Moving Loads – Dynamic Analysis and Identification Techniques: Structures and Infrastructures Book Series, Vol. 8, 1st Edition (Paperback) book cover

Moving Loads – Dynamic Analysis and Identification Techniques

Structures and Infrastructures Book Series, Vol. 8, 1st Edition

By Siu-Seong Law, Xin-Qun Zhu

CRC Press

332 pages

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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.

Table of Contents

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 Moving Force, Moving Mass and Moving Oscillator Multi-span Beam Timoshenko Beam Beam with Crack Prestressed Beam

1.3.2 Continuous Plate under Moving Loads Plate Models Moving Forces Quarter-truck Model 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 Loads

2.1 Introduction

2.2 Multi-span Continuous Beam

2.2.1 The Exact Solution Free Vibration Dynamic Behavior under Moving Loads

2.2.2 Solution with Assumed Modes Assumed Modes for a Uniform Beam Assumed Modes for a Non-uniform Beam

2.2.3 Precise Time Step Integration versus Newmark-Beta Method Newmark-Beta Method 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 Natural Frequency of Orthotropic Plates 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

4.2.3 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 Equation of Motion and Modal Superposition Force Identification from Bending Moments Identification from Accelerations 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 Identification from Strains Identification from Accelerations Statement of the Problem

6.3.2 Plate Model Identification from Strains Identification from Accelerations Computation Algorithm

6.3.3 Numerical Examples Beam Model Two-dimensional Plate Model

6.3.4 Laboratory Studies Beam Model 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 Single-Force Identification Two-Forces Identification

7.2.6 Experiment and Results Single-Force Identification 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 Validation of Method II Study on the Effects of Sensor Type and Location Further Studies on the Sensor Location Effect and Velocity Measurement Effect of the Aspect Ratio of the Bridge Deck Further Studies on the Effect of Noise in Different Types of Measurements

7.3.4 Experimental Studies Experimental Set-up Axle Loads and Wheel Loads Identification

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 Generalized Orthogonal Function Expansion Moving Force Identification Theory

8.3.2 Plate Model

8.4 Applications

8.4.1 Identification with a Beam Model Single-Span Beam Two-Span Continuous Beam

8.4.2 Identification with a Plate Model Study on the Noise Effect Identification with Incomplete Modal Information Effects of Travel Path Eccentricity

8.5 Laboratory Studies

8.5.1 Beam Model Experimental Setup and Measurements Force Identification

8.5.2 Plate Model Experimental Set-up Wheel Load Identification 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 Predictive Analysis Interpretive Analysis

9.2.2 Interpretive Method II

9.2.3 Regularization Method Equation of Motion Vehicle Axle Load Identification from Strain Measurements 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 Access to Available Data 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 Dynamic Axle Load Identification Study 1: Effect of Number of Modes Study 2: Effect of Measuring Locations Study 3: Effect of Load Eccentricities

10.4.2 Wheel Load Identification Study 4: Effect of Measuring Locations Study 5: Effect of Load Eccentricities 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 Zeroth-Order Hold Discrete Method in Response Analysis Triangle First-Order Hold Discrete Method

10.5.2 First Order Regularization versus Zeroth Order Regularization Tikhonov Regularization 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

About the Authors/Editor

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.

About the Series

Structures and Infrastructures

Book Series Editor: Prof. Dan M. Frangopol, Lehigh University, PA, USA

Our knowledge to model, analyze, design, maintain, manage and predict the life-cycle performance of structures and infrastructures is continually growing. However, the complexity of these systems continues to increase and an integrated approach is necessary to understand the effect of technological, environmental, economical, social and political interactions on the life-cycle performance of engineering structures and infrastructures. In order to accomplish this, methods have to be developed to systematically analyze structure and infrastructure systems, and models have to be formulated for evaluating and comparing the risks and benefits associated with various alternatives. We must maximize the life-cycle benefits of these systems to serve the needs of our society by selecting the best balance of the safety, economy and sustainability requirements despite imperfect information and knowledge.

In recognition of the need for such methods and models, the aim of this book series is to present research, developments, and applications written by experts on the most advanced technologies for analyzing, predicting and optimizing the performance of structures and infrastructures such as buildings, bridges, dams, underground construction, offshore platforms, pipelines, naval vessels, ocean structures, nuclear power plants, and also airplanes, aerospace and automotive structures.

The scope of this book series covers the entire spectrum of structures and infrastructures. Thus it includes, but is not restricted to, mathematical modeling, computer and experimental methods, practical applications in the areas of assessment and evaluation, construction and design for durability, decision making, deterioration modeling and aging, failure analysis, field testing, structural health monitoring, financial planning, inspection and diagnostics, life-cycle analysis and prediction, loads, maintenance strategies, management systems, nondestructive testing, optimization of maintenance and management, specifications and codes, structural safety and reliability, system analysis, time-dependent performance, rehabilitation, repair, replacement, reliability and risk management, service life prediction, strengthening and whole life costing.

This book series is intended researchers, practitioners, and students world-wide with a background in civil, aerospace, mechanical, marine and automotive engineering, as well as people working in infrastructure maintenance, monitoring, management and cost analysis of structures and infrastructures. Some volumes are monographs defining the current state of the art and/or practice in the field, and some are textbooks to be used in undergraduate (mostly seniors), graduate and postgraduate courses. This book series is affiliated to Structure and Infrastructure Engineering (Taylor & Francis, ), an international peer-reviewed journal which is included in the Science Citation Index.
If you like to contribute to this series as an author or editor, please contact the Series Editor ( or the Publisher ( A book proposal form can be downloaded at

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Subject Categories

BISAC Subject Codes/Headings:
TECHNOLOGY & ENGINEERING / Construction / General