Table Of ContentMoving Loads –
Dynamic Analysis and Identification
Techniques
StructuresandInfrastructuresSeries
ISSN1747-7735
BookSeriesEditor:
Dan M. Frangopol
ProfessorofCivilEngineeringand
FazlurR.KhanEndowedChairofStructuralEngineeringandArchitecture
DepartmentofCivilandEnvironmentalEngineering
CenterforAdvancedTechnologyforLargeStructuralSystems(ATLSSCenter)
LehighUniversity
Bethlehem,PA,USA
Volume 8
Moving Loads –
Dynamic Analysis and Identification
Techniques
1 2
Siu-Seong Law and Xin-Qun Zhu
1CivilandStructuralEngineeringDepartment,HongKongPolytechnic
University,Kowloon,HongKong
2SchoolofEngineering,UniversityofWesternSydney,Australia
Colophon
Book Series Editor:
Dan M. Frangopol
VolumeAuthors:
Siu-Seong Law and Xin-Qun Zhu
Taylor & Francis is an imprint of theTaylor & Francis Group,
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British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Law,Siu-Seong.
Moving loads :dynamic analysis and identification techniques / Siu-Seong Law,
Xin-Qun Zhu.
p. cm.—(Structures and infrastructures series,ISSN 1747-7735 ;v. 8)
ISBN 978-0-415-87877-7 (hardback)—ISBN 978-0-203-84142-6 (eBook) 1. Live
loads. 2. Structural dynamics. I. Zhu,Xin-Qun. II. Title.
TA654.3.L39 2011
624.1(cid:1)72—dc22
2010052748
Publishedby: CRCPress/Balkema
P.O.Box447,2300AKLeiden,TheNetherlands
e-mail:[email protected]
www.crcpress.com–www.taylorandfrancis.co.uk–www.balkema.nl
ISBN 978-0-415-87877-7 (Hbk)
ISBN 978-0-203-84142-6 (eBook)
Structures and Infrastructures Series:ISSN 1747-7735
Volume 8
Table of Contents
Editorial XI
AbouttheBookSeriesEditor XIII
Preface XV
Dedication XIX
Acknowledgements XXI
AbouttheAuthors XXIII
Chapter1 Introduction 1
1.1 Overview 1
1.2 BackgroundoftheMovingLoadProblem 1
1.3 ModelsfortheVehicle–BridgeSystem 3
1.3.1 ContinuousBeamunderMovingLoads 3
1.3.1.1 MovingForce,MovingMassandMoving
Oscillator 3
1.3.1.2 Multi-spanBeam 4
1.3.1.3 TimoshenkoBeam 5
1.3.1.4 BeamwithCrack 5
1.3.1.5 PrestressedBeam 6
1.3.2 ContinuousPlateunderMovingLoads 6
1.3.2.1 PlateModels 7
1.3.2.2 MovingForces 7
1.3.2.3 Quarter-truckModel 8
1.3.2.4 Half-truckModel 8
1.4 DynamicAnalysisoftheVehicle–BridgeSystem 9
1.4.1 MethodsbasedonModalSuperpositionTechnique 9
1.4.2 MethodsbasedontheFiniteElementMethod 10
1.5 TheLoadIdentificationTechniques 12
1.5.1 TheWeigh-In-MotionTechnique 12
1.5.2 TheForceIdentificationTechniques 13
1.5.3 TheMovingForceIdentificationTechniques 14
1.6 ProblemStatementontheMovingLoadIdentification 17
1.7 ModelCondensationTechniques 20
1.8 Summary 24
VI Table of Contents
Part I – Moving Load Problems
Chapter2 DynamicResponseofMulti-spanContinuousBeamsunder
MovingLoads 27
2.1 Introduction 27
2.2 Multi-spanContinuousBeam 27
2.2.1 TheExactSolution 28
2.2.1.1 FreeVibration 28
2.2.1.2 DynamicBehaviorunderMovingLoads 30
2.2.2 SolutionwithAssumedModes 31
2.2.2.1 AssumedModesforaUniformBeam 34
2.2.2.2 AssumedModesforaNon-uniformBeam 34
2.2.3 PreciseTimeStepIntegrationversusNewmark-BetaMethod 35
2.2.3.1 Newmark-BetaMethod 35
2.2.3.2 PreciseTimeStepIntegrationMethod 36
2.3 Multi-spanContinuousBeamwithElasticBearings 39
2.3.1 FreeVibration 41
2.3.2 DynamicBehaviorunderMovingLoads 43
2.4 Summary 44
Chapter3 DynamicResponseofOrthotropicPlatesunderMovingLoads 45
3.1 Introduction 45
3.2 OrthotropicPlatesunderMovingLoads 45
3.2.1 FreeVibration 45
3.2.2 DynamicBehaviorunderMovingLoads 49
3.2.3 NumericalSimulation 50
3.2.3.1 NaturalFrequencyofOrthotropicPlates 50
3.2.3.2 SimplySupportedBeam-SlabTypeBridgeDeck
underMovingLoads 50
3.3 Multi-spanContinuousOrthotropicPlateunderMovingLoads 54
3.3.1 DynamicBehaviorunderMovingLoads 55
3.3.2 ModalAnalysisofMulti-spanContinuousPlates 56
3.3.3 NumericalExamples 58
3.4 Summary 58
Chapter4 ApplicationofVehicle–BridgeInteractionDynamics 59
4.1 Introduction 59
4.2 BridgeDynamicResponse 59
4.2.1 VehicleandBridgeModels 60
4.2.2 Vehicle–BridgeInteraction 60
4.2.3 RoadSurfaceRoughness 65
4.2.4 BrakingofVehicle 65
4.2.5 ComputationalAlgorithm 66
4.2.6 NumericalSimulation 66
4.3 DynamicLoadsonContinuousMulti-LaneBridgeDecksfrom
MovingVehicles 75
4.3.1 BridgeModel 76
Table of Contents VII
4.3.2 VehicleModel 76
4.3.3 Vehicle–BridgeInteraction 78
4.4 ImpactFactors 79
4.4.1 DynamicLoadingfromaSingleVehicle 80
4.4.2 DynamicLoadingfromMultipleVehicles 85
4.5 Summary 88
Part II – Moving Load Identification Problems
Chapter5 MovingForceIdentificationinFrequency–TimeDomain 91
5.1 Introduction 91
5.2 MovingForceIdentificationinFrequency–TimeDomain 91
5.2.1 EquationofMotion 91
5.2.2 IdentificationfromAccelerations 93
5.2.3 SolutioninTimeDomain 95
5.2.4 IdentificationfromBendingMomentsandAccelerations 96
5.2.5 RegularizationoftheSolution 97
5.3 NumericalExamples 99
5.3.1 SingleForceIdentification 100
5.3.2 TwoForcesIdentification 101
5.4 LaboratoryExperimentswithTwoMovingLoads 106
5.4.1 ExperimentalSetup 106
5.4.2 ExperimentalProcedure 108
5.4.3 ExperimentalResults 109
5.5 Summary 112
Chapter6 MovingForceIdentificationinTimeDomain 113
6.1 Introduction 113
6.2 MovingForceIdentification–TheTimeDomainMethod(TDM) 114
6.2.1 Theory 114
6.2.1.1 EquationofMotionandModalSuperposition 114
6.2.1.2 ForceIdentificationfromBendingMoments 115
6.2.1.3 IdentificationfromAccelerations 117
6.2.1.4 IdentificationfromBendingMomentsandAccelerations 119
6.2.2 SimulationStudies 120
6.2.3 ExperimentalStudies 122
6.2.4 Discussions 125
6.3 MovingForceIdentification–ExactSolutionTechnique(EST) 125
6.3.1 BeamModel 125
6.3.1.1 IdentificationfromStrains 126
6.3.1.2 IdentificationfromAccelerations 128
6.3.1.3 StatementoftheProblem 129
6.3.2 PlateModel 129
6.3.2.1 IdentificationfromStrains 131
6.3.2.2 IdentificationfromAccelerations 133
6.3.2.3 ComputationAlgorithm 134
VIII Table of Contents
6.3.3 NumericalExamples 135
6.3.3.1 BeamModel 135
6.3.3.2 Two-dimensionalPlateModel 144
6.3.4 LaboratoryStudies 153
6.3.4.1 BeamModel 153
6.3.4.2 PlateModel 159
6.4 Summary 171
Chapter7 MovingForceIdentificationinStateSpace 173
7.1 Introduction 173
7.2 MethodI–SolutionbasedonDynamicProgramming 173
7.2.1 State–SpaceModel 173
7.2.2 FormulationofMatrixGforTwoMovingLoadsIdentification 175
7.2.3 ProblemStatement 175
7.2.4 ComputationAlgorithm 176
7.2.5 NumericalExamples 178
7.2.5.1 Single-ForceIdentification 179
7.2.5.2 Two-ForcesIdentification 180
7.2.6 ExperimentandResults 183
7.2.6.1 Single-ForceIdentification 184
7.2.6.2 Two-ForcesIdentification 186
7.2.7 DiscussionsonthePerformanceofMethodI 187
7.3 MethodII–SolutionbasedonRegularizationAlgorithm 188
7.3.1 DiscreteTimeState–SpaceModel 188
7.3.2 MovingLoadIdentification 192
7.3.3 NumericalStudies 193
7.3.3.1 ValidationofMethodII 195
7.3.3.2 StudyontheEffectsofSensorTypeandLocation 196
7.3.3.3 FurtherStudiesontheSensorLocationEffectand
VelocityMeasurement 198
7.3.3.4 EffectoftheAspectRatiooftheBridgeDeck 201
7.3.3.5 FurtherStudiesontheEffectofNoisein
DifferentTypesofMeasurements 202
7.3.4 ExperimentalStudies 204
7.3.4.1 ExperimentalSet-up 204
7.3.4.2 AxleLoadsandWheelLoadsIdentification 204
7.3.5 ComparisonoftheTwoState–SpaceApproaches 206
7.4 Summary 208
Chapter8 MovingForceIdentificationwithGeneralized
OrthogonalFunctionExpansion 209
8.1 Introduction 209
8.2 OrthogonalFunctions 209
8.2.1 SeriesExpansion 209
8.2.2 GeneralizedOrthogonalFunction 210
8.2.3 WaveletDeconvolution 211
8.3 MovingForceIdentification 213
Table of Contents IX
8.3.1 BeamModel 213
8.3.1.1 GeneralizedOrthogonalFunctionExpansion 213
8.3.1.2 MovingForceIdentificationTheory 214
8.3.2 PlateModel 215
8.4 Applications 217
8.4.1 IdentificationwithaBeamModel 217
8.4.1.1 Single-SpanBeam 217
8.4.1.2 Two-SpanContinuousBeam 219
8.4.2 IdentificationwithaPlateModel 221
8.4.2.1 StudyontheNoiseEffect 222
8.4.2.2 IdentificationwithIncompleteModalInformation 223
8.4.2.3 EffectsofTravelPathEccentricity 224
8.5 LaboratoryStudies 226
8.5.1 BeamModel 226
8.5.1.1 ExperimentalSetupandMeasurements 226
8.5.1.2 ForceIdentification 226
8.5.2 PlateModel 227
8.5.2.1 ExperimentalSet-up 227
8.5.2.2 WheelLoadIdentification 228
8.5.2.3 EffectofUnequalNumberofModesinthe
ResponseandintheIdentification 230
8.6 Summary 233
Chapter9 MovingForceIdentificationbasedonFiniteElementFormulation 235
9.1 Introduction 235
9.2 MovingForceIdentification 235
9.2.1 InterpretiveMethodI 235
9.2.1.1 PredictiveAnalysis 235
9.2.1.2 InterpretiveAnalysis 236
9.2.2 InterpretiveMethodII 236
9.2.3 RegularizationMethod 239
9.2.3.1 EquationofMotion 239
9.2.3.2 VehicleAxleLoadIdentificationfromStrain
Measurements 241
9.2.3.3 RegularizationAlgorithm 243
9.3 NumericalExamples 244
9.3.1 EffectofDiscretizationoftheStructureandSamplingRate 244
9.3.2 EffectofNumberofSensorsandNoiseLevel 245
9.4 LaboratoryVerification 247
9.4.1 ExperimentalSet-up 247
9.4.2 IdentificationfromMeasuredStrains 247
9.5 ComparativeStudies 249
9.5.1 EffectofNoiseLevel 249
9.5.2 EffectofModalTruncation 251
9.5.3 EffectofNumberofMeasuringPoints 253
9.5.4 EffectofSamplingFrequency 253
9.6 Summary 255
X Table of Contents
Chapter10 ApplicationofVehicle–BridgeInteractionForceIdentification 257
10.1 MeritsandDisadvantagesofDifferentMovingForceIdentification
Techniques 257
10.2 PracticalIssuesontheVehicle–BridgeInteractionForceIdentification 258
10.2.1 BridgeWeigh-In-Motion 258
10.2.2 MovingForceIdentificationTechniques 259
10.2.2.1 AccesstoAvailableData 260
10.2.2.2 AccuracyofAvailableData 261
10.3 FurtherComparisonoftheFEMFormulationandtheESTMethod
intheVehicle–BridgeInteractionIdentification 262
10.3.1 EffectofRoadSurfaceRoughnessandMovingSpeed 263
10.3.2 IdentificationofMovingLoadsonaBridgeDeckwith
VaryingSpeeds 264
10.3.3 IdentificationwithIncompleteVehicleSpeedInformation 267
10.4 DynamicAxleandWheelLoadIdentification 268
10.4.1 DynamicAxleLoadIdentification 268
10.4.1.1 Study1:EffectofNumberofModes 268
10.4.1.2 Study2:EffectofMeasuringLocations 270
10.4.1.3 Study3:EffectofLoadEccentricities 273
10.4.2 WheelLoadIdentification 275
10.4.2.1 Study4:EffectofMeasuringLocations 275
10.4.2.2 Study5:EffectofLoadEccentricities 278
10.4.2.3 Study6:EffectofNumberofModes 280
10.5 ModificationsandSpecialTopicsontheMovingLoadIdentification
Techniques 281
10.5.1 FirstOrderHoldDiscreteversusZerothOrderHoldDiscrete 282
10.5.1.1 Zeroth-OrderHoldDiscreteMethodin
ResponseAnalysis 282
10.5.1.2 TriangleFirst-OrderHoldDiscreteMethod 284
10.5.2 FirstOrderRegularizationversusZerothOrderRegularization 286
10.5.2.1 TikhonovRegularization 286
10.5.2.2 First-OrderTikhonovRegularization 287
10.6 Summary 288
Chapter11 ConcludingRemarksandFutureDirections 289
11.1 StateoftheArt 289
11.2 FutureDirections 290
11.2.1 EffectofUncertaintiesonMovingForceIdentification 290
11.2.2 MovingForceIdentificationwithComplexStructures 291
11.2.3 IntegratedBridgeWeigh-In-MotionwithStructuralHealth
Monitoring 291
References 293
SubjectIndex 305
StructuresandInfrastructuresSeries 307
