Contents -- Preface -- Acknowledgments -- List of Symbols -- 1. Introduction -- 1.1 Major Considerations -- 1.2 Vehicle Models -- 1.3 Bridge Models -- 1.4 Railway Bridges and Vehicles -- 1.5 Methods of Solution -- 1.6 Impact Factor and Speed Parameter -- 1.7 Concluding Remarks -- Part I Moving Load Problems -- 2. Impact Response of Simply-Supported Beams -- 2.1 Introduction -- 2.2 Simple Beam Subjected to a Single Moving Load -- 2.3 Impact Factor for Midpoint Displacement -- 2.4 Impact Factor for Midpoint Bending Moment -- 2.5 Impact Factor for End Shear Force -- 2.6 Simple Beam Subjected to a Series of Moving Loads -- 2.6.1 Modeling of Wheel Loads of a Train -- 2.6.2 Method of Solution -- 2.6.3 Phenomenon of Resonance -- 2.6.4 Phenomenon of Cancellation -- 2.6.5 Optimal Design Criteria -- 2.7 Illustrative Examples -- 2.7.1 Comparison with Finite Element Solutions -- 2.7.2 Effects of Moving Masses and Damping -- 2.7.3 Effect of Span to Car Length Ratio -- 2.8 Concluding Remarks -- 3. Impact Response of Railway Bridges with Elastic Bearings -- 3.1 Introduction -- 3.2 Equation of Motion -- 3.3 Fundamental Frequency of the Beam -- 3.4 Dynamic Response Analysis -- 3.5 Phenomena of Resonance and Cancellation -- 3.6 Effect of Structural Damping -- 3.7 Envelope Formula for Resonance Response -- 3.8 Impact Factor and Envelope Impact Formulas -- 3.9 Numerical Examples -- 3.9.1 Phenomenon of Resonance -- 3.9.2 Effect of Structural Damping -- 3.9.3 Envelope Impact Formula -- 3.10 Concluding Remarks -- 4. Mechanism of Resonance and Cancellation for Elastically-Supported Beams -- 4.1 Introduction -- 4.2 Formulation of the Theory -- 4.2.1 Assumed Modal Shape of Vibration -- 4.2.2 Single Moving Load -- 4.2.3 A Series of Moving Loads -- 4.3 Conditions of Resonance and Cancellation -- 4.4 Mechanism of Resonance and Cancellation.

4.5 Field Measurement of Vibration of Railway Bridges -- 4.6 Concluding Remarks -- 5. Curved Beams Subjected to Vertical and Horizontal Moving Loads -- 5.1 Introduction -- 5.2 Governing Differential Equations -- 5.3 Curved Beam Subjected to a Single Moving Load -- 5.3.1 Vertical Moving Load -- 5.3.2 Horizontal Moving Load -- 5.4 Unified Expressions for Vertical and Radial Vibrations -- 5.5 Solutions for Multi Moving Loads -- 5.6 Conditions of Resonance and Cancellation -- 5.7 Numerical Examples -- 5.7.1 Comparison of Analytic with Finite Element Solutions -- 5.7.2 Phenomenon of Cancellation Under Single or Multi Moving Masses -- 5.7.3 Phenomenon of Resonance Under Multi Moving Masses -- 5.7.4 I-S Plot - Impact Effect Caused by Moving Loads -- 5.8 Concluding Remarks -- Part II Interaction Dynamics Problems -- 6. Vehicle-Bridge Interaction Element Based on Dynamic Condensation -- 6.1 Introduction -- 6.2 Equations of Motion for the Vehicle and Bridge -- 6.3 Element Equations in Incremental Form -- 6.4 Equivalent Stiffness Equation for Vehicles -- 6.5 Vehicle-Bridge Interaction Element -- 6.6 Incremental Dynamic Analysis with Iterations -- 6.6.1 Equivalent Stiffness Equations for VBI System -- 6.6.2 Procedure of Iterations -- 6.7 Numerical Verification -- 6.7.1 Simple Beam Subjected to Moving Sprung Mass -- 6.7.2 Simple Beam Subjected to Moving Train -- 6.7.3 Free-Fixed Beam with Various Models for Moving Vehicles -- 6.8 Parametric Studies -- 6.8.1 Models for Bridge, Train and Rail Irregularities -- 6.8.2 Moving Load versus Sprung Mass Model -- 6.8.3 Effect of Rail Irregularities -- 6.8.4 Effect of Ballast Stiffness -- 6.8.5 Effect of Vehicle Suspension Stiffness -- 6.8.6 Effect of Vehicle Suspension Damping -- 6.9 Concluding Remarks -- 7. Vehicle-Bridge Interaction Element Considering Pitching Effect -- 7.1 Introduction.

7.2 Equations of Motion for the Vehicle and Bridge -- 7.3 Rigid Vehicle-Bridge Interaction Element -- 7.4 Equations of Motion for the VBI System -- 7.5 Numerical Studies -- 7.5.1 Simple Beam Traveled by a Two-Axle System -- 7.5.2 Simple Beam Traveled by a Train Consisting of Five Identical Cars -- 7.5.3 Riding Comfort in the Presence of Track Irregularities -- 7.5.4 Effect of Elasticity of the Suspension System -- 7.5.5 Effect of Damping of the Suspension System -- 7.5.6 Effect of Track Irregularity -- 7.6 Concluding Remarks -- 8. Modeling of Vehicle-Bridge Interactions by the Concept of Contact Forces -- 8.1 Introduction -- 8.2 Vehicle Equations and Contact Forces -- 8.3 Solution of Contact Forces from Vehicle Equations -- 8.4 VBI Element Considering Vertical Contact Forces Only -- 8.5 VBI Element Considering General Contact Forces -- 8.6 System Equations and Structural Damping -- 8.7 Procedure of Time-History Analysis for VBI Systems -- 8.8 Numerical Examples and Verification -- 8.8.1 Cantilever Beam Subjected to a Moving Load -- 8.8.2 Cantilever Beam Subjected to a Moving Mass -- 8.8.3 Simple Beam Subjected to a Moving Sprung Mass -- 8.8.4 Simple Beam Subjected to a Moving Rigid Bar Supported by Spring-Dashpot Units -- 8.8.5 Bridge Subjected to a Vehicle in Deceleration -- 8.8.6 Bridges Subjected to a Train Consisting of 10 Identical Cars -- 8.9 Concluding Remarks -- 9. Vehicle-Rails-Bridge Interaction - Two-Dimensional Modeling -- 9.1 Introduction -- 9.2 Train and Bridge Models and Minimal Bridge Segment -- 9.3 Vehicle's Equations of Motion and Contact Forces -- 9.4 Rails and Bridge Element Equations -- 9.4.1 Central Finite Rail (CFR) Element and Bridge Element -- 9.4.2 Left Semi-Infinite Rail (LSR) Element -- 9.4.3 Right Semi-Infinite Rail (RSR) Element -- 9.5 VRI Element Considering Vertical Contact Forces Only.

9.6 VRI Element Considering General Contact Forces -- 9.7 System Equations and Structural Damping -- 9.8 Shift of Bridge Segment and Renumbering of Nodal Degrees of Freedom -- 9.9 Verification of Proposed Procedure -- 9.10 Numerical Studies -- 9.10.1 Steady-State Responses of the Train, Rails and Bridge -- 9.10.2 Impact Response of Rails and Bridge Under Various Train Speeds -- 9.10.3 Response of Train to Track Irregularity and Riding Comfort of Train -- 9.10.4 Effect of the Track System -- 9.11 Concluding Remarks -- 10. Vehicle-Rails-Bridge Interaction - Three-Dimensional Modeling -- 10.1 Introduction -- 10.2 Three-Dimensional Models for Train, Track and Bridge -- 10.3 Vehicle Equations and Contact Forces -- 10.4 Equations for the Rail and Bridge Elements -- 10.4.1 Central Finite Rail (CFR) Element for Track A -- 10.4.2 Central Finite Rail (CFR) Element for Track B -- 10.4.3 The Bridge Element -- 10.4.4 Left Semi-Infinite Rail (LSR) Element for Track A -- 10.4.5 Right Semi-Infinite Rail (RSR) Element for Track A -- 10.4.6 Left Semi-Infinite Rail (LSR) Element for Track B -- 10.4.7 Right Semi-Infinite Rail (RSR) Element for Track B -- 10.5 VRI Element Considering Vertical and Lateral Contact Forces -- 10.6 VRI Element Considering General Contact Forces -- 10.7 System Equations and Structural Damping -- 10.8 Simulation of Track Irregularities -- 10.9 Verification of the Proposed Theory and Procedure -- 10.10 Dynamic Characteristics of Train-Rails-Bridge Systems -- 10.10.1 Properties of the Railway Vehicles and Bridge -- 10.10.2 Natural Frequencies of the Railway Vehicles and Bridge -- 10.10.3 Dynamic Interactions Between the Train and Bridge -- 10.10.4 Train-Rails-Bridge Interaction Considering Track Irregularities -- 10.11 Dynamic Effects Induced by Trains at Different Speeds -- 10.12 Response Induced by Two Trains in Crossing.

10.13 Criteria for Derailment and Safety Assessment of Trains -- 10.14 Concluding Remarks -- 11. Stability of Trains Moving over Bridges Shaken by Earthquakes -- 11.1 Introduction -- 11.2 Analysis Model for Train-Rails-Bridge System -- 11.3 Railway-Bridge System with Ground Motions -- 11.3.1 Central Finite Rail (CFR) Element for Track A -- 11.3.2 Central Finite Rail (CFR) Element for Track B -- 11.3.3 Bridge Element -- 11.3.4 Left Semi-Infinite Rail (LSR) Element for Tracks A and B -- 11.3.5 Right Semi-Infinite Rail (RSR) Element for Tracks A and B -- 11.4 Method of Analysis -- 11.5 Description of Input Earthquake Records -- 11.6 Train Resting on Railway Bridge under Earthquake -- 11.6.1 Responses of Bridge and Train Car -- 11.6.2 Contact Forces between Wheels and Rails -- 11.6.3 Maximum YQ Ratio for Wheelsets in Earthquake -- 11.6.4 Stability of an Idle Train under Earthquakes of Various Intensities -- 11.7 Trains Moving over Railway Bridges under Earthquakes -- 11.7.1 Responses of Bridge and Train Car -- 11.7.2 Maximum YQ Ratio for Moving Trains in Earthquake -- 11.7.3 Stability Assessment of Moving Trains in Earthquake -- 11.8 Concluding Remarks -- Appendix A Derivation of Response Function P1 in Eq. (2.55) -- Appendix B Newmark's β Method -- Appendix C Vertical Frequency of Vibration of Curved Beam -- Appendix D Horizontal Frequency of Vibration of Curved Beam -- Appendix E Derivation of Residual Vibration for Curved Beam in Eq. (5.53) -- Appendix F Beam Element and Structural Damping Matrix -- F.1 Equation of Motion for Beam Element -- F.2 Structural Damping Matrix -- Appendix G Partitioned Matrices and Vector for Vehicle, Eq. (9.4) -- Appendix H Related Matrices and Vectors for CFR Element -- Appendix I Related Matrices and Vectors for 3D Vehicle Model -- Appendix J Mass and Stiffness Matrices for Rail and Bridge Elements.

J.1 Mass and Stiffness Matrices of the CFR Element for Both Tracks.