1st Edition

# Moving Loads – Dynamic Analysis and Identification Techniques Structures and Infrastructures Book Series, Vol. 8

By Siu-Seong Law, Xin-Qun Zhu Copyright 2011
332 Pages
by CRC Press

332 Pages
by CRC Press

332 Pages
by CRC Press

Also available as eBook on:

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.

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
1.5.1 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 Loads
2.1 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 Response
4.2.1 Vehicle and Bridge Models
4.2.2 Vehicle–Bridge Interaction
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
4.5 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 125
6.3.1.1 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 Model
6.3.3.2 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
7.3.3 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-up
7.3.5 Comparison of the Two State–Space Approaches
7.4 Summary

Chapter 8 Moving Force Identification with Generalized Orthogonal Function Expansion
8.1 Introduction
8.2 Orthogonal Functions
8.2.1 Series Expansion
8.2.2 Generalized Orthogonal Function
8.2.3 Wavelet Deconvolution
8.3 Moving Force Identification
8.3.1 Beam Model
8.3.1.1 Generalized Orthogonal Function Expansion
8.3.1.2 Moving Force Identification Theory
8.3.2 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 Model
8.4.2.1 Study on the Noise Effect
8.4.2.2 Identification with Incomplete Modal Information
8.4.2.3 Effects of Travel Path Eccentricity
8.5 Laboratory Studies
8.5.1 Beam Model
8.5.1.1 Experimental Setup and Measurements
8.5.1.2 Force Identification
8.5.2 Plate Model
8.5.2.1 Experimental Set-up
8.5.2.3 Effect of Unequal Number of Modes in the Response and in the Identification
8.6 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 Analysis
9.2.2 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 Rate
9.3.2 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 Points
9.5.4 Effect of Sampling Frequency
9.6 Summary

Chapter 10 Application of Vehicle–Bridge Interaction Force Identification
10.1 Merits and Disadvantages of Different Moving Force Identification Techniques
10.2 Practical Issues on the Vehicle–Bridge Interaction Force Identification
10.2.1 Bridge Weigh-In-Motion
10.2.2 Moving Force Identification Techniques
10.2.2.2 Accuracy of Available Data
10.3 Further Comparison of the FEM Formulation and the EST Method in the Vehicle–Bridge Interaction Identification
10.3.1 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 Information
10.4 Dynamic Axle and Wheel Load Identification
10.4.1.1 Study 1: Effect of Number of Modes
10.4.1.2 Study 2: Effect of Measuring Locations
10.4.1.3 Study 3: Effect of Load Eccentricities
10.4.2.1 Study 4: Effect of Measuring Locations
10.4.2.2 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 Analysis
10.5.1.2 Triangle First-Order Hold Discrete Method
10.5.2 First Order Regularization versus Zeroth Order Regularization
10.5.2.1 Tikhonov Regularization
10.5.2.2 First-Order Tikhonov Regularization
10.6 Summary

Chapter 11 Concluding Remarks and Future Directions
11.1 State of the Art
11.2 Future Directions
11.2.1 Effect of Uncertainties on Moving Force Identification
11.2.2 Moving Force Identification with Complex Structures
11.2.3 Integrated Bridge Weigh-In-Motion with Structural Health Monitoring References

Subject Index

### Biography

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.