Dynamic Response Analysis and Moving Load Identification Techniques

The interaction phenomenon is very common between different components of a mechanical system. It is a natural phenomenon and is found with the impact force in aircraft landing; the estimation of degree of ripeness of an apple from impact on a beam; the interaction of the magnetic head of a computer disk leading to miniature development of modern computer; etc. Uncertainty in some of them would lead to inaccurate analysis results on the behavior of the structure. The interaction force is difficult to measure unless instruments have been installed during construction for this purpose. Some of the interaction problems are difficult to quantify due to the lack of thorough knowledge on the interaction behavior. Analytical skills are required to estimate the interaction forces of the mechanical system in order to enable advanced developments in different areas of modern technology.
This volume provides a comprehensive treatment on this topic with the vehicle-bridge system for an illustration of the moving load problem. It covers a whole range of topics, including mathematical concepts of the moving load problems with continuous beams and plates, vehicle-bridge interaction dynamics, weigh-in-motion techniques, moving load identification algorithms in the frequency-time domain, in the time domain and in the state space domain, techniques based on the generalized orthogonal function expansion and on the finite element formulation. The methods and algorithms can be implemented for on-line identification of the interaction forces.
This book is intended for structural engineers and advanced students who wish to explore the benefit of interaction phenomenon and techniques for identification of such interaction forces. It is also recommended for researchers and decision makers working on the operation and maintenance of major infrastructures and engineering facilities.

Siu-Seong Law is is an Associate Professor with the Civil and Structural Engineering Department of the Hong Kong Polytechnic University, prior to which he spent several years in the civil engineering industry with especial experience with long-span bridges.

Chapter 1 Introduction 1.1 Overview 1.2 Background of the Moving Load Problem 1.3 Models for the Vehicle–Bridge System 1.3.1 Continuous Beam under Moving Loads 1.3.1.1 Moving Force, Moving Mass and Moving Oscillator 1.3.1.2 Multi-span Beam 1.3.1.3 Timoshenko Beam 1.3.1.4 Beam with Crack 1.3.1.5 Prestressed Beam 1.3.2 Continuous Plate under Moving Loads 1.3.2.1 Plate Models 1.3.2.2 Moving Forces 1.3.2.3 Quarter-truck Model 1.3.2.4 Half-truck Model 1.4 Dynamic Analysis of the Vehicle–Bridge System 1.4.1 Methods based on Modal Superposition Technique 1.4.2 Methods based on the Finite Element Method 1.5 The Load Identification Techniques 1.5.1 The Weigh-In-Motion Technique 1.5.2 The Force Identification Techniques 1.5.3 The Moving Force Identification Techniques 1.6 Problem Statement on the Moving Load Identification 1.7 Model Condensation Techniques 1.8 Summary
Part I – Moving Load Problems
Chapter 2 Dynamic Response of Multi-span Continuous Beams under Moving Loads2.1 Introduction 2.2 Multi-span Continuous Beam 2.2.1 The Exact Solution 2.2.1.1 Free Vibration 2.2.1.2 Dynamic Behavior under Moving Loads 2.2.2 Solution with Assumed Modes 2.2.2.1 Assumed Modes for a Uniform Beam 2.2.2.2 Assumed Modes for a Non-uniform Beam 2.2.3 Precise Time Step Integration versus Newmark-Beta Method 2.2.3.1 Newmark-Beta Method 2.2.3.2 Precise Time Step Integration Method 2.3 Multi-span Continuous Beam with Elastic Bearings 2.3.1 Free Vibration 2.3.2 Dynamic Behavior under Moving Loads 2.4 Summary
Chapter 3 Dynamic Response of Orthotropic Plates under Moving Loads 3.1 Introduction 3.2 Orthotropic Plates under Moving Loads 3.2.1 Free Vibration 3.2.2 Dynamic Behavior under Moving Loads 3.2.3 Numerical Simulation 3.2.3.1 Natural Frequency of Orthotropic Plates 3.2.3.2 Simply Supported Beam-Slab Type Bridge Deck under Moving Loads 3.3 Multi-span Continuous Orthotropic Plate under Moving Loads 3.3.1 Dynamic Behavior under Moving Loads 3.3.2 Modal Analysis of Multi-span Continuous Plates 3.3.3 Numerical Examples 3.4 Summary
Chapter 4 Application of Vehicle–Bridge Interaction Dynamics 4.1 Introduction 4.2 Bridge Dynamic Response4.2.1 Vehicle and Bridge Models 4.2.2 Vehicle–Bridge Interaction 4.2.3 Road Surface Roughness 4.2.4 Braking of Vehicle 4.2.5 Computational Algorithm 4.2.6 Numerical Simulation 4.3 Dynamic Loads on Continuous Multi-Lane Bridge Decks from Moving Vehicles 4.3.1 Bridge Model 4.3.2 Vehicle Model 4.3.3 Vehicle–Bridge Interaction 4.4 Impact Factors 4.4.1 Dynamic Loading from a Single Vehicle 4.4.2 Dynamic Loading from Multiple Vehicles 4.5 Summary
Part II – Moving Load Identification Problems
Chapter 5 Moving Force Identification in Frequency–Time Domain 5.1 Introduction 5.2 Moving Force Identification in Frequency–Time Domain 5.2.1 Equation of Motion 5.2.2 Identification from Accelerations 5.2.3 Solution in Time Domain 5.2.4 Identification from Bending Moments and Accelerations 5.2.5 Regularization of the Solution 5.3 Numerical Examples 5.3.1 Single Force Identification 5.3.2 Two Forces Identification 5.4 Laboratory Experiments with Two Moving Loads 5.4.1 Experimental Setup 5.4.2 Experimental Procedure 5.4.3 Experimental Results 5.5 Summary
Chapter 6 Moving Force Identification in Time Domain 6.1 Introduction 6.2 Moving Force Identification – The Time Domain Method (TDM) 6.2.1 Theory 6.2.1.1 Equation of Motion and Modal Superposition 6.2.1.2 Force Identification from Bending Moments 6.2.1.3 Identification from Accelerations 6.2.1.4 Identification from Bending Moments and Accelerations 6.2.2 Simulation Studies 6.2.3 Experimental Studies 6.2.4 Discussions 6.3 Moving Force Identification – Exact Solution Technique (EST) 6.3.1 Beam Model 1256.3.1.1 Identification from Strains 6.3.1.2 Identification from Accelerations 6.3.1.3 Statement of the Problem 6.3.2 Plate Model 6.3.2.1 Identification from Strains 6.3.2.2 Identification from Accelerations 6.3.2.3 Computation Algorithm 6.3.3 Numerical Examples 6.3.3.1 Beam Model6.3.3.2 Two-dimensional Plate Model 6.3.4 Laboratory Studies 6.3.4.1 Beam Model 6.3.4.2 Plate Model 6.4 Summary
Chapter 7 Moving Force Identification in State Space 7.1 Introduction 7.2 Method I – Solution based on Dynamic Programming 7.2.1 State–Space Model 7.2.2 Formulation of Matrix G for Two Moving Loads Identification 7.2.3 Problem Statement 7.2.4 Computation Algorithm 7.2.5 Numerical Examples 7.2.5.1 Single-Force Identification 7.2.5.2 Two-Forces Identification 7.2.6 Experiment and Results 7.2.6.1 Single-Force Identification 7.2.6.2 Two-Forces Identification 7.2.7 Discussions on the Performance of Method I 7.3 Method II – Solution based on Regularization Algorithm 7.3.1 Discrete Time State–Space Model 7.3.2 Moving Load Identification 7.3.3 Numerical Studies 7.3.3.1 Validation of Method II 7.3.3.2 Study on the Effects of Sensor Type and Location 7.3.3.3 Further Studies on the Sensor Location Effect and Velocity Measurement 7.3.3.4 Effect of the Aspect Ratio of the Bridge Deck 7.3.3.5 Further Studies on the Effect of Noise in Different Types of Measurements 7.3.4 Experimental Studies 7.3.4.1 Experimental Set-up7.3.4.2 Axle Loads and Wheel Loads Identification7.3.5 Comparison of the Two State–Space Approaches 7.4 Summary
Chapter 8 Moving Force Identification with Generalized Orthogonal Function Expansion8.1 Introduction 8.2 Orthogonal Functions8.2.1 Series Expansion 8.2.2 Generalized Orthogonal Function 8.2.3 Wavelet Deconvolution 8.3 Moving Force Identification8.3.1 Beam Model 8.3.1.1 Generalized Orthogonal Function Expansion 8.3.1.2 Moving Force Identification Theory8.3.2 Plate Model 8.4 Applications 8.4.1 Identification with a Beam Model 8.4.1.1 Single-Span Beam 8.4.1.2 Two-Span Continuous Beam 8.4.2 Identification with a Plate Model8.4.2.1 Study on the Noise Effect 8.4.2.2 Identification with Incomplete Modal Information8.4.2.3 Effects of Travel Path Eccentricity 8.5 Laboratory Studies 8.5.1 Beam Model8.5.1.1 Experimental Setup and Measurements 8.5.1.2 Force Identification 8.5.2 Plate Model 8.5.2.1 Experimental Set-up 8.5.2.2 Wheel Load Identification 8.5.2.3 Effect of Unequal Number of Modes in the Response and in the Identification8.6 Summary
Chapter 9 Moving Force Identification based on Finite Element Formulation 9.1 Introduction 9.2 Moving Force Identification 9.2.1 Interpretive Method I 9.2.1.1 Predictive Analysis 9.2.1.2 Interpretive Analysis9.2.2 Interpretive Method II 9.2.3 Regularization Method 9.2.3.1 Equation of Motion 9.2.3.2 Vehicle Axle Load Identification from Strain Measurements 9.2.3.3 Regularization Algorithm 9.3 Numerical Examples 9.3.1 Effect of Discretization of the Structure and Sampling Rate9.3.2 Effect of Number of Sensors and Noise Level 9.4 Laboratory Verification 9.4.1 Experimental Set-up 9.4.2 Identification from Measured Strains 9.5 Comparative Studies 9.5.1 Effect of Noise Level 9.5.2 Effect of Modal Truncation 9.5.3 Effect of Number of Measuring Points9.5.4 Effect of Sampling Frequency 9.6 Summary
Chapter 10 Application of Vehicle–Bridge Interaction Force Identification10.1 Merits and Disadvantages of Different Moving Force Identification Techniques10.2 Practical Issues on the Vehicle–Bridge Interaction Force Identification10.2.1 Bridge Weigh-In-Motion 10.2.2 Moving Force Identification Techniques10.2.2.1 Access to Available Data10.2.2.2 Accuracy of Available Data10.3 Further Comparison of the FEM Formulation and the EST Method in the Vehicle–Bridge Interaction Identification10.3.1 Effect of Road Surface Roughness and Moving Speed 10.3.2 Identification of Moving Loads on a Bridge Deck with Varying Speeds 10.3.3 Identification with Incomplete Vehicle Speed Information10.4 Dynamic Axle and Wheel Load Identification 10.4.1 Dynamic Axle Load Identification 10.4.1.1 Study 1: Effect of Number of Modes10.4.1.2 Study 2: Effect of Measuring Locations 10.4.1.3 Study 3: Effect of Load Eccentricities 10.4.2 Wheel Load Identification 10.4.2.1 Study 4: Effect of Measuring Locations10.4.2.2 Study 5: Effect of Load Eccentricities 10.4.2.3 Study 6: Effect of Number of Modes 10.5 Modifications and Special Topics on the Moving Load Identification Techniques 10.5.1 First Order Hold Discrete versus Zeroth Order Hold Discrete 10.5.1.1 Zeroth-Order Hold Discrete Method in Response Analysis10.5.1.2 Triangle First-Order Hold Discrete Method 10.5.2 First Order Regularization versus Zeroth Order Regularization10.5.2.1 Tikhonov Regularization 10.5.2.2 First-Order Tikhonov Regularization10.6 Summary
Chapter 11 Concluding Remarks and Future Directions11.1 State of the Art 11.2 Future Directions 11.2.1 Effect of Uncertainties on Moving Force Identification11.2.2 Moving Force Identification with Complex Structures 11.2.3 Integrated Bridge Weigh-In-Motion with Structural Health Monitoring ReferencesSubject Index