample is available these estimates are uncertain That uncertainty may be described using the multivari ate Gaussian distribution More specifically one may sample values of 𝜂 that are also consistent with observed time series and repeat step 6 for those new parameter val ues a number nboot of times The result of step 7 is nboot empirical approximations to the distribution of the peak For clarity Figure C7 shows only nboot 50 replicates of the distribution of the peak for the example data set Typically 1000 replicates say may be used The bar shown in Figure C 7 depicts an 80 confidence interval for the mean which is calculated from 1000 replicates This technique is an approximation to a bootstrap algorithm 16 20 Discussion of Results The dashed line in Figure C8 shows the peak estimated by GpotMax applied to the entire time series of duration 100 seconds This estimate is k k k k 450 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 4 3 2 00 05 10 15 20 5 80 CI for Mean Peak Value Bootstrap Replicates of the Distribution of the Peak Density 6 7 Figure C7 Replicates of the distribution of the peak starting with the series shown in Figure C4a The short horizontal line shows an 80 confidence interval for the mean of the distribution of the peak value 3 GpotMax 100s6 Peak Pressure Coefficient GpotMax 100s extrapolation GpotMax 100s Mean of 100s6 extrapolations to 100s Observed 100s Observed 100s6 4 Data Block Number 5 6 1 2 0 1 2 3 4 Figure C8 Comparison of estimates based on six equal data blocks and on global analysis using GpotMax k k k k Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 451 close to the observed peak of the time series shown by the solid line The squares show the results of six analyses performed on six partitions of the same time series each of length 100 s6 The GpotMax estimates closely track the observed peaks ie the circles for each of the partitions For each partition GpotMax may also be used to calculate the mean of the distribution of the peak for a duration of 100 seconds shown by the triangles in Figure C8 The six individual partitions can yield estimates that differ by as much as approximately 25 from the estimate based on the entire 100second time series However the average of these six estimates shown by the dashed line is reasonably close to the global estimate and the observed 100second peak As noted earlier the full version of the algorithm based on Eq C1 referred to as FpotMax does not assume that the tail length parameter of the distribution of peaks is zero It is shown in 17 that the estimates by GpotMax and FpotMax of the distri butions of the peaks are similar for five representative pressure taps of the building model examined herein and are close to the observed peaks GpotMax rather than FpotMax may therefore be used in practice unless there were one or two very large peaks relative to other threshold crossings C3 Dependence of Peak Estimates by BLUE Upon Number of Partitions Peaks were estimated using the epochal method for two probabilities of nonexceedance p 078 and p 05704 The latter corresponds to the mean of the Gumbel distribution while the former is commonly used by wind tunnel operators 18 and is close to the number 080 specified in the ISO 4354 19 For a number of partitions 10 n 24 the estimated peaks for tap 708 wind direction 360 varied between 372 and 420 for p 078 and between 348 and 382 for p 057 For comparison the single GpotMax and FpotMax estimates were 341 and 335 respectively and the observed peak was 324 C4 Summary Current procedures for estimating peaks of pressure time series have drawbacks that motivated the development of the new procedure one advantage of which is that it typ ically results in an extreme value data set larger than is the case for epochal procedures The translation procedure has the drawback that it depends upon the estimate of the marginal distribution of a nonGaussian time series which is typically difficult to per form reliably The epochal procedure used in conjunction with the BLUE estimation of the Gumbel parameters depends in some cases very significantly upon the number of partitions being used The procedure described in this Appendix is based on a Poisson process model for quantities y that exceed a specified threshold u of the time series being considered The estimate depends upon the choice of the threshold A criterion is available that allows the analyst to make an optimal choice according to a specified metric of the thresh old value k k k k 452 Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks Two versions of the proposed procedure are available One version denoted by Fpot Max includes estimation of a tail length parameter resulting in a tail of the Fréchet or the reverse Weibull distribution type The second version denoted by GpotMax assumes that the tail length parameter vanishes resulting in a tail of the Gumbel distribution type Typically GpotMax results in fully satisfactory estimates and should in practice be used for structural design applications which include the analysis of wind speed time series of time series of pressure coefficients or of wind effects such as internal forces demandtocapacity indexes interstory drift and floor accelerations References 1 Sadek F and Simiu E 2002 Peak nonGaussian wind effects for databaseassisted lowrise building design Journal of Engineering Mechanics 128 5 530539 https wwwnistgovwind 2 Eaton KJ and Mayne JR 1975 The measurement of wind pressures on twostory houses at Aylesbury Journal of Industrial Aerodynamics 1 67109 3 Simiu E Pintar AL Duthinh D and Yeo D 2017 Wind load factors for use in the wind tunnel procedure ASCEASME Journal of Risk and Uncertainty in Engi neering Systems Part A Civil Engineering 3 4 04017007 httpswwwnistgov wind 4 Lieblein J Efficient Methods of Extreme Value Methodology NBSIR74602 National Bureau of Standards Washington DC 1974 httpswwwnistgovwind 5 Core Team R 2015 R A Language and Environment for Statistical Comput ing Vienna Austria Available wwwRprojectorg R Foundation for Statistical Computing 6 Pintar A potMax Estimating the distribution of the maximum of a time series using peaksoverthreshold models Available httpsgithubcomusnistgovpotMax Aug 16 2017 2016 7 Ho TCE Surry D Morrish D and Kopp GA 2005 The UWO contribution to the NIST aerodynamic database for wind loads on low buildings Part 1 archiving format and basic aerodynamic data Journal of Wind Engineering and Industrial Aerodynamics 93 1 130 For the aerodynamic data see httpswwwnistgov wind 8 Pintar A L Simiu E Lombardo F T and Levitan M Maps of NonHurricane NonTornadic Winds Speeds With Specified Mean Recurrence Intervals for the Con tiguous United States Using a TwoDimensional Poisson Process Extreme Value Model and Local Regression NIST Special Publication 500301 httpsnvlpubsnist govnistpubsSpecialPublicationsNISTSP500301pdf 2015 9 Pickands J III The twodimensional Poisson process and extremal processes Jour nal of Applied Probability 8 745756 1971 10 Smith RL 1989 Extreme value analysis of environmental time series an applica tion to trend detection in groundlevel Ozone Statistical Science 4 367393 11 Smith RL 2004 Statistics of extremes with applications in environment insur ance and finance In Extreme Values in Finance Telecommunications and the Environment ed B Finkenstädt and H Rootzén 178 Chapman HallCRC Ch 1 k k k k Appendix C PeaksOverThreshold PoissonProcess Procedure for Estimating Peaks 453 12 Coles S 2004 The use and misuse of extreme value models in practice In Extreme Values in Finance Telecommunications and the Environment ed B Finkenstädt and H Rootzén 79100 Chapman HallCRC Ch 2 13 MannshardtShamseldin EC Smith RL Sain SR et al 2010 Downscaling extremes a comparison of extreme value distributions in pointsource and gridded precipitation data The Annals of Applied Statistics 4 484502 14 Pickands J III Bayes quantile estimation and threshold selection for the General ized Pareto family in Proceedings of the Conference on Extreme Value Theory and Applications Gaithersburg MD 1993 J Galambos J Lechner and E Simiu ed Boston MA Kluwer Academic Publishers 1994 15 Casella G and Berger RL 2002 Statistical Inference vol 2 Pacific Grove CA Duxbury 16 Yeo D 2014 Generation of large directional wind speed data sets for estimation of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 17 Duthinh D Pintar AL and Simiu E 2017 Estimating peaks of stationary ran dom processes a peaksoverthreshold approach ASCE ASME Journal of Risk and Uncertainty in Engineering Systems Part A Civil Engineering 3 4 04017028 httpswwwnistgovwind 18 Peng X Yang L Gurley K Prevatt D and Gavanski E Prediction of peak wind loads on lowrise building Proceedings of the 12th Americas Conference on Wind Engineering Seattle WA 2013 19 International Standard ISO 4354 20090601 2nd ed Wind actions on struc tures Annex D informative Aerodynamic pressure and force coefficients Geneva Switzerland p 22 20 Efron B and Tibshirani RJ An Introduction to the Bootstrap CRC Press 1994 k k k k 455 Appendix D Structural Dynamics FrequencyDomain Approach D1 Introduction The mathematical model for windinduced dynamic response is Newtons second law that is an ordinary secondorder differential equation In Part II of the book the solu tion to this equation was obtained by timedomain methods This approach is currently feasible because i forcing functions can be obtained as functions of time from simulta neously measured aerodynamic time histories and ii computer capabilities allow the ready solution of the differential equations of motion of the dynamical systems of inter est Neither of these two capabilities was available until relatively recently For this rea son the differential equations were transformed via Fourier transformation into more tractable algebraic functions in the frequency domain and forcing functions were thus defined via spectral and crossspectral densities Frequencydomain solutions of struc tural dynamics problems remain useful for certain applications and can provide helpful insights into windinduced structural dynamics Section D2 presents the building blocks of the frequencydomain approach for the singledegreeoffreedom system Section D3 presents basic results obtained for con tinuously distributed linear systems Section D4 is an interesting application of those results the determination of the alongwind response of a tall building with rectangular shape in plan to wind normal to one of its faces D2 The SingleDegreeofFreedom Linear System Consider the singledegreeoffreedom motion of a particle of mass M subjected to a timedependent force Ft The particle is restrained by an elastic spring with stiffness k Its motion is damped by a viscous damper with coefficient c The particles displacement xt is opposed by i a restoring force kx and ii a damping force c dxdt c x where the stiffness k and the damping coefficient c are assumed to be constant Newtons second law states that the product of the particles mass by its acceleration Mx equals the total force applied to the particle The equation of motion of the system is Mx c x kx Ft D1 With the notations n1 kM2𝜋 and 𝜁1 c2 kM where n1 denotes the fre quency of vibration of the oscillator1 and 𝜁1 is the damping ratio ie the ratio of the 1 The quantity 2𝜋n is called circular frequency and is commonly denoted by 𝜔 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 456 Appendix D Structural Dynamics damping c to the critical damping 𝜁cr 2 kM beyond which the systems motion would no longer be oscillatory Eq D1 becomes x 2𝜁2𝜋n1 x 2𝜋n12 x Ft M D2 For structures 𝜁1 is typically small in the order of 1 D21 Response to a Harmonic Load In the particular case of a harmonic load Ft F0 cos2𝜋nt it can be verified by substi tution that the steadystate solution of Eq D2 is xt HnF0 cos2𝜋nt 𝜃 D3 where Hn 1 4𝜋2n2 1M1 nn122 4𝜁2 1nn1212 D4 𝜃 tan1 2𝜁1nn1 1 nn12 D5 The quantity 𝜃 is the phase angle and Hn is the systems mechanical admittance func tion or mechanical amplification factor For n n1 that is if the frequency of the harmonic forcing function coincides with the frequency of vibration of the oscillator the amplitude of the response is largest and is inversely proportional to the damping ratio 𝜁1 In this case the motion exhibits resonance In the particular case Ft F0 sin2𝜋nt the steady state response can be written as xt HnF0 sin2𝜋nt 𝜃 D6 D22 Response to an Arbitrary Load Let the system described by Eq D2 be subjected to the action of a load equal to the unit impulse function 𝛿t acting at time t 0 that is to a load defined as follows Figure D1 𝛿t 0 for t 0 D7 lim Δt0 Δt 0 𝛿tdt 1 for t 0 D8 The response of the system to the load 𝛿t depends on time and is denoted by Gt Δt t δt δ0 lim δtdt 1 Δt0 Δt 0 Figure D1 Unit impulse function k k k k Appendix D Structural Dynamics 457 Figure D2 Load Ft t τ Ft t 0 τ An arbitrary load Ft Figure D2 may be described as a sum of elemental impulses of magnitude F𝜏 d𝜏 each acting at time 𝜏 Since the system is linear the response at time t to each such impulse is Gt 𝜏F𝜏 d𝜏 The total response is xt t Gt 𝜏F𝜏d𝜏 D9 The limits of the integral indicate that all the elemental impulses that have acted before time t have been taken into account Denoting 𝜏 t 𝜏 Eq D9 becomes xt 0 G𝜏Ft 𝜏d𝜏 D10 Let Ft F0 cos 2𝜋nt It follows from Eqs D3 and D10 that Hn cos 𝜃 0 G𝜏 cos 2𝜋n𝜏d𝜏 D11a Hn sin 𝜃 0 G𝜏 sin 2𝜋n𝜏d𝜏 D11b Equations D11a and D11b yield Eqs D12a and D12b whose summation yields Eq D13 H2ncos2 𝜃 0 0 G𝜏1 cos 2𝜋n𝜏1G𝜏2 cos 2𝜋n𝜏2d𝜏1d𝜏2 D12a H2nsin2𝜃 0 0 G𝜏1 sin 2𝜋n𝜏1G𝜏2 sin 2𝜋n𝜏2d𝜏1d𝜏2 D12b H2n 0 0 G𝜏1G𝜏2 cos 2𝜋n 𝜏1 𝜏2d𝜏1d𝜏2 D13 D23 Response to a Stationary Random Load Now let the load Ft be a stationary process with spectral density SFn Using Eqs B20 B21 and D10 we obtain the spectral density of the system response as follows Sxn 2 Rx𝜏 cos 2𝜋n𝜏d𝜏 k k k k 458 Appendix D Structural Dynamics 2 lim T 1 T T2 T2 xtxt 𝜏dt cos 2𝜋n𝜏d𝜏 2 lim T 1 T T2 T2 dt 0 G𝜏1Ft 𝜏1d𝜏1 0 G𝜏2Ft 𝜏 𝜏2d𝜏2 cos 2𝜋n𝜏d𝜏 2 0 G𝜏1 0 G𝜏2 RF𝜏 𝜏1 𝜏2 cos 2𝜋n𝜏d𝜏 d𝜏2 d𝜏1 2 0 0 G𝜏1G𝜏2 cos 2𝜋n𝜏1 𝜏2d𝜏1d𝜏2 RF𝜏 𝜏1 𝜏2 cos 2𝜋n𝜏 𝜏1 𝜏2d𝜏 𝜏1 𝜏2 2 0 0 G𝜏1G𝜏2 sin 2𝜋n𝜏1 𝜏2d𝜏1d𝜏2 RF𝜏 𝜏1 𝜏2 sin 2𝜋n𝜏 𝜏1 𝜏2d𝜏 𝜏1 𝜏2 D14 where in the last step the following identity is used cos2𝜋n𝜏 cos 2𝜋n𝜏 𝜏1 𝜏2 𝜏1 𝜏2 D15 From Eqs B20 B23 D12ab and D13 there follows Sxn H2nSFn D16 This relation between frequencydomain forcing and response is useful in applications D3 Continuously Distributed Linear Systems D31 Normal Modes and Frequencies Generalized Coordinates Mass and Force D311 Modal Equations of Motion A linearly elastic structure with continuously distributed mass per unit length mz and low damping can be shown to vibrate in resonance with the exciting force if the latter has certain sharply defined frequencies called the structures natural frequencies of vibra tion Associated with each natural frequency is a mode or modal shape of the vibrating structure The first four normal modes xiz i 1 2 3 4 of a vertical cantilever beam with running coordinate z are shown in Figure 113 The natural modes and frequencies are structural properties independent of the loads A deflection xzt along a principal axis of a continuous system due to time dependent forcing can in general be written in the form xz t i xiz𝜉it D17 where the functions 𝜉it are called the generalized coordinates of the system and xiz denotes the modal shape in the ith mode of vibration For a building similar expressions k k k k Appendix D Structural Dynamics 459 hold for deflections yzt in the direction of its second principal axis and for horizontal torsional angles 𝜑zt For structures whose centers of mass and elastic centers do not coincide the x y and 𝜑 motions are coupled as is shown in Section D4 which presents the development of the equations of motion for this general case In this section we limit ourselves to presenting the modal equations of motion corre sponding to the particular case of translational motion along a principal axis x 𝜉it 2𝜁i2𝜋ni 𝜉t 2𝜋ni2 𝜉it Qit Mi i 1 2 3 D18 where 𝜁i ni Mi and Qi are the ith mode damping ratio natural frequency generalized mass and generalized force respectively Mi H 0 xiz2mzdz D19 Qi H 0 pz txizdz D20 where mz is the mass of the structure per unit length pz t is the load acting on the structure per unit length and H is the structures height For a concentrated load acting at z z1 pz t Ft𝛿z z1 D21 where 𝛿z z1 is defined with a change of variable as in Eq D8 Qit lim Δz0 z1Δz z1 pz txizdz xiz1Ft D22 D32 Response to a Concentrated Harmonic Load If a concentrated load Ft F0 cos 2𝜋nt D23 is acting on the structure at a point of coordinate z1 by virtue of Eq D22 the general ized force in the ith mode is Qit F0 xiz1 cos 2𝜋nt D24 and the steadystate solutions of Eq D18 are similar to the solution Eq D3 of a singledegreeoffreedom system under a harmonic load 𝜉it F0 xiz1Hin cos2𝜋nt 𝜃i D25 where Hin 1 4𝜋2n2 i Mi1 nn122 4𝜁2 i nni212 D26 𝜃i tan1 2𝜁inni 1 nni2 D27 k k k k 460 Appendix D Structural Dynamics The response of the structure at a point of coordinate z is then xz t F0 i xizxiz1Hin cos2𝜋nt 𝜃i D28 It is convenient to write Eq D28 in the form xz t F0Hz z1 n cos2𝜋nt 𝜃z z1 n D29 where as follows immediately from Eqs B4a and B4b Hz z1 n i xizxiz1Hin cos 𝜃i 2 i xizxiz1Hin sin 𝜃i 2 12 D30 𝜃z z1 n tan1 i xizxiz1Hin sin 𝜃i i xizxiz1Hin cos 𝜃i D31 Similarly the steady state response at a point of coordinate z to a concentrated load Ft F0 sin 2𝜋nt D32 acting at a point of coordinate z1 can be written as xz t F0Hz z1 n sin2𝜋nt 𝜃z z1 n D33 D33 Response to a Concentrated Stationary Random Load Let the response at a point of coordinate z to a concentrated unit impulsive load 𝛿t acting at time t 0 at a point of coordinate z1 be denoted Gz z1 t Following the same reasoning that led to Eq D10 the response xzt to an arbitrary load Ft acting at a point of coordinate z1 is xz t 0 Gz z1 𝜏Ft 𝜏d𝜏 D34 Note the complete similarity of Eqs D29 D33 and D34 to Eqs D3 D6 and D10 respectively Therefore the same steps that led to Eq D16 yield the relation between the spectra of the random forcing and the response Sxz z1 n H2z z1 nSFn D35 D34 Response to Two Concentrated Stationary Random Loads Let xzt denote the response at a point of coordinate z to two stationary loads F1t and F2t acting at points with coordinates z1 and z2 respectively The autocovariance of xzt is see Eq B21 Rxz 𝜏 lim T 1 T T2 T2 xz txz t 𝜏dt k k k k Appendix D Structural Dynamics 461 lim T 1 T T2 T2 0 Gz z1 𝜏1F1t 𝜏1d𝜏1 0 Gz z2 𝜏1F2t 𝜏1d𝜏1 0 Gz z1 𝜏2F1t 𝜏 𝜏2d𝜏2 0 Gz z2 𝜏2F2t 𝜏 𝜏2d𝜏2 dt 0 Gz z1 𝜏1 0 Gz z1 𝜏2RF1𝜏 𝜏1 𝜏2d𝜏2 d𝜏1 0 Gz z2 𝜏1 0 Gz z2 𝜏2RF2𝜏 𝜏1 𝜏2d𝜏2 d𝜏1 0 Gz z1 𝜏1 0 Gz z2 𝜏2RF1F2𝜏 𝜏1 𝜏2d𝜏2 d𝜏1 0 Gz z2 𝜏1 0 Gz z1 𝜏2RF1F2𝜏 𝜏1 𝜏2d𝜏2 d𝜏1 D36 The spectral density of the displacement xzt is Sxz n 2 Rxz 𝜏 cos 2𝜋n𝜏d𝜏 2 Rxz 𝜏 cos 2𝜋n𝜏 𝜏1 𝜏2 𝜏1 𝜏2d𝜏 𝜏1 𝜏2 D37 Substitute the righthand side of Eq D36 for Rxz 𝜏 in Eq D37 Using the relations Hz zi n cos 𝜃z zi n 0 Gz zi 𝜏 cos 2𝜋n𝜏d𝜏 D38a Hz zi n sin 𝜃z zi n 0 Gz zi 𝜏 sin 2𝜋n𝜏d𝜏 D38b which are similar to Eqs D11a and D11b and Hz z1 nHz z2 n cos𝜃z z1 n 𝜃z z2 n 0 0 Gz z1 𝜏1Gz z2 𝜏2 cos 2𝜋n𝜏1 𝜏2d𝜏1d𝜏2 D39a Hz z1 nHz z2 n sin𝜃z z1 n 𝜃z z2 n 0 0 Gz z1 𝜏1Gz z2 𝜏2 sin 2𝜋n𝜏1 𝜏2d𝜏1d𝜏2 D39b which are derived from Eqs D38a and D38b and following the steps that led to Eq D16 there results Sxz n H2z z1 n SF1n H2z z2 n SF2n 2Hz z1 n Hz z2 nSC F1F2n cos𝜃z z1 n 𝜃z z2 n SQ F1F2n sin𝜃z z1 n 𝜃z z2 n D40 k k k k 462 Appendix D Structural Dynamics where SC F1F2n and SQ F1F2n are the cospectrum and quadrature spectrum of the forces F1t and F2t defined by Eqs B33 and B34 respectively D35 Effect of the Correlation of the Loads upon the Magnitude of the Response Let two stationary random loads F1t F2t act at points of coordinates z1 and z2 respectively The loads F1t and F2t are perfectly correlated By definition in this case SC F1F2n SC F1n and SQ F1F2n 0 Eqs B21 and B29 B20 and B33 B23 and B34 From Eq D40 Sxz n H2z z1 n H2z z2 n 2Hz z1 n Hz z2 n cos𝜃z z1 n 𝜃z z2 nSF1n D41 If z1 z2 Sxz n 4H2z z1 nSF1n D42 Consider now two loads F1t and F2t for which the crosscovariance RF1F2𝜏 0 Then by Eqs B33 and B34 SC F1F2n SQ F1F2n 0 D43 and if SF1n SF2n Sxz n H2z z1 n H2z z2 n SF1n D44 If z1 z2 Sxz n 2H2z z1 n SF1n D45 The spectrum of the response to the action of the two uncorrelated loads is in this case only half as large as in the case of the perfectly correlated loads D36 Distributed Stationary Random Loads The spectral density of the response to a distributed stationary random load can be obtained by generalizing Eq D40 to the case where an infinite number of elemental loads rather than two concentrated loads are acting on the structure Thus if the load is distributed over an area A and if it is noted that in the absence of torsion the mechan ical admittance functions are independent of the acrosswind coordinate y the spectral density of the alongwind fluctuations may be written as Sxz n AA Hz z1 nHz z2 nSC p 1p 2n cos𝜃z z1 n 𝜃z z2 n SQ p 1p 2n sin𝜃z z1 n 𝜃z z2 ndA1A2 D46 k k k k Appendix D Structural Dynamics 463 where p 1 and p 2 denote pressures acting at points of coordinates y1 z1 and y2 z2 respectively It can be verified that from Eq D46 there follows2 Sxz n 1 16𝜋4 i j xizxjz n2 i n2 j MiMj 1 1 nni22 4𝜁2 i nni21 nnj22 4𝜁2 j nnj2 1 n ni 2 1 n nj 2 4𝜁i𝜁j n ni n nj AA xiz1xjz2SC p 1p 2ndA1dA2 2𝜁j n nj 1 n ni 2 2𝜁i n ni 1 n nj 2 AA xiz1xjz2SQ p 1p 2ndA1dA2 D47 If the damping is small and the resonant peaks are well separated the crossterms in Eq D47 become negligible and Sxz n i x2 i zAAxiz1xiz2SC p 1p 2ndA1dA2 16𝜋4n4 i M2 i 1 nni22 4𝜁2 i nni2 D48 D4 Example AlongWind Response To illustrate the application of the material presented in Section D36 we consider the alongwind response of tall buildings subjected to pressures per unit area py z t pz py z t Figure D3 Mean Response The alongwind deflection induced by the mean pressures pz is xz B i H 0 pzxizdz 4𝜋2n2 i Mi xiz D49 Consider the case of loading induced by wind with longitudinal speed Uzt Uz uz t normal to a building face The sum of the mean pressures pz acting on the wind ward and leeward faces of the building is then pz 12 𝜌Cw ClBU 2z D50 2 By using Eqs D30 and D31 D26 and D27 and B4ab For a derivation of Eq D47 in terms of complex variables see 1 k k k k 464 Appendix D Structural Dynamics x z y B D H pyztdA Figure D3 Schematic view of a building where 𝜌 is the air density Cw and Cl are the values averaged over the building width B of the mean positive pressure coefficient on the windward face and of the negative pressure coefficient on the leeward face respectively and Uz is the mean wind speed at elevation z in the undisturbed oncoming flow Equation D49 then becomes xz 1 2𝜌Cw ClB i H 0 U 2zxizdz 4𝜋2n2 i Mi xiz D51 Fluctuating Response Deflections and Accelerations The cospectrum of the pressures at points M1 M2 of coordinates y1 z1 y2 z2 respectively can be written as SC p 1p 2 S12 p z1 nS12 p z2 nCohy1 y2 z1 z2 nNn D52 where S12 p z n is the spectral density of the fluctuating pressures at point Pi i 1 2 Cohy1 y2 z1 z2 n is the coherence of pressures both of which are acting on the same building face and Nn is the coherence of pressures one of which is acting on the wind ward face while the other is acting on the leeward face of the building By definition if both P1 and P2 are on the same building face Nn 1 Since pz t 1 2𝜌CUz uz t2 D53 where C which is equal to Cw or Cl depending upon whether the pressure acts on the windward or leeward face is the average pressure coefficient Spzi n 𝜌2C2U 2ziSuzi n D54 where we used the fact that u2 is small in relation to 2Uzuz k k k k Appendix D Structural Dynamics 465 Equation D48 then becomes Sxz n 𝜌2 16𝜋4 i x2 i zC2 w 2CwClNn C2 l n4 i M2 i 1 nni22 4𝜁2 i nni2 B 0 B 0 H 0 H 0 xiz1xiz2Uz1Uz2 S12 u z1S12 u z2Cohy1 y2 z1 z2 ndy1dy2dz1dz2 D55 The coherence Cohy1 y2 z1 z2 n may be expressed as in Chapter 2 A simple ten tative expression for the function Nn a measure of the coherence between pressures on the windward and leeward faces is Nn 1 for nUzD 02 D56a Nn 0 for nUzD 02 D56b where D is the depth of the building Figure D3 The mean square value of the fluctuating alongwind deflection is Eq B15 𝜎2 xz 0 Sxz ndn D57 From Eq B16b it follows that the mean square value of the alongwind accelera tion is 𝜎2 xz 16𝜋4 0 n4Sxz ndn D58 The expected value of the largest peak of the fluctuating alongwind deflection occur ring in the time interval T is xmax Kxz𝜎xz D59 where see Eqs B52 and B47 Kxz 2 ln vxzT12 0577 2 ln vxzT12 D60 vxz 0 n2Sxz ndn 0 Sxz ndn 12 D61 Similarly the largest peak of the alongwind acceleration is approximately xmaxz Kzz𝜎xz D62 Kxz 2 ln vxzT12 0577 2 ln vxzT12 D63 vxz 0 n6Sxz ndn 0 n4Sxz ndn 12 D64 It can be shown that the mean square value of the deflection may be written approxi mately as a sum of two terms the background term that entails no resonant amplifica tion and is due to the quasistatic effect of the fluctuating pressures and the resonant term which is associated with resonant amplification due to force components with k k k k 466 Appendix D Structural Dynamics frequencies equal or close to the fundamental natural frequency of the structure and is inversely proportional to the damping ratio 2 p 212 References 1 Robson JD 1964 An Introduction to Random Vibration New York Elsevier 2 Simiu E and Scanlan RH 1996 Wind Effects on Structures 3rd ed New York Wiley k k k k 467 Appendix E Structural Reliability E1 Introduction The objective of structural reliability is to develop criteria resulting in acceptably low probabilities that structures will fail to perform adequately under dead live and envi ronmental loads Adequate performance is defined as the nonexceedance of specified limit states The following are examples of limit states Demandtocapacity indexes DCIs may not significantly exceed unity strength limit state Buildings essential from a community resilience point of view eg hospitals police stations fire stations power plants must not collapse under loads induced by extreme events collapse limit state Interstory drift may not exceed a specified limit dependent upon type of cladding andor partitions serviceability limit state Accelerations may not exceed a specified peak or rms value serviceability limit state The performance of equipment essential to the building functionality must not be affected by the occurrence of an extreme event serviceability limit state Cladding performance must not result in damage to the structures contents service ability limit state Other limit states may be specified depending upon the building its contents and its functions Associated with the exceedance of any limit state is a minimum allowable mean recurrence interval MRI The more severe the consequences of exceeding the limit state the larger are the minimum allowable MRIs Building codes specify strength limit states For example the ASCE Standard 716 specifies a 700year MRI of the event that the strength of structural members of typi cal structures will be exceeded for critical structures whose failure would cause loss of life the Standard specifies a higher MRIs The specified MRIs are not based on explicit estimates of failure probabilities but rather on professional consensus based on experi ence intuition or belief Limit states not related to life safety and associated MRIs may be established by agreement among the owner the designer and the insurer although some nonstructural limit states may require compliance with regulatory requirements In the early phases of its development it was believed that structural reliability could assess the performance of any structural system by performing the following steps Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 468 Appendix E Structural Reliability i clear and unambiguous definition of limit states ii specification of design criteria on acceptable probabilities of exceedance of the limit states and iii checking whether for the structure being designed those criteria are satisfied The clear definition of certain limit states can be a difficult task For redundant structural systems as opposed to individual structural members structural safety assessments via reliability calculations are typically not possible in the present state of the art In addition probability distribution tails which determine failure probabilities are in many cases unknown Finally the specification of the acceptable failure proba bility for any limit state can be a complex economic or political issue that exceeds the bounds of structural engineering In view of apparently insuperable difficulties inherent in the original goals of struc tural reliability the discipline has settled for more modest goals Under the demand inherent in the wind and gravity loads with specified MRIs andor affected by their respective load factors each member cross section must experience DCIs lower than or approximately equal to unity Past experience with wind effects on buildings suggests that the memberbymember approach just described is safe1 even though it does not provide any explicit indication of the probability of exceedance of the incipient collapse limit state Improved forecasting capabilities which allow sufficient time for evacuation have resulted in massively reduced loss of life due to hurricanes particularly in developed countries The motivation to perform research into failure limit states has been far stronger for seismic regions than for regions with strong winds and the ASCE 7 Standard specifies seismic design criteria based on nonlinear analyses consistent with the requirement that the structure not collapse under a Maximum Considered Earthquake with a 2500year MRI The development of similar design criteria and research into nonlinear structural behavior are only beginning to be performed for structures subjected to wind loads Such development is necessary among other reasons because evacuation can be impractical or hampered by traffic problems hence the need for certain structures to be capable of safely surviving strong winds Section E2 explains why the use of probability distributions of demand and capacity may be problematic in structural engineering practice Subsequent sections are devoted to Load and Resistance Factor Design LRFD and its limitations Section E3 structural strength reserve Section E4 design MRIs for multihazard regions Section E5 The calibration of design MRIs for structures experiencing significant dynamic effects or for which errors in the estimation of extreme wind effects are significantly larger than the typical errors accounted for in the ASCE 716 Standard is considered in Chapters 7 and 12 E2 The Basic Problem of Structural Safety Assume that the probability distribution of the demand Q and the capacity R Pq r ProbQ q R r E1 is known The probability that q Q q dq and r R r dr is f q rdq ds The prob ability of failure is the probability that r q the shaded area in Figure E1 Pf 0 dq q 0 f q rdr E2 1 The degree to which this is the case depends upon the structures strength reserve see Section E5 k k k k Appendix E Structural Reliability 469 Q R dq q q f qr Figure E1 Domain of integration for calculation of probability of failure Since the loads and resistances are independent ie f qr f Qqf Rr Pf 0 fQq q 0 fRrdr dq 0 fQqFRq dq E3a b where f Q is the probability density function of the demand load and f R and FR are the probability density and the cumulative distribution functions of the capacity resistance respectively The integrand of Eq E3a b depends upon the upper and lower tail of the distributions f Q and FR respectively Figure E2 Typically it is not possible to ascertain what those distributions are For this reason a fully probabilistic approach to the estimation of structural reliabilities is in most cases not feasible E3 FirstOrder SecondMoment Approach Load and Resistance Factors The first ordersecond moment FOSM approach considered in this section was devel oped primarily in the 1970s following the realization that structural reliability theory based on explicit estimation of failure probabilities is not achievable in practice E31 Failure Region Safe Region and Failure Boundary Consider a member subjected to a load Q and let the load that induces a given limit state eg first yield be denoted by R Both Q and R are random variables that define the load space Failure occurs for any pair of values for which R Q 0 E4 The safe region is defined by the inequality R Q 0 E5 The failure boundary separates the failure and the safe regions and is defined by the relation R Q 0 E6 FR fQ QR Figure E2 Probability density function f Q and cumulative distribution function FR k k k k 470 Appendix E Structural Reliability Relations similar to Eqs E4E6 hold in the load effect space defined by the variables Qe and Re where Qe is an effect eg the stress induced in a member by the load Q and Re is the corresponding limit state eg the yield stress The failure boundary is then Re Qe 0 E7 Henceforth we use for simplicity the notations Q R for both the load space and the load effect space In general Q and R are functions of independent random variables X1 X2 Xn eg terrain roughness aerodynamic coefficients wind speeds natural frequencies damping ratios strength called basic variables that is Q QX1 X2 Xm E8 R RXm1 Xm2 Xn E9 Substitution of Eqs E8 and E9 into Eq E6 yields the failure boundary in the space of the basic variables defined by the equation Figure E3 gX1 X2 Xn 0 E10 It can be useful in applications to map the failure region the safe region and the failure boundary onto the space of variables Y 1 and Y 2 defined by transformations Y1 ln R E11 Y2 ln Q E12 On the failure boundary R Q so in the coordinates Y 1 Y 2 the failure boundary is Y 1 Y 2 E32 Safety Indexes Denote by S the failure boundary in the space of the reduced variables xi red Xi Xi𝜎xiwhere the variables Xi are mutually independent and Xi and 𝜎xi are respectively the mean and standard deviation of Xi The subscript red stands for reduced The reliability index denoted by 𝛽 is defined as the shortest distance in this space between the origin ie the image in the space of the reduced variables of Q R Failure region gX1Xn 0 Safe region gX1Xn 0 Failure boundary gX1Xn 0 Figure E3 Safe region failure region and failure boundary k k k k Appendix E Structural Reliability 471 the point with coordinates Xi and the failure boundary S The point on the boundary S closest to the origin and its image in the space of the original basic variables Xi are called the design point For any given structural problem the numerical value of the safety index depends upon the set of variables being considered Assume that the load Q and resistance R follow the normal distribution It is conve nient to express the random variables in nondimensional terms as follows qred Q Q 𝜎Q rred R R 𝜎R E13a b The failure surface Eq E6 has the following expression in the space of the reduced coordinates 𝜎Ssred 𝜎Qqred R Q 0 E14 The coordinates of points A B in Figure E4 are respectively R Q 𝜎Q 0 and 0 R Q 𝜎R E15 The slope of failure surface line is 𝛼 tan1OBOA tan1𝜎Q𝜎R E16 The slope of line L normal to the failure surface is 1tan1𝜎Q𝜎R The design point D is the intersection of the failure surface and line L Its coordinates q red r red are q red 𝛽 sin 𝛼 R Q 𝜎Q 𝜎2 R 𝜎2 Q r red 𝛽 cos 𝛼 R Q 𝜎R 𝜎2 R 𝜎2 Q E17a b O B A Failure surface S R Q 0 L OD β rred qred D Design point α Dq red red r β sin α β cos α OA R Q σQ OB R Q σR tanα σQ σR Figure E4 Index 𝛽 in the space of the reduced variables qred and rred k k k k 472 Appendix E Structural Reliability where the distance 𝛽 between the origin and the failure surface in the space of reduced variables is defined as the safety index From Eqs E17 if follows that 𝛽 R Q 𝜎2 R 𝜎2 Q12 E18 Example E1 Assume that the resistance is deterministic that is R R The mapping of the failure boundary Q R 0 onto the space of the reduced variate qred Q Q𝜎Q is a point q red such that Q R that is q red R Q𝜎Q Figure E5 The asterisk denotes the design point The origin in that space is the point for which qred 0 and corresponds to Q Q The distance 𝜎r 0 between the origin and the failure boundary is the safety index 𝛽 R Q𝜎Q since in this case in Eq E18 The case Q R 0 load larger than resistance corre sponds to failure In the space of the reduced variable failure occurs for qred q red that is qred 𝛽 The larger the ratio 𝛽 R Q𝜎Q the smaller is the probability of failure The relia bility index thus provides an indication on a members safety However this indication is largely qualitative unless information is available on the probability distribution of the variate Q Instead of operating in the load space R Q consider the failure boundary in the trans formed space defined by Eqs E11 and E12 If Q and R are assumed to be mutually independent and lognormally distributed the distribution of Y 1 Y 2 and Y 2 Y 1 ie lnQ lnR and lnRQ respectively will be normal Following the same steps as in the normal distribution case but applying them to the variables Y 1 and Y 2 the safety index becomes 𝛽 Y 1 Y 2 𝜎2 Y1 𝜎2 Y212 E19 Expansion in a Taylor series yields the expression Y1 ln R R R 1 R 1 2R R2 1 R 2 E20 qred Failure boundary 0 Design point qred β Figure E5 Index 𝛽 for member with random load and deterministic resistance k k k k Appendix E Structural Reliability 473 O B A D In Qred In Rred β sin α β cos α L OD β ln Rred ln Qred Failure surface ln R ln Q 0 OA ln R ln Q VQ D Design point tan α VQVR α OB ln R ln Q VR Figure E6 Index 𝛽 for member with random load and random resistance in the space of the reduced variables lnRred and lnQred and a similar expression for Y 2 Averaging these expressions neglecting second and higher order terms and using the notations 𝜎RR VR 𝜎QQ VQ the safety index can be expressed as 𝛽 ln R ln Q V 2 R V 2 Q12 E21 Figure E6 is the counterpart of Figure E4 obtained by substituting in Eq E18 ln R ln Q VR VQ for R Q 𝜎R 𝜎Q respectively Note The approach wherein only means and standard deviations or coefficients of variation are used is called the firstorder second moment FOSM approach E33 Reliability Indexes and Failure Probabilities The probability of failure is Pf ProbR Q 0 Probg 0 E22 If the variates R and Q are normally distributed the probability distribution of R Q g is also normal It follows then from Eq E22 that Pf Fg0 E23 Figure E7 where Fg is the Gaussian cumulative distribution of g or Pf Prg 0 Φ 0 g 𝜎g Φ R Q 𝜎2 R 𝜎2 Q12 Φ𝛽 1 Φ𝛽 E24abcde k k k k 474 Appendix E Structural Reliability 0 g fgg βσg Pf Fg0 g Figure E7 Probability distribution function f g g of variate g R Q The probability of failure is equal to the area under the curve f g g for g 0 where Φ is the standard normal cumulative distribution function and 𝛽 is defined by Eq E18 If the variates R and Q are lognormally distributed meaning that their logarithms are normally distributed the probability of failure is Pf Prln R ln Q 0 1 Φ ln R ln Q 𝜎2 ln R 𝜎2 ln Q12 1 Φ𝛽 E25abc where the fraction in Eq E25b is equal to the safety index defined in Eq E21 The usefulness of Eq E24e and E25c is limited by the fact that typically neither the load nor the resistance is normally or lognormally distributed E34 Partial Safety Factors Load and Resistance Factor Design Consider a structure characterized by a set of variables with means Xi and standard deviations 𝜎i and design points X i i 1 2 n in the space of the original variables By definition X i Xi 𝜎Xix i red E26 Equation E26 can be written in the form X i 𝛾XiXi E27 where 𝛾Xi 1 VXix i red E28 and VXi 𝜎XiXi the asterisk denotes the design point Let i 1 2 X1 Q X2 R and 𝛾X1 𝛾Q 𝛾X2 𝜑R The quantities 𝛾Q 𝜑R are called the load and the resistance factor respectively We consider now the case Y 1 ln R and Y 2 ln Q on which current design practice is based The counterpart to Eqs E26 is ln Q red ln Q ln Q VQ E29 k k k k Appendix E Structural Reliability 475 Since lnQ 𝛽sinα where tan 𝛼 V QV R see Figure E6 and 𝛽 is defined by Eq E21 it follows that lnQQ VQ𝛽sin𝛼 E30 see Figure E6 Therefore Q Q expVQ𝛽sin𝛼 E31 Since Q 𝛾QQ the load factor is 𝛾Q expVQ𝛽sin𝛼 E32 Similarly the resistance factor is 𝜑R expVR𝛽cos𝛼 E33 In Eqs E29E33 𝛽 is defined by Eq E21 The following linear approximation to Eq E26 has been developed for use in standards 1 𝛾Q 1 055𝛽VQ E34 Equation E34 can in many instances be a poor approximation to Eq E32 E35 Calibration of Safety Index 𝜷 Wind Directionality and Mean Recurrence Intervals of Wind Effects Because the approach to the calculation of the safety index by methods that presuppose the universal validity of the lognormal distribution can be unsatisfactory load factors specified explicitly or implicitly in the ASCE 7 Standard have been calibrated against past practice using uncertainty estimates and engineering judgment see Chapters 7 and 12 If wind directionality is considered by explicitly taking into account the directional distribution of the wind speeds at the building site rather than by using wind direction ality factors as specified by the ASCE 7 Standard MRIs of the design wind effects are no longer equal to the MRIs of the design wind speeds see Chapter 13 for details E4 Structural Strength Reserve The design of structural members by LRFD methods ensures that they do not experi ence unacceptable behavior as they attain the respective strength limit states However it is desirable that even if those limit states are exceeded the performance of the struc ture remains acceptable in some sense A structure with large strength reserve is one for which this is the case for wind effects with MRIs significantly larger than the MRIs inducing strength limit states Strength reserve can be assessed by estimating MRIs of incipient collapse or other appropriate performance measures Sections E41 and E42 provide such estimates for portal frames for a single wind direction and by considering the effect of all wind direc tions respectively and note a thorough study of postelastic behavior of tall buildings subjected to wind 2 k k k k 476 Appendix E Structural Reliability E41 Portal Frame Ultimate Capacity Under Wind with Specified Direction For lowrise industrial steel buildings with gable roofs and portal frames nonlinear pushover studies have been conducted in which the buildings were subjected to two sets of wind pressures 3 One set consisted of wind pressures based on aerodynamic information specified for lowrise structures in the ASCE 7 Standard The second set consisted of simultaneous wind pressures measured and recorded in the wind tunnel at a large number of taps on the building models surface The structural design of the frames was based on ASCE 7 Standard loads and the Allowable Stress Design approach The objectives of the studies were i to compare the strength reserve levels estimated by using a the simplified wind loads inherent in the ASCE Standard and b recorded time series of wind tunnel pressures and ii to examine the degree to which the strength reserve can be increased by the adoption of alternative designs The following alternative features of the lateral bracing and joint stiffening were considered 1 a 25 m spacing and b 6 m spacing of lateral bracing of rafter bottom flanges 2 Knee a horizontal and vertical stiffeners and b horizontal vertical and diagonal stiffeners 3 Ridge a without and b with vertical stiffener at ridge Strength analyses were performed for the load combinations involving wind Calcula tions were performed of the ratio 𝜆 between ultimate and allowable wind load for each load combination being considered the ultimate wind load corresponding to incipient failure through local or global instability as determined by using a finite element analysis program Reducing the distance between bracings of the rafters lower flanges increased the strength reserve more effectively than providing diagonal stiffeners in the knee joint Figure E8 Significant differences were found between the values of 𝜆 obtained under loading by pressures specified in the ASCE 7 Standard provisions and loading by the more realistic pressures measured in the wind tunnel For details see 4 E42 Portal Frame Ultimate Capacity Estimates Based on MultiDirectional Wind Speeds The following methodology was developed for the estimation of MRIs of ultimate wind effects by accounting for wind directionality 5 a b Figure E8 Local buckling in knee a with and b without diagonal stiffener industrial building steel portal frame k k k k Appendix E Structural Reliability 477 1 Using recorded wind tunnel pressure data obtain the loads that induce peak internal forces axial forces bending moments shear forces at a number of cross sections deemed to be critical Obtain the loads corresponding to a unit wind speed at 10 m above ground over open terrain for say 16 or 36 wind directions spanning the 360 range These loads multiplied by the square of wind speeds U considered in design are used in step 2 2 Using nonlinear finite element analyses determine the wind speed from each direc tion 𝜃i that causes the frame to experience incipient failure defined as the onset of deformations that increase so fast under loads that implicit nonlinear finite element analyses fail to converge to a solution 3 From available wind climatological data create by simulation time series of direc tional wind speeds with length td that exceeds the anticipated MRIs of the failure events 6 4 Count the number nf of cases in which directional wind speeds in the time series created in step 3 exceed the directional wind speeds determined in step 2 to pro duce incipient failure events The MRI in years of the failure event is estimated as N tdnf This methodology was applied to an industrial lowbuilding portal frame located in a hurricaneprone region The frame was strengthened by triangular stiffeners at the column supports and by haunches and horizontal vertical and diagonal stiffeners at the knee joints Owing to such strengthening the estimated failure MRI was in this case quite high 100000 years corresponding to a nominal 11000 probability that the frame will fail during a 100year life E43 Nonlinear Analysis of Tall Buildings Under Wind Loads An extensive study of postelastic behavior of highrise buildings subjected to wind loads is presented in 2 which incorporates and adapts methods and results obtained for structures that behave nonlinearly under seismic loads Future research may consider the possibility that under the strong wind loading inducing nonlinear behavior in struc tural members tall buildings might experience aeroelastic or to introduce a new but apposite term aeroplastic effects E5 Design Criteria for MultiHazard Regions E51 Strong Winds and Earthquakes Structures in regions subjected to both strong earthquakes and strong winds are cur rently designed by considering separately loads induced by earthquakes and by winds and basing the final design on the more demanding of those loads The rationale for this approach has been that the probability of simultaneous occurrence of both earth quakes and high winds is negligibly small It is shown in this section that implicit in this approach are probabilities of failure that can be greater by a factor of up to two than their counterparts for structures exposed to wind only or to earthquakes only An intuitive illustration of this statement follows Assume that a motorcycle racer applies for insurance against personal injuries The insurer will calculate a rate k k k k 478 Appendix E Structural Reliability commensurate with the probability that the racer will be injured in a motorcycle accident Assume now that the motorcycle racer is also a highwire artist The insur ance rate would then be increased since the probability that an injury will occur during a specified period of time either in a motorcycle or highwire accident will be greater than the probability associated with risk due to only one of these types of accident This is true even though the nature of the injuries in the two types of event may differ This argument is expressed formally as Ps1 s2 Ps1 Ps2 E35 where Ps1 annual probability of event s1 injury due to a motorcycle accident Ps2 annual probability of event s2 injury due to a highwire accident and Ps1 s2 probability of injury due to a motorcycle or a highwire accident Equation E35 is applicable to structures as well particularly to members experienc ing large demands under lateral loads eg columns in lower floors For details and case studies see 7 8 E52 Winds and Storm Surge Unlike earthquakes and windstorms winds and storm surge are not independent events Therefore for some applications it is necessary to consider their simultaneous effects This entails the following steps i select a stochastic set of hurricane storm tracks in the region of interest ii use the selected storm tracks to generate time histories of wind speeds and corresponding time histories of storm surge heights at sites affected by those wind speeds iii use those time histories to calculate time series of wind and storm surge effects and iv obtain from those time series estimates of joint effects of wind and storm surge with the mean recurrence intervals of interest 9 10 In this approach the calculations are performed in the load effect space An important factor in the estimation of storm surge heights is the bathymetry at and near the site of interest To be realistic storm surge intensities must be based on current information on local bathymetry which can change significantly over time References 1 Ravindra MKG Theodore V and Cornell CA 1978 Wind and snow load factors for use in LRFD Journal of the Structural Division 104 14431457 2 Mohammadi A Wind performancebased design of highrise buildings Doctoral dissertation Department of Civil and Environmental Engineering Florida International University 2016 3 Jang S Lu LW Sadek F and Simiu E 2002 Databaseassisted wind load capacity estimates for lowrise steel frames Journal of Structural Engineering 128 15941603 4 Duthinh D and Fritz WP 2007 Safety evaluation of lowrise steel structures under wind loads by nonlinear databaseassisted technique Journal of Structural Engineering 133 587594 httpswwwnistgovwind k k k k Appendix E Structural Reliability 479 5 Duthinh D Main JA Wright AP and Simiu E 2008 Lowrise steel structures under directional winds mean recurrence interval of failure Journal of Structural Engineering 134 13831388 6 Yeo D 2014 Generation of large directional wind speed data sets for estimation of wind effects with long return periods Journal of Structural Engineering 140 04014073 httpswwwnistgovwind 7 Duthinh D and Simiu E 2010 Safety of structures in strong winds and earth quakes multihazard considerations Journal of Structural Engineering 136 330333 httpswwwnistgovwind 8 Crosti C Duthinh D and Simiu E 2010 Risk consistency and synergy in multihazard design Journal of Structural Engineering 137 16 9 Phan L T Simiu E McInerney M A Taylor A A and Powell M D Method ology for the Development of Design Criteria for Joint Hurricane Wind Speed and Storm Surge Events Proof of Concept NIST Technical Note 1482 National Insti tute of Standards and Technology Gaithersburg MD 2007 httpswwwnistgov wind 10 Phan L T Slinn D N and Kline S W Introduction of Wave Setup Effects and Mass Flux to the Sea Lake and Overland Surges from Hurricanes SLOSH Model NISTIR 7689 National Institute of Standards and Technology Gaithersburg MD 2010 httpswwwnistgovwind k k k k 481 Appendix F World Trade Center Response to Wind A Skidmore Owings and Merrill Report Note The material that follows reproduces NIST document NCSTAR12 Appendix D dated April 13 2004 httpwtcnistgovNCSTAR1NCSTAR12indexhtm submitted by Skidmore Owings and Merrill LLP Chicago Illinois wtcnistgov The documents listed in Sections F1 F2 and F3 are not in the public domain but are believed to be obtainable under the provisions of the Freedom of Information Act The material illustrates difficulties encountered by practicing structural engineers in evaluating wind engineering laboratory reports and contains useful comments on the state of the art in wind engineering at the time of its writing The text that follows is identical to the text of the Skidmore Owings and Merrill report except for numbering of the headings F1 Overview F11 Project Overview The objectives for Project 2 of the WTC Investigation include the development of reference structural models and design loads for the WTC Towers These will be used to establish the baseline performance of each of the towers under design gravity and wind loading conditions The work includes expert review of databases and baseline structural analysis models developed by others as well as the review and critique of the wind loading criteria developed by NIST F12 Report Overview This report covers work on the development of wind loadings associated with Project 2 This task involves the review of wind loading recommendations developed by NIST for use in structural analysis computer models The NIST recommendations are derived from wind tunnel testingwind engineering reports developed by independent wind engineering consultants in support of insurance litigation concerning the WTC towers The reports were provided voluntarily to NIST by the parties to the insurance litigation As the third party outside experts assigned to this Project SOMs role during this task was to review and critique the NISTdeveloped wind loading criteria for use in computer analysis models This critique was based on a review of documents provided by NIST specifically the wind tunnelwind engineering reports and associated correspondence from independent wind engineering consultants and the resulting interpretation and recommendations developed by NIST Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 482 Appendix F World Trade Center Response to Wind F2 NISTSupplied Documents F21 Rowan Williams Davies Irwin RWDI Wind Tunnel Reports Final Report WindInduced Structural Responses World Trade Center Tower 1 New York New York Project Number 021310A October 4 2002 Final Report WindInduced Structural Responses World Trade Center Tower 2 New York New York Project Number021310B October 4 2002 F22 Cermak Peterka Petersen Inc CPP Wind Tunnel Report WindTunnel Tests World Trade Center New York NY CPP Project 022420 August 2002 F23 Correspondence Letter dated October 2 2002 from Peter IrwinRWDI to Matthys LevyWeidlinger Associates Re Peer Review of Wind Tunnel Tests World Trade Center RWDI Reference 021310 Weidlinger Associates Memorandum dated March 19 2003 from Andrew Cheung to Najib Abboud Re Errata to WAI Rebuttal Report Letter dated September 12 2003 from Najib N AbboudHartWeidlinger to S Shyam Sunder and Fahim SadekNIST Re Responses to NISTs Questions on WindInduced Structural Responses World Trade Center Project Number 021310A and 021310B October 2002 by RWDI Prepared for HartWeidlinger Letter dated April 6 2004 From Najib N Abboud Weidlinger Associates To Fahim Sadek and Emil Simiu Re Response to NISTs question dated March 30 2004 regarding Final Report Wind Induced Structural Responses World Trade Center Tower 2 RWDI Oct 4 2002 F24 NIST Report Estimates of Wind Loads on the WTC Towers Emil Simiu and Fahim Sadek April 7 2004 F3 Discussion and Comments F31 General This report covers a review and critique of the NIST recommended wind loads derived from wind load estimates provided by two independent private sector wind engineer ing groups RWDI and CPP These wind engineering groups performed wind tunnel testing and wind engineering calculations for various private sector parties involved in insurance litigation concerning the destroyed WTC Towers in New York There are substantial disparities greater than 40 in the predictions of base shears and base k k k k Appendix F World Trade Center Response to Wind 483 overturning moments between the RWDI and CPP wind reports NIST has attempted to reconcile these differences and provide wind loads to be used for the baseline structural analysis F32 Wind Tunnel Reports and Wind Engineering The CPP estimated wind base moments far exceed the RWDI estimates These differ ences far exceed SOMs experience in wind force estimates for a particular building by independent wind tunnel groups In an attempt to understand the basis of the discrepancies NIST performed a critique of the reports Because the wind tunnel reports only summarize the wind tunnel test data and wind engineering calculations precise evaluations are not possible with the provided information For this reason NIST was only able to approximately evaluate the differences NIST was able to numerically estimate some corrections to the CPP report but was only able to make some qualitative assessments of the RWDI report It is important to note that wind engineering is an emerging technology and there is no consensus on certain aspects of current practice Such aspects include the correlation of wind tunnel tests to fullscale building behavior methods and compu tational details of treating local statistical historical wind data in overall predictions of structural response and types of suitable aeroelastic models for extremely tall and slender structures It is unlikely that the two wind engineering groups involved with the WTC assessment would agree with NIST in all aspects of its critique This presumptive disagreement should not be seen as a negative but reflects the state of wind tunnel practice It is to be expected that wellqualified experts will respectfully disagree with each other in a field as complex as wind engineering SOMs review of the NIST report and the referenced wind tunnel reports and correspondence has only involved discussions with NIST it did not involve direct communication with either CPP or RWDI SOM has called upon its experience with wind tunnel testing on numerous tall building projects in developing the following comments F321 CPP Wind Tunnel Report The NIST critique of the CPP report is focused on two issues a potential overestimation of the wind speed and an underestimation of load resulting from the method used for integrating the wind tunnel data with climatic data NIST made an independent estimate of the wind speeds for a 720year return period These more rare wind events are dominated by hurricanes that are reported by rather broad directional sectors 225 The critical direction for the towers is from the azimuth direction of 205210 This wind direction is directly against the nominal south face of the towers the plan north of the site is rotated approximately 30 degrees from the true north and generates dominant crosswind excitation from vortex shedding The nearest sector data are centered on azimuth 2025 SSW and 225 SW There is a substantial drop 12 in the NIST wind velocity from the SSW sector to the SW sector The change in velocity with direction is less dramatic in the CCP 720year velocities or in the ARA hurricane wind roses included in the RWDI report This sensitivity to directionality is a cause for concern in trying to estimate a wind speed for a particular direction However it should be noted that the magnitude of the NIST interpolated estimated velocity for the 210 azimuth direction is similar to the ARA wind rose The reduction of forces has k k k k 484 Appendix F World Trade Center Response to Wind been estimated by NIST based on a square of the velocity however a power of 23 may be appropriate based on a comparison of the CPP 50year nominal and 720year base moments and velocities The NIST critique of the CPP use of sector by sector approach of integrating wind tunnel and climatic data is fairly compelling The likelihood of some degree of under estimation is high but SOM is not able to verify the magnitude of error 15 which is estimated by NIST This estimate would need to be verified by future research as noted by NIST F322 RWDI Wind Tunnel Report The NIST critique of RWDI has raised some issues but has not directly estimated the effects These concerns are related to the wind velocity profiles with height used for hurricanes and the method used for upcrossing NIST questioned the profile used for hurricanes and had an exchange of correspon dence with RWDI While RWDIs written response is not sufficiently quantified to permit a precise evaluation of NISTs concerns significant numerical corroboration on this issue may be found in the April 6 letter Question 2 from N Abboud Weidlinger Associates to F Sadek and E Simiu NIST NIST is also concerned about RWDIs upcrossing method used for integrating wind tunnel test data and climatic data This method is computationally complex and verifica tion is not possible because sufficient details of the method used to estimate the return period of extreme events are not provided F323 Building Period used in Wind Tunnel Reports SOM noted that both wind tunnel reports use fundamental periods of vibrations that exceed those measured in the actual north tower buildings The calculation of building periods are at best approximate and generally underestimate the stiffness of a building thus overestimating the building period The wind load estimates for the WTC tow ers are sensitive to the periods of vibration and often increase with increased period as demonstrated by a comparison of the RWDI base moments with and without PDelta effects Although SOM generally recommends tall building design and analysis be based on PDelta effects in this case even the first order period analysis without PDelta exceeds the actual measurements It would have been desirable for both RWDI and CPP to have used the measured building periods F324 NYCBC Wind Speed SOM recommends that the wind velocity based on a climatic study or ASCE 702 wind velocity be used in lieu of the New York City Building Code NYCBC wind velocity The NYCBC wind velocity testing approach does not permit hurricanes to be accommo dated by wind tunnel testing as intended by earlier ASCE 7 fastest mile versions because it is based on a method that used an importance factor to correct 50year wind speeds for hurricanes Because the estimated wind forces are not multiplied by an importance factor this hurricane correction is incorporated in analytical methods of determining wind forces but is lost in the wind tunnel testing approach of determining wind forces k k k k Appendix F World Trade Center Response to Wind 485 F325 Incorporating Wind Tunnel Results in Structural Evaluations It is expected that ASCE 7 load factors will also be used for member forces for evaluating the WTC towers Unfortunately the use of ASCE 7 with wind tunnelproduced loadings is not straightforward Neither wind tunnel report gives guidance on how to use the provided forces with ASCE 7 load factors The ASCE 7 load factors are applied to the nominal wind forces and according to the ASCE 7 commentary are intended to scale these lower forces up to wind forces asso ciated with long return period wind speeds The approach of taking 500year return period wind speeds and dividing the speeds by the square root of 15 to create a nominal design wind speed determining the building forces from these reduced nominal design wind speeds and then magnifying these forces by a load factor often 16 is at best convoluted For a building that is as aerodynamically active as the WTC an approach of directly determining the forces at the higher long return period wind speeds would be preferred The CPP data did provide the building forces for their estimates of both 720years a load factor of 16 and the reduced nominal design wind speeds A compar ison of the wind forces demonstrates the potential error in using nominal wind speeds in lieu of directly using the underlying long period wind speeds It should also be noted that the analytical method of calculating wind forces in ASCE 7 provides an importance factor of 115 for buildings such as the WTC in order to provide more conservative designs for buildings with high occupancies Unfortunately no similar clear guidance is provided for high occupancy buildings where the wind loads are determined by wind tunnel testing Utilizing methods provided in the ASCE 7 Commentary would suggest that a return period of 1800 years with wind tunnelderived loads would be comparable to the ASCE 7 analytical approach to determining wind loads for a high occupancy building It would be appropriate for the wind tunnel private sector laboratories or NIST as future research beyond the scope of this project to address how to incorporate wind tunnel loadings into an ASCE 7based design F326 Summary The NIST review is critical of both the CPP and RWDI wind tunnel reports It finds substantive errors in the CPP approach and questions some of the methodology used by RWDI It should be noted that boundary layer wind tunnel testing and wind engineering is still a developing branch of engineering and there is not industrywide consensus on all aspects of the practice For this reason some level of disagreement is to be expected Determining the design wind loads is only a portion of the difficulty As a topic of future research beyond the scope of this project NIST or wind tunnel private sector laboratories should investigate how to incorporate these wind tunnelderived results with the ASCE 7 Load Factors F33 NIST Recommended Wind Loads NIST recommends a wind load that is between the RWDI and CPP estimates The NIST recommended values are approximately 83 of the CPP estimates and 115 of k k k k 486 Appendix F World Trade Center Response to Wind the RWDI estimates SOM appreciates the need for NIST to reconcile the disparate wind tunnel results It is often that engineering estimates must be done with less than the desired level of information In the absence of a wind tunnel testing and wind engineering done to NIST specifications NIST has taken a reasonable approach to estimate appropriate values to be used in the WTC study However SOM is not able to independently confirm the precise values developed by NIST The wind loads are to be used in the evaluation of the WTC structure It is therefore recommended that NIST provide clear guidelines on what standards are used in the evaluations and how they are to incorporate the provided wind loads k k k k 487 Index a Accelerations building 217 464 and human discomfort 226 Acrosswind response 287 chimneys towers and stacks 292 318 323 suspendedspan bridges 335 338 tall buildings 321 Added mass 309 378 Addition of probabilities 412 Adiabatic lapse rate 6 Admittance mechanical 252 254 456 462 Advection turbulent energy 39 Aerodynamic damping 289 294 309 319 321 341 negative 305 323 positive 322 Aerodynamic derivatives flutter 305 motional 311 Scanlan 308 310 steadystate 300 Aerodynamic loads 171 183 Aerodynamics bluff body 73 bridge deck improvement of 335 tall buildings improvement of 326 Aeroelastic behavior 170 283 300 312 Aeroelastic instability 283 Aeroelastic testing 105 385 Airsupported structures 151 386 Air viscosity 75 Alleviation of windinduced response 325 344 Alongwind response 321 455 463 bridges 338 tall buildings 207 Angle of attack 92 125 297 309 316 341 Animation wind pressures 97 ANSIANS232011 68 70 401 Antenna dishes 350 Anticyclonic circulations 10 Arrival rate of see Rate of arrival ASCE 7 Standard 10 21 27 31 46 58 66 102 122 173 192 209 211 273 468 ASOS see Automated Surface Observing Systems Aspect ratio 146 188 192 316 349 396 Atmospheric boundary layer 8 17 circulations 3 hydrodynamics 7 motions 19 pressure 5 10 13 66 78 108 123 184 389 thermodynamics 3 turbulence 35 wind tunnel simulation 120 Autocorrelation 36 437 446 Autocovariance function 437 460 Automated Surface Observing Systems ASOS 17 58 60 Averaging times wind speeds 17 30 33 57 215 b Balance frictionless wind 8 Barotropic flows 51 Baseball aerodynamics 80 Base pressure 91 Bayes rule 413 Bénard 85 Bernoulli equation 76 80 89 Blockage numerical simulation 141 wind tunnel testing 127 Wind Effects on Structures Modern Structural Design for Wind Fourth Edition Emil Simiu and DongHun Yeo 2019 John Wiley Sons Ltd Published 2019 by John Wiley Sons Ltd k k k k 488 Index Bluff body aerodynamics see Aerodynamics Bora winds 11 Boundary layers atmospheric 8 17 depth 8 22 44 108 internal 46 laminar 123 thickness of 20 turbulent 96 Boussinesq approximation 143 145 Bridge decks buffeting response 312 338 flutter 305 galloping 297 torsional divergence 303 vortexinduced response 287 Bridge response in turbulent flow 339 Brighton Chain Pier failure 283 BruntVäisäla frequency 24 Buffeting of bridges 312 338 of tall buildings 463 Buffon on pedestrianlevel winds 227 Buildings databaseassisted design 171 equivalent static wind loads 219 c Cable bundled 297 roofs 385 vibration 344345 Calibration tubing systems 130 Capping inversion 7 17 20 21 25 Center aerodynamic 378 elastic 198 262 306 mass 171 174 179 198 306 Central limit theorem 421 CFD see Computational Fluid Dynamics Change of terrain roughness 45 Chimneys 315 acrosswind response 315 318 323 Chinook winds 10 Circular frequency 36 197 298 307 455 Circulations atmospheric 3 Cladding design 186 188 192 Climatology 62 66 Clusters 446 Coefficient of variation CoV 157 205 419 Coefficients aerodynamic 292 307 312 372 drag and lift force 91 moment 91 pressure 90 Coherence 45 316 337 340 378 439 464 Comfort criteria 225 227 Computational Fluid Dynamics 25 45 73 135 204 Computational Wind Engineering 73 135 232 268 385 Condensation 4 7 11 12 Conditional probabilities 68 412 Conditions boundary 138 initial 137 Confidence intervals 425 extreme wind predictions 64 67 Confidence level 426 Construction stage bridges 331 Continuity equation of 19 74 136 139 Coriolis effects and wind tunnel testing 106 forces 8 78 parameter 8 19 22 25 27 44 106 Correlation see Autocorrelation Crosscorrelation crosscovariance function Correlation coefficient 419 Cospectrum 439 462 464 CourantFriedrichsLewy CFL condition 139 Critical divergence velocity 303 Critical flutter velocity 312 341 Critical region flow about cylinders 91 Crosscorrelation crosscovariance function 361 438 462 Crossspectral density function 44 315 340 Crossspectrum 44 315 340 see also Crossspectral density function of turbulence fluctuations 44 Cumulative distribution functions 55 162 214 416 443 473 CWE see Computational Wind Engineering Cyclones extratropical 10 45 215 tropical 10 Cyclostrophic equation 393 wind 13 66 Cylinders flow past 84 90 96 123 148 287 297 d DAD see Databaseassisted design Dampers 200 226 251 326 344 455 Damping aerodynamic see Aerodynamic damping k k k k Index 489 critical 197 456 negative 305 323 ratio 128 197 200 204 254 288 292 298 318 322 338 344 376 382 455 470 tall buildings 171 207 Databaseassisted design 171 flexible buildings 267 interpolation procedures 263 NISTUWO database 100 183 189 259 444 rigid buildings 259 Tokyo Polytechnic University TPU database 100 183 190 259 273 DCI see Demandtocapacity index Debris windborne 404 Declustering 426 446 De Haan estimation 65 426 Demandtocapacity index 171 174 211 217 Density function probability spectral see Spectral density function Depressurization during tornado passage 394 Derivatives material 75 motional see Flutter substantial 75 DES see Detached Eddy Simulation Design wind effects 175 203 209 Detached Eddy Simulation 146 Deviating force 7 Deviatoric stress 75 SGS 143 Diffusion turbulent energy 6 39 Dimensional analysis 105 Directional data simulation of 66 Directionality effects of wind 158 databaseassisted design approach 173 214 outcrossing approach 434 sectorbysector approach 212 Directionality factor 215 457 uncertainty in 164 Directional wind speed data 58 60 175 261 268 427 Direct Numerical Simulation DNS 140 Discomfort windinduced 225 building occupant 226 in pedestrian area 227 Dissipation turbulent energy 39 140 Distribution function cumulative 55 162 214 416 443 473 Distribution probability 415 Fréchet see Fréchet distribution Gaussian see Normal distribution Generalized Extreme Value see Generalized Extreme Value GEV distribution Generalized Pareto see Generalized Pareto distribution geometric 420 Gumbel see Gumbel distribution joint 417 of largest values lognormal see Lognormal distribution mixed 57 normal see Normal distribution of peaks in random signals 441 Poisson see Poisson distribution reverse Weibull see Reverse Weibull distribution Type I see Gumbel distribution Type II see Fréchet distribution Type III see Reverse Weibull distribution Divergence 303 DNS see Direct Numerical Simulation Downdraft thunderstorms 12 49 Drag 8 91 312 331 349 361 374 377 400 coefficient see Coefficients aerodynamic drag and lift force Drift interstory 217 Duration storm 33 36 163 Dynamic pressure 76 90 162 Dynamic response 195 Dynamics structural 455 e Earthquakes and winds multihazard regions 428 Eddies turbulent 36 38 140 142 146 Eddy conduction 6 Eddy viscosity 21 143 Effective wind loads 171 196 276 Efficiency estimator 425 Ekman layer turbulent 21 Ekman spiral 21 Elastic center eccentricity of 197 Energy cascade 38 141 Energy dissipation 39 140 289 Energy production 40 Energy spectrum turbulent 39 140 Energy turbulent kinetic 39 140 142 Enhanced Fujita EF scale 13 16 68 Ensemble 144 433 k k k k 490 Index Epochal approach estimation of extreme speeds 61 63 424 Equivalent static wind loads 174 219 Ergodic processes 433 Errors acknowledged 151 aerodynamic interpolation 206 convergence 149 discretization 149 dynamic response 204 estimates extreme wind effects 157 extreme wind estimation 159 iterative 149 modeling 158 215 physical modeling 149 programminguser 149 round off 149 sampling 61 62 64 unacknowledged 151 Escarpments flow over 46 Estimates extreme wind speeds 63 65 Estimators efficiency of 425 Exceedance probabilities 55 Exposure categories 31 Extratropical storms 36 49 55 159 Extreme Value EV distribution joint Generalized Extreme Value distribution 423 Generalized Pareto distribution 423 reverse Weibull see Reverse Weibull distribution Type I see Gumbel distribution Type II see Fréchet distribution Type III see Reverse Weibull distribution Extreme wind speeds and effects 55 211 nonparametric methods for estimating 215 428 parametric methods for estimating 214 426 Eye hurricane 10 49 Eyewall 10 48 f Fastestmile wind 18 33 484 Fetch 20 46 120 Finite Difference Method FDM 136 Finite Element Method FEM 136 Finite Volume Method FVM 137 First gust 12 49 Firstorder second moment reliability 469 Flachsbart 99 170 Flexible buildings 225 267 321 455 Flow reattachment 80 87 92 93 123 312 reversal 80 230 separation 80 85 87 91 93 96 123 124 146 147 161 171 289 312 344 367 Flutter 305 aerodynamic derivatives flutter 305 analysis threedimensional 338 critical velocity see Critical flutter velocity formulation of problem for twodimensional bridge 306 Scanlan flutter derivatives 308 310 torsional 342 turbulent flowinduced flutter 312 vortexinduced oscillation and flutter 305 Foehn winds 10 Fourier integrals 433 Fourier series 433 Fourier transform pair 434 438 Fréchet distribution 62 423 445 452 Free atmosphere 8 20 Frequency circular see Circular frequency natural 253 288 305 338 380 458 reduced 106 108 185 288 306 331 Friction effect on air flow 8 Frictionless wind balance Friction velocity 22 25 32 44 Froude number 106 385 g Galloping 297 of coupled bars 300 power line 297 suspensionspan bridge 305 308 309 311 wake 297 Gaussian distribution see Normal distribution Generalized coordinates 197 199 200 255 319 322 338 458 Generalized Extreme Value GEV distribution 423 Generalized force 459 Generalized mass 191 Generalized Pareto distribution 423 Geostrophic height 31 Geostrophic wind 22 GlauertDen Hartog criterion 297 Goodness of fit 425 Gradient height 8 31 404 Gradient velocity 8 20 k k k k Index 491 Grids structured 137 unstructured 137 Gumbel distribution 62 422 Gust front see First gust Gust speeds 17 57 Guyed towers 367 h Harmonic load response to 456 Helicopter landing decks offshore platforms 367 HFFB see HighFrequency Force Balance HighFrequency Force Balance 128 Hills wind flow over 47 Histograms 415 Hourly wind speed 18 32 Hshaped cross section 311 Human response to vibrations 226 Hurricaneborne missile speeds 405 Hurricanes 10 definition of 10 estimation of extreme winds in 60 simulations of 58 60 structure of 10 48 turbulence intensities in 48 uncertainties in hurricane wind speeds 160 wind flows in 10 Hybrid RANSLES 146 Hydrodynamics atmospheric 7 i Impulse function unit see Unit impulse function Incompressible flow 73 76 136 139 Independence stochastic 414 Inertial subrange spectra in 39 40 141 Influence coefficients 181 Instabilities aeroelastic 283 Integral turbulence scale closedform expression for 42 definition 36 dependence on height 37 43 relation to turbulence spectrum 42 Intensity function 445 Intensity turbulence see Turbulence intensities Interaction equations 172 Internal boundary layer 46 Internal pressures 100 390 394 Interstory drift 217 Inviscid fluids 78 Isobars 7 Isotropy local 39 j Jeteffect winds 11 Jet wall 49 Joint probability distribution 417 k KeuleganCarpenter number 379 Kinematic viscosity 76 Kolmogorov 39 l Lapse rate see Adiabatic lapse rate Large Eddy Simulation 142 wallmodeled 144 Largest values see Extreme value EV distribution joint LES see Large Eddy Simulation Liebleins method BLUE best linear unbiased estimator 63 Lift 89 287 307 308 coefficients see Coefficients aerodynamic drag and lift force Load and resistance factor design LRFD 203 262 468 475 Load factors wind see Wind load factor Location parameter 163 Lockin vortexinduced 287 Logarithmic law 27 range of validity 27 Lognormal distribution 421 Lowfrequency turbulence and flow simulation 123 Lowrise buildings 259 m Mature stage of storm 13 Mean recurrence intervals 55 214 424 design 173 208 428 468 Mean return period 55 214 424 Mean turbulent field closure 38 Mean value 419 Mean velocity profiles see Wind speed profiles Median 419 Microburst 12 115 Micrometeorology xix Missiles hurricaneborne 405 k k k k 492 Index Missiles contd tornadoborne 399 Mixed wind climates 56 Modal shapes 128 171 199 207 293 319 322 338 458 Modes normal see Normal modes Modes of vibration 129 198 344 Molecular conduction 6 Moments method of 63 425 Monin coordinate 40 Monte Carlo methods 59 66 204 211 215 402 427 449 Morison equation 378 MRI see Mean recurrence intervals Multidegreeoffreedom systems 197 255 Multihazard regions design criteria for 477 Multiplication rule probabilities 413 n National Building Code of Canada 31 Natural frequencies of vibration 38 107 198 305 376 458 Net pressures 13 80 184 192 Neutral stratification 6 17 25 Newtonian fluids 75 Nondirectional wind speeds 58 60 212 215 Nonlinear response 180 476 477 Nonparametric statistical estimates 215 428 multiple hazards 428 single hazard 428 Nonparametric statistics extreme winds and effects 61 66 172 211 215 260 Normal distribution 421 Normal modes 199 322 458 Noslip condition 138 o Occupant comfort tall buildings 226 Ocean winds over 29 Offshore structures 367 Orthogonality of normal modes 200 255 Outcrossing approach to wind directionality 439 Outer layer atmospheric 22 p Parametric estimates extreme values 214 426 Parent population 422 Parsevals equality 435 PΔPδ effects see Secondorder effects Peak gust speed 16 18 58 Peak pressures comparisons between measurements of 125 Peaks in random signals 441 Peaksoverthreshold approach estimation of extreme speeds 61 Poisson process estimation of peaks 444 Pedestrian discomfort 227 Percentage point function 422 Poisson distribution 421 Power law wind profiles 30 Power lines 297 Pressure coefficients 90 defect in hurricanes 10 distributions on bluff bodies 92 99 124 drop in tornadoes 389 dynamic see Dynamic pressure gradient force 7 19 76 80 393 internal 100 scanning systems 129 131 Probability density function 415 Probability distribution see Distribution probability Probability theory 411 Production turbulent energy 39 Profiles wind speed see Wind speed profiles q Quadrature spectrum 45 438 462 Quartering winds 370 Quasistatic response 465 r Rain 193 344 Randomness 151 411 Random processes 433 Random signals 433 Random variables 433 RANS see ReynoldsAveraged NavierStokes Simulation Rate of arrival 59 172 214 261 268 421 424 427 428 445 Reattachment flow 80 87 92 93 123 312 Reduced frequency 106 108 185 288 306 331 Reduced velocity 106 300 323 340 Reliability structural see Structural reliability Residual stress tensor 143 k k k k Index 493 Resonance 456 Resonant amplification Resonant response 38 40 195 324 465 Response background 465 fluctuating 464 in frequency domain 455 nonlinear see Nonlinear response quasistatic see Quasistatic response resonant see Resonant response surfaces 212 tall buildings preliminary estimates 267 in time domain 197 Return period 56 226 420 Reversal flow see Flow reversal Reverse Weibull distribution 423 452 ReynoldsAveraged NavierStokes Simulation unsteady RANS URANS 140 144 Reynolds number definition 80 dependence of Strouhal number on 87 effect on aerodynamics bodies with sharp edges 124 effect on drag bodies with round edges 91 93 327 violation of in the wind tunnel 123 161 Reynolds stress tensor 142 145 Ridges wind flow over 47 Rigid buildings see Lowrise buildings Rigid portal frames 259 Roof airsupported 386 Rossby number similarity 106 Roughness length 22 28 Roughness regimes wind speeds in different 31 Roughness terrain see Terrain Roughness s SaffirSimpson scale 12 conversion of to speeds above open terrain 34 Sampling errors in extreme speeds estimation 64 Scale parameter 62 422 Scanlan flutter derivatives 308 310 Scruton number 291 336 344 Secondorder effects 180 Section models bridge testing 332 Selfexcited motions 73 283 305 Semisubmersible platforms 368 Separation flow see Flow separation Serviceability requirements tall buildings 173 267 467 SGS see Subgrid scale Shape parameter see Tail length parameter Shear stress 19 22 27 75 139 Shrouds 326 Similarity requirements wind tunnel testing 105 Simulation of random processes Monte Carlo see Monte Carlo methods Simultaneous pressure measurements 129 200 Singledegreeoffreedom systems 196 252 455 Skidmore Owings and Merrill wind load factor 481 World Trade Center 481 Slender towers 315 Snow deposition 108 119 Solar heating of Earths surface 3 Spatial coherence 96 123 259 378 Spatially averaged pressure coefficients 186 Spatial smoothing 66 Spectral density function definition 435 of lateral velocity fluctuations 38 of longitudinal velocity fluctuations 44 of multidegreeoffreedom system response 458 of onedegreeoffreedom system response 457 463 onesided 437 of turbulent energy see Energy spectrum turbulent twosided 437 of vertical velocity fluctuations 44 Spectrum see Spectral density function Speedup effects 46 Splitter plates 87 Spoiler devices 325 Stable stratification 6 7 24 Stacks 99 292 315 325 Stagnation pressure 77 Standard deviation 419 Stationarity statistical 49 437 Stationary random signal 433 Statistics 411 Stiffness matrix 179 198 Stochastic 415 Storm surge 478 Straight winds 8 10 k k k k 494 Index Strake systems 325 345 Stratification flow 17 25 atmospheric 7 conventionally neutral 24 neutral see Neutral stratification stable see Stable stratification truly neutral 24 unstable see Unstable stratification Strength reserve 475 Stress tensor 75 Strouhal number 86 88 106 287 Structural dynamics 455 frequencydomain analysis 455 timedomain analysis 197 Structural engineering tasks 173 175 Structural reliability 467 Subgrid scale 142 Supercritical range flow about cylinder in 91 Superstations 66 Surface drag coefficients 29 Surface layer atmospheric 21 22 35 40 wind profile 27 Surface roughness effect on pressure 93 124 Surface shear 22 Surface wind in built environment 229 Suspendedspan bridge see Bridge decks buffeting response Sustained wind speeds 18 32 Synoptic storms 10 34 48 57 t Tacoma narrows bridge 283 311 333 342 Tail length parameter Generalized Extreme Value distribution 423 Generalized Pareto distribution 424 Type II distribution 423 Type III distribution 62 423 Tall buildings see Flexible buildings Taylors hypothesis 37 Tensile membrane structures 385 Tension leg platforms 376 Tensor stress see Stress tensor Terrain exposure ASCE Standard 31 Terrain roughness 31 Tests statistical 425 Theodorsen 305 307 308 Thermodynamics atmospheric 3 Threshold 61 65 226 423 426 443 Thunderstorms 12 49 55 58 66 159 TMD devices see Tuned mass dampers Topographic effects ASCE Standard 46 Topographic factor 47 Tornadoborne missile speeds 399 Tornadoes 13 66 68 389 399 simulators 113 115 Torsional deformation 197 divergence 303 flutter 342 occupant discomfort due to 226 response flexible buildings 197 tuned mass dampers 16 Total probability theorem of 413 Towers with circular cross section 292 315 trussed 361 Tropical cyclones 10 extreme winds 62 statistics 60 structure of 10 48 Trussed frameworks 349 Tuned mass dampers 251 Turbulence atmospheric see Atmospheric boundary layer turbulence crossspectrum see Crossspectrum effect on aerodynamics 89 92 93 122 effect on bridge stability 312 in flows with stable stratification 136 integral scales see Integral turbulence scale intensities see Turbulence intensities mechanical 24 modeled 140 142 144 146 resolved 140 142 simulation of 105 135 smallest scales 39 140 spectrum see Spectral density function Turbulence intensities 35 122 Type I extreme value distribution see Gumbel distribution Type II extreme value distribution see Fréchet distribution Type III extreme value distribution see Reverse Weibull distribution u Ultimate structural capacity 476 Uncertainties aleatory 151 epistemic 151 individual 159 k k k k Index 495 overall 159 in pressure coefficients 161 quantification 148 157 204 in wind effects estimation 159 Underlying distribution 422 Unit impulse function 456 Unstable stratification 6 Upcrossing rate mean 439 Updrafts 7 10 115 v Variance 419 Variation coefficient of see Coefficient of variation Veering angle 9 22 25 Velocity defect law 22 Velocity fluctuations coherence 45 cospectrum 45 crossspectrum 44 lateral 44 longitudinal 36 40 120 quadrature spectrum 45 spectrum 38 vertical 44 Velocity profiles see Wind speed profiles Verification and Validation VV 148 Vibrations cable see Cable vibration human response to 226 Viscosity air 75 eddy see Eddy viscosity kinematic see Kinematic viscosity units 75 water 75 Viscous effects 39 79 80 84 138 von Kármán constant 23 29 von Kármán spectrum 41 von Kármán vortex trail 85 Vortex flow 77 229 232 formations twodimensional flow 77 shedding 86 96 106 287 292 305 315 335 trail behind cylinder 85 Vortexinduced lift 287 lockin 287 oscillations 287 response alleviation of 325 bridges 335 stacks 315 towers 292 twodimensional flow 287 w Wake galloping 297 Wake in twodimensional flow 82 Wall jet 49 Wall law of 22 Wall unit 142 144 146 Wavelength 37 Wavenumber 37 140 141 Weibull distribution see Reverse Weibull distribution Windborne debris 404 Wind directionality factor 212 216 475 Winddriven rain intrusion 108 118 183 Wind effects matrix of 214 Wind engineering tasks 173 175 Wind load factor 208 design MRI see Mean recurrence intervals design Wind loads peaks 443 Wind pressures fluctuating 97 Wind speed data 57 data sets description and access to 58 directional 58 micrometeorological homogeneity of 57 nondirectional 58 Wind speed profiles 20 in different roughness regimes 31 logarithmic law 27 see Logarithmic law near a change of surface roughness 45 near change of roughness 45 neutral stratification 20 nonhorizontal terrain 46 over hills 46 over water 29 34 power law 30 see Power law wind profiles Wind speeds in different roughness regimes relation between 31 Wind speeds matrix of 213 Windstorms 10 Wind tunnel procedure ASCE Standard 122 Wind tunnel testing 105 blockage 127 dimensional analysis 105 effect of incoming turbulence 122 lowrise buildings 123 125 k k k k 496 Index Wind tunnel testing contd similarity requirements 105 suspendedspan bridges 331 tall buildings 128 types of 108 variation of results among laboratories 122 485 violation of Reynolds number in 123 151 161 World Trade Center 481 estimated response to wind 483 z Zero plane displacement 30 COLOCAR NOME DA INSTITUIÇÃO VIBRAÇÕES INDUZIDAS PELO VENTO EM ESTRUTURAS NOME DATA SUMÁRIO 1 INTRODUÇÃO3 2 A ESTRUTURA4 3 CONCEITOS APLICADOS7 3 CONCLUSÃO19 4 REFERÊNCIAS20 3 1 INTRODUÇÃO Este trabalho irá explorar os diversos conceitos verificados em sala de aula de forma prática Para tanto será realizada uma simulação de um prédio hipotético neste caso um armazém com 40m de altura onde serão utilizados dados reais dessa localização hipotética para analisar o impacto das vibrações induzidas pelo vento em sua estrutura Analisando as normas vigentes em conjunto com essas informações é possível entender um pouco melhor e de forma mais palpável os conceitos visto em sala Após a simulação será analisada a potencial aplicabilidade de uma medida mitigadora para a vibração 4 2 A ESTRUTURA A estrutura modelada para essa simulação é de um armazém do tipo autoportante de 40m de altura Esse tipo de armazém é composto por racks de aço e fechamento de painel térmico isoportante Esse tipo de estrutura é comumente utilizado devido ao seu bom aproveitamento de espaços Através da verticalização da armazenagem dos produtos a estrutura torna a movimentação e o acesso aos paletes muito mais fáceis e ágeis garantindo uma grande seletividade de produtos Além disso suporta fechamento lateral e superior de prédios dispensando um armazém de alvenaria reduzindo custos A estrutura é composta por um conjunto de montantes que são compostos por colunas e contraventados por travessas e diagonais Nestes montantes são encaixados os planos de armazenagem através de longarinas com conexões semirrígidas denominadas garras A estrutura é fixada através de chumbadores químicos e nivelada com auxílio do grout uma argamassa com alta resistência Figura 1 Representação esquemática do nivelamento com grout Fonte Autor 5 Figura 2 Autoportante sem fechamento lateral visão interna dos racks Fonte Autor Figura 3 Autoportante com fechamento lateral Fonte Autor 6 O Fechamento da estrutura é feito através de telhas e a parte estrutural da cobertura é composta por treliças fixadas a cada linha de montantes assegurando uma perfeita união entre os elementos servindo também para fixação dos perfis de suporte da telha as tesouras Essa permite a passagem de dutos de ventilação sprinklers e outras estruturas pertinentes ao projeto Figura 4 Fechamento superior e treliças Fonte Autor 7 3 CONCEITOS APLICADOS As estruturas de grande porte como o sistema de armazenagem autoportante descrito no item 2 são frequentemente sujeitas a forças dinâmicas induzidas pelo vento O comportamento dessas estruturas sob a ação do vento é um tópico complexo que envolve a interação de vários fenômenos de vibração O objetivo deste tópico é correlacionar conceitos avançados de vibrações com os cálculos e modelagens matemáticas utilizados na simulação O vento interage com a estrutura criando forças dinâmicas que podem induzir vibrações em componentes estruturais verticais e horizontais Essas vibrações são resultado de fenômenos como desprendimento de vórtices flutuação de pressão e forças de arrasto e sustentação As vibrações mecânicas induzidas pelo vento em estruturas são um fenômeno complexo que pode comprometer a integridade e a funcionalidade de edificações A análise dessas vibrações é crucial para garantir a segurança das estruturas especialmente aquelas de grande porte Utilizando o livro Wind Effects on Structures Modern Structural Design for Wind como referência este documento detalha os conceitos avançados de vibrações mecânicas e suas implicações na engenharia estrutural A pressão dinâmica do vento é um conceito fundamental na análise de vibrações mecânicas induzidas pelo vento em estruturas Este conceito é essencial para calcular as forças atuantes em uma edificação devido à ação do vento garantindo a segurança e a integridade da construção A pressão dinâmica do vento p é a pressão exercida pelo vento em movimento sobre a superfície de uma estrutura Ela é calculada com base na densidade do ar ρ e na velocidade do vento v A fórmula para calcular a pressão dinâmica do vento é p1 2 ρ v 2 Esta pressão é aplicada às superfícies da estrutura gerando forças que podem induzir vibrações Na simulação realizada a pressão dinâmica do vento foi calculada conforme a norma NBR 61231988 Esta norma fornece diretrizes para determinar os coeficientes de pressão externa e interna bem como a velocidade 8 característica do vento utilizando diagramas de isopletas e fatores topográficos estatísticos e de rugosidade Os dados para as simulações foram obtidos majoritariamente da NBR 6123 Inicialmente determinouse a velocidade característica do vento através do diagrama de isopletas fornecidos na norma NBR 61231988 Depois aplicouse a fórmula acima para calcular a pressão dinâmica Posteriormente foram determinados os coeficientes de pressão externa seguindo a mesma norma levando em consideração a direção do vento 0 90 180 e 270 e fatores como topografia e rugosidade do terreno O sentido do vento considerado na simulação segue a Figura 5 Figura 5 Sentido do vento considerado nas simulações Fonte Autor Da norma NBR61231988 foram extraídos dados de pressão externa interna velocidade característica do vento utilizando o diagrama de isopletas e fatores topográficos estatísticos e rugosidade Imagem abaixo ilustrando sentido do vento considerado em projeto Foram considerados dois coeficientes de pressão interna para o projeto em questão Cpi 02 e Cpi 03 Para o fator topográfico considerouse S1 100 9 O fator de rugosidade é do tipo categoria IV classe C por ser localizado no norte de minas assim S2 1 075 S2 2 086 S2 3 09 S2 4 094 A representação dos coeficientes de pressão externa pode ser verificada nas imagens a seguir Figura 6 Coeficientes de pressão externa a 0 Fonte Autor Figura 7 Coeficientes de pressão externa a 901 10 Fonte Autor Figura 8 Coeficientes de pressão externa a 902 11 Fonte Autor Figura 9 Coeficientes de pressão externa a 180 Fonte Autor Figura 10 Coeficientes de pressão externa a 270 12 Fonte Autor O fator estatístico S3 utilizado foi de 095 a velocidade característica do vento de acordo com o gráfico de isopletas foi de V0 35ms Por fim as cargas de vento são q1 028 kNm² q2 037 kNm² q3 041 kNm² q4 044 kNm² Outro ponto importante de verificação é a equação do movimento que é uma resposta dinâmica de uma estrutura sujeita a forças de vento é governada pela equação de movimento para um sistema massamolaamortecedor m uc uk uFt onde m é a massa da estrutura c é o coeficiente de amortecimento k é a rigidez u é o deslocamento e Ft é a força induzida pelo vento Esta equação é resolvida numericamente em softwares como o SAP2000 para determinar os deslocamentos e 13 frequências naturais da estrutura Para o armazém da simulação não houve deslocamentos fora da norma Outro ponto crítico é relativo ao desprendimento de vórtices O desprendimento de vórtices é um fenômeno aerodinâmico que ocorre quando um fluido como o ar flui ao redor de um corpo como uma estrutura ou um edifício criando vórtices alternados Este fenômeno pode induzir vibrações significativas em estruturas afetando sua estabilidade e integridade A frequência dessas forças é determinada pela frequência de Strouhal f s f s St v d onde St é o número de Strouhal e d é a largura característica da estrutura A ressonância ocorre se a frequência de Strouhal coincide com uma das frequências naturais da estrutura amplificando significativamente as vibrações Quando a frequência de Strouhal coincide com uma das frequências naturais da estrutura ocorre o fenômeno de ressonância amplificando as vibrações induzidas Isso pode levar a oscilações severas e potencialmente danosas comprometendo a integridade estrutural Este fenômeno é particularmente relevante para estruturas esbeltas e altas como torres e arranhacéus A análise de flambagem também é relevante no contexto do desprendimento de vórtices A análise de flambagem é crítica para garantir a estabilidade da estrutura sob cargas de compressão A carga crítica de flambagem elástica por torção Ncr T é dada por Ncr T 1 r0 2G I T π 2ECw LeT 2 onde r0 é o raio de giração G é o módulo de cisalhamento I T é o momento de inércia de torção E é o módulo de elasticidade Cwé o módulo de resistência à torção de guerra e LeT é o comprimento efetivo para flambagem Todos os cálculos realizados de momento fletor de flambagem e flambagem elástica e por flexão estavam dentro das normas 14 Com essas informações foi possível rodar as simulações com o auxílio do software SAP2000 O SAP2000 é um software de análise estrutural e design utilizado amplamente na engenharia civil e oferece uma variedade de ferramentas para modelagem análise e design de estruturas complexas Este software é especialmente útil em projetos que demandam análises dinâmicas e estáticas detalhadas incluindo aquelas que envolvem vibrações induzidas pelo vento Nele são lançadas todas as combinações de ações de forças do vento e também adicionado toda geometria de perfis utilizadas no projeto Abaixo algumas imagens do software mencionado Figura 11 Modelagem da estrutura no SAP 2000 1 Fonte Autor 15 Figura 12 Modelagem da estrutura no SAP 2000 2 Fonte Autor As imagens abaixo mostram alguns dos deslocamentos estruturais da estrutura autoportante obtidos nas simulações Figura 12 Deslocamento e impacto da vibração mecânica obtido na simulação 1 Fonte Autor 16 Figura 13 Deslocamento e impacto da vibração mecânica obtido na simulação 2 Fonte Autor Figura 14 Deslocamento e impacto da vibração mecânica obtido na simulação 3 Fonte Autor 17 Figura 15 Deslocamento e impacto da vibração mecânica obtido na simulação 4 Fonte Autor Todos os deslocamentos estão de acordo com o deslocamento limite imposto pela norma ABNT NBR 1552422007 e BS EN 155122009 Isso significa que as medidas adequadas de escolha de material e de controle de vibração foram aplicadas Caso fosse verificado alguma situação anormal algum efeito de vibração induzido pelo vendo técnicas como como amortecedores de massa sintonizada Tuned Mass Dampers TMDs e a modificação da rigidez estrutural são frequentemente usadas Amortecedores de massa sintonizada TMDs são dispositivos passivos que consistem em uma massa adicional conectada à estrutura através de molas e amortecedores O princípio de funcionamento dos TMDs é baseado na absorção de energia das vibrações da estrutura reduzindo a amplitude das oscilações Quando a estrutura vibra devido a forças externas como o vento a massa do TMD oscila em oposição às vibrações da estrutura Isso cria uma força de amortecimento que dissipa a energia vibracional reduzindo a resposta dinâmica da 18 estrutura A frequência natural do TMD é ajustada sintonizada para coincidir com a frequência de ressonância da estrutura maximizando a eficiência na redução das vibrações Os TMDs são amplamente utilizados em edifícios altos torres pontes e outras estruturas esbeltas Eles são particularmente eficazes na mitigação de vibrações induzidas pelo vento e por eventos sísmicos A instalação de TMDs pode ser feita durante a fase de construção ou adicionada posteriormente se forem observadas vibrações excessivas Quando o TMD não é aplicável podese aplicar oura medida de contenção a modificação da rigidez estrutural Esse método envolve alterar a distribuição de rigidez de uma estrutura para alterar suas frequências naturais e modos de vibração Isso pode ser feito adicionando ou reforçando elementos estruturais como vigas colunas contraventamentos ou utilizando materiais com diferentes propriedades de rigidez Ao modificar a rigidez da estrutura as frequências naturais são ajustadas de modo a evitar a coincidência com as frequências de excitação como aquelas induzidas pelo vento Isso reduz a possibilidade de ressonância e consequentemente as amplitudes das vibrações Esse método é aplicável em edifícios pontes e outras estruturas que experienciam vibrações indesejadas Este método é particularmente útil quando uma estrutura não pode acomodar TMDs devido às limitações de espaço ou design Comparando ambas as técnicas é possível identificar que ambas as técnicas têm suas vantagens e limitações Os TMDs são eficazes na redução de vibrações em estruturas existentes e podem ser ajustados para diferentes condições de carga No entanto eles podem ser caros e requerem manutenção regular Por outro lado a modificação da rigidez estrutural é uma solução mais permanente e pode ser integrada ao design inicial da estrutura No entanto pode ser uma intervenção mais invasiva e complexa em estruturas existentes 19 3 CONCLUSÃO Durante a execução desse estudo foi possível ter uma visão muito melhor e mais assertiva dos conceitos aprendidos em sala de aula Foi possível entender as particularidades estruturais de construções modulares de aço e foi também entender o processo de estudo do impacto do vento desde a formulação da hipótese até a coleta de dados a execução e interpretação da simulação e a contenção e potenciais medidas mitigadoras O processo foi bastante enriquecedor do ponto de vista educativo e mais ainda do ponto de vista prático pois o tema é de extrema complexidade e que pode ser evitado com um bom estudo prévio e um bom projeto estrutural 20 4 REFERÊNCIAS SIMIU E SCANLAN R H Wind Effects on Structures Modern Structural Design for Wind 4th ed New York John Wiley Sons 2021