Vehicle-Bridge Interaction Dynamics

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Vehicle-BridgeInteraction DynamicsWith Applications to High-Speed Raileawy

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Vehicle-BridgeInteraction DynamicsWith Applications to High-Speed Railways

Y. B. Yang

J. D. Yau

Y. S. Wu

National Taiwan University, Taiwan

Tamkang University, Taiwan

Sinotech Engineering Consultants, Ltd., Taiwan

World ScientificWNEW JERSEY · LONDON · SINGAPORE · BEIJING · ASHANGHAI · HONG KONG · TAIPEI · CHENNAI

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VEHICLE–BRIDGE INTERACTION DYNAMICS: WITH APPLICATIONS TOHIGH-SPEED RAILWAYS

December 2, 2005 9:20 Vehicle–Bridge Interaction Dynamics bk04-001

Contents

Preface xv

Acknowledgments xxi

List of Symbols xxiii

1. Introduction 1

1.1 Major Considerations . . . . . . . . . . . . . . . . . 11.2 Vehicle Models . . . . . . . . . . . . . . . . . . . . 51.3 Bridge Models . . . . . . . . . . . . . . . . . . . . . 91.4 Railway Bridges and Vehicles . . . . . . . . . . . . 121.5 Methods of Solution . . . . . . . . . . . . . . . . . 151.6 Impact Factor and Speed Parameter . . . . . . . . 191.7 Concluding Remarks . . . . . . . . . . . . . . . . . 22

Part I Moving Load Problems 25

2. Impact Response of Simply-Supported Beams 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 272.2 Simple Beam Subjected to a Single Moving Load . 302.3 Impact Factor for Midpoint Displacement . . . . . 362.4 Impact Factor for Midpoint Bending Moment . . . 402.5 Impact Factor for End Shear Force . . . . . . . . . 432.6 Simple Beam Subjected to a Series of

Moving Loads . . . . . . . . . . . . . . . . . . . . . 45

v


vi Vehicle–Bridge Interaction Dynamics

2.6.1 Modeling of Wheel Loads of a Train . . . . 452.6.2 Method of Solution . . . . . . . . . . . . . 482.6.3 Phenomenon of Resonance . . . . . . . . . 542.6.4 Phenomenon of Cancellation . . . . . . . . 562.6.5 Optimal Design Criteria . . . . . . . . . . 57

2.7 Illustrative Examples . . . . . . . . . . . . . . . . . 582.7.1 Comparison with Finite Element Solutions 592.7.2 Effects of Moving Masses and Damping . . 622.7.3 Effect of Span to Car Length Ratio . . . . 63

2.8 Concluding Remarks . . . . . . . . . . . . . . . . . 67

3. Impact Response of Railway Bridges with Elastic Bearings 69

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 693.2 Equation of Motion . . . . . . . . . . . . . . . . . . 713.3 Fundamental Frequency of the Beam . . . . . . . . 733.4 Dynamic Response Analysis . . . . . . . . . . . . . 753.5 Phenomena of Resonance and Cancellation . . . . . 773.6 Effect of Structural Damping . . . . . . . . . . . . 823.7 Envelope Formula for Resonance Response . . . . . 873.8 Impact Factor and Envelope Impact Formulas . . . 903.9 Numerical Examples . . . . . . . . . . . . . . . . . 91

3.9.1 Phenomenon of Resonance . . . . . . . . . 913.9.2 Effect of Structural Damping . . . . . . . . 933.9.3 Envelope Impact Formula . . . . . . . . . 96


4. Mechanism of Resonance and Cancellation forElastically-Supported Beams 101

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1014.2 Formulation of the Theory . . . . . . . . . . . . . . 103

4.2.1 Assumed Modal Shape of Vibration . . . . 1034.2.2 Single Moving Load . . . . . . . . . . . . . 1054.2.3 A Series of Moving Loads . . . . . . . . . . 106

4.3 Conditions of Resonance and Cancellation . . . . . 1084.4 Mechanism of Resonance and Cancellation . . . . . 112


Contents vii

4.5 Field Measurement of Vibration ofRailway Bridges . . . . . . . . . . . . . . . . . . . . 118


5. Curved Beams Subjected to Vertical and HorizontalMoving Loads 125

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1255.2 Governing Differential Equations . . . . . . . . . . 1275.3 Curved Beam Subjected to a Single

Moving Load . . . . . . . . . . . . . . . . . . . . . 1295.3.1 Vertical Moving Load . . . . . . . . . . . . 1295.3.2 Horizontal Moving Load . . . . . . . . . . 135

5.4 Unified Expressions for Vertical andRadial Vibrations . . . . . . . . . . . . . . . . . . . 138

5.5 Solutions for Multi Moving Loads . . . . . . . . . . 1405.6 Conditions of Resonance and Cancellation . . . . . 1435.7 Numerical Examples . . . . . . . . . . . . . . . . . 144

5.7.1 Comparison of Analytic withFinite Element Solutions . . . . . . . . . . 144

5.7.2 Phenomenon of Cancellation UnderSingle or Multi Moving Masses . . . . . . . 146

5.7.3 Phenomenon of Resonance Under MultiMoving Masses . . . . . . . . . . . . . . . 149

5.7.4 I–S Plot — Impact Effect Caused byMoving Loads . . . . . . . . . . . . . . . . 150


Part II Interaction Dynamics Problems 153

6. Vehicle–Bridge Interaction Element Based onDynamic Condensation 155

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1556.2 Equations of Motion for the Vehicle and Bridge . . 1576.3 Element Equations in Incremental Form . . . . . . 1616.4 Equivalent Stiffness Equation for Vehicles . . . . . 1636.5 Vehicle–Bridge Interaction Element . . . . . . . . . 165


viii Vehicle–Bridge Interaction Dynamics

6.6 Incremental Dynamic Analysis with Iterations . . . 1696.6.1 Equivalent Stiffness Equations for

VBI System . . . . . . . . . . . . . . . . . 1696.6.2 Procedure of Iterations . . . . . . . . . . . 171

6.7 Numerical Verification . . . . . . . . . . . . . . . . 1756.7.1 Simple Beam Subjected to Moving

Sprung Mass . . . . . . . . . . . . . . . . . 1766.7.2 Simple Beam Subjected to

Moving Train . . . . . . . . . . . . . . . . 1796.7.3 Free-Fixed Beam with Various Models

for Moving Vehicles . . . . . . . . . . . . . 1806.8 Parametric Studies . . . . . . . . . . . . . . . . . . 182

6.8.1 Models for Bridge, Train andRail Irregularities . . . . . . . . . . . . . . 183

6.8.2 Moving Load versus SprungMass Model . . . . . . . . . . . . . . . . . 184

6.8.3 Effect of Rail Irregularities . . . . . . . . . 1866.8.4 Effect of Ballast Stiffness . . . . . . . . . . 1886.8.5 Effect of Vehicle Suspension Stiffness . . . 1916.8.6 Effect of Vehicle Suspension Damping . . . 194


7. Vehicle–Bridge Interaction Element ConsideringPitching Effect 199

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1997.2 Equations of Motion for the Vehicle and Bridge . . 2027.3 Rigid Vehicle–Bridge Interaction Element . . . . . 2077.4 Equations of Motion for the VBI System . . . . . . 2137.5 Numerical Studies . . . . . . . . . . . . . . . . . . . 217

7.5.1 Simple Beam Traveled by aTwo-Axle System . . . . . . . . . . . . . . 217

7.5.2 Simple Beam Traveled by a TrainConsisting of Five Identical Cars . . . . . . 219

7.5.3 Riding Comfort in the Presence ofTrack Irregularities . . . . . . . . . . . . . 223


Contents ix

7.5.4 Effect of Elasticity of theSuspension System . . . . . . . . . . . . . 223

7.5.5 Effect of Damping of theSuspension System . . . . . . . . . . . . . 226

7.5.6 Effect of Track Irregularity . . . . . . . . . 2297.6 Concluding Remarks . . . . . . . . . . . . . . . . . 229

8. Modeling of Vehicle–Bridge Interactions by theConcept of Contact Forces 233

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2338.2 Vehicle Equations and Contact Forces . . . . . . . 2368.3 Solution of Contact Forces from

Vehicle Equations . . . . . . . . . . . . . . . . . . . 2408.4 VBI Element Considering Vertical Contact

Forces Only . . . . . . . . . . . . . . . . . . . . . . 2428.5 VBI Element Considering General

Contact Forces . . . . . . . . . . . . . . . . . . . . 2448.6 System Equations and Structural Damping . . . . . 2458.7 Procedure of Time-History Analysis for

VBI Systems . . . . . . . . . . . . . . . . . . . . . 2478.8 Numerical Examples and Verification . . . . . . . . 249

8.8.1 Cantilever Beam Subjected to aMoving Load . . . . . . . . . . . . . . . . . 249

8.8.2 Cantilever Beam Subjected to aMoving Mass . . . . . . . . . . . . . . . . . 252

8.8.3 Simple Beam Subjected to a MovingSprung Mass . . . . . . . . . . . . . . . . . 254

8.8.4 Simple Beam Subjected to a MovingRigid Bar Supported bySpring-Dashpot Units . . . . . . . . . . . . 257

8.8.5 Bridge Subjected to a Vehiclein Deceleration . . . . . . . . . . . . . . . . 262

8.8.6 Bridges Subjected to a Train Consistingof 10 Identical Cars . . . . . . . . . . . . . 266



x Vehicle–Bridge Interaction Dynamics

9. Vehicle–Rails–Bridge Interaction —Two-Dimensional Modeling 271

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2719.2 Train and Bridge Models and Minimal

Bridge Segment . . . . . . . . . . . . . . . . . . . . 2739.3 Vehicle’s Equations of Motion and

Contact Forces . . . . . . . . . . . . . . . . . . . . 2779.4 Rails and Bridge Element Equations . . . . . . . . 279

9.4.1 Central Finite Rail (CFR) Element andBridge Element . . . . . . . . . . . . . . . 279

9.4.2 Left Semi-Infinite Rail (LSR) Element . . 2839.4.3 Right Semi-Infinite Rail (RSR) Element . 285

9.5 VRI Element Considering Vertical ContactForces Only . . . . . . . . . . . . . . . . . . . . . . 286

9.6 VRI Element Considering GeneralContact Forces . . . . . . . . . . . . . . . . . . . . 287

9.7 System Equations and Structural Damping . . . . . 2899.8 Shift of Bridge Segment and Renumbering of

Nodal Degrees of Freedom . . . . . . . . . . . . . . 2929.9 Verification of Proposed Procedure . . . . . . . . . 2939.10 Numerical Studies . . . . . . . . . . . . . . . . . . . 295

9.10.1 Steady-State Responses of the Train,Rails and Bridge . . . . . . . . . . . . . . . 296

9.10.2 Impact Response of Rails and BridgeUnder Various Train Speeds . . . . . . . . 299

9.10.3 Response of Train to Track Irregularityand Riding Comfort of Train . . . . . . . . 303

9.10.4 Effect of the Track System . . . . . . . . . 3079.11 Concluding Remarks . . . . . . . . . . . . . . . . . 308

10. Vehicle–Rails–Bridge Interaction —Three-Dimensional Modeling 311

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 31110.2 Three-Dimensional Models for Train, Track

and Bridge . . . . . . . . . . . . . . . . . . . . . . . 313


Contents xi

10.3 Vehicle Equations and Contact Forces . . . . . . . 31410.4 Equations for the Rail and Bridge Elements . . . . 326

10.4.1 Central Finite Rail (CFR) Element forTrack A . . . . . . . . . . . . . . . . . . . 327

10.4.2 Central Finite Rail (CFR) Element forTrack B . . . . . . . . . . . . . . . . . . . 332

10.4.3 The Bridge Element . . . . . . . . . . . . . 33410.4.4 Left Semi-Infinite Rail (LSR) Element

for Track A . . . . . . . . . . . . . . . . . 33710.4.5 Right Semi-Infinite Rail (RSR) Element

for Track A . . . . . . . . . . . . . . . . . 34010.4.6 Left Semi-Infinite Rail (LSR) Element

for Track B . . . . . . . . . . . . . . . . . 34210.4.7 Right Semi-Infinite Rail (RSR) Element

for Track B . . . . . . . . . . . . . . . . . 34310.5 VRI Element Considering Vertical and Lateral

Contact Forces . . . . . . . . . . . . . . . . . . . . 34310.6 VRI Element Considering General

Contact Forces . . . . . . . . . . . . . . . . . . . . 34710.7 System Equations and Structural Damping . . . . . 34910.8 Simulation of Track Irregularities . . . . . . . . . . 35410.9 Verification of the Proposed Theory

and Procedure . . . . . . . . . . . . . . . . . . . . . 36110.10 Dynamic Characteristics of

Train–Rails–Bridge Systems . . . . . . . . . . . . . 36610.10.1 Properties of the Railway Vehicles

and Bridge . . . . . . . . . . . . . . . . . . 36610.10.2 Natural Frequencies of the Railway

Vehicles and Bridge . . . . . . . . . . . . . 36710.10.3 Dynamic Interactions Between the

Train and Bridge . . . . . . . . . . . . . . 36710.10.4 Train–Rails–Bridge Interaction

Considering Track Irregularities . . . . . . 37210.11 Dynamic Effects Induced by Trains at

Different Speeds . . . . . . . . . . . . . . . . . . . . 38410.12 Response Induced by Two Trains in Crossing . . . 390


xii Vehicle–Bridge Interaction Dynamics

10.13 Criteria for Derailment and Safety Assessmentof Trains . . . . . . . . . . . . . . . . . . . . . . . . 399


11. Stability of Trains Moving over Bridges Shaken byEarthquakes 409

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 40911.2 Analysis Model for Train–Rails–Bridge System . . 41111.3 Railway–Bridge System with Ground Motions . . . 414

11.3.1 Central Finite Rail (CFR) Element forTrack A . . . . . . . . . . . . . . . . . . . 414

11.3.2 Central Finite Rail (CFR) Element forTrack B . . . . . . . . . . . . . . . . . . . 418

11.3.3 Bridge Element . . . . . . . . . . . . . . . 41911.3.4 Left Semi-Infinite Rail (LSR) Element

for Tracks A and B . . . . . . . . . . . . . 42011.3.5 Right Semi-Infinite Rail (RSR) Element

for Tracks A and B . . . . . . . . . . . . . 42311.4 Method of Analysis . . . . . . . . . . . . . . . . . . 42411.5 Description of Input Earthquake Records . . . . . . 42611.6 Train Resting on Railway Bridge

under Earthquake . . . . . . . . . . . . . . . . . . . 43511.6.1 Responses of Bridge and Train Car . . . . 43611.6.2 Contact Forces between Wheels and Rails 44311.6.3 Maximum YQ Ratio for Wheelsets

in Earthquake . . . . . . . . . . . . . . . . 44611.6.4 Stability of an Idle Train under

Earthquakes of Various Intensities . . . . . 44811.7 Trains Moving over Railway Bridges

under Earthquakes . . . . . . . . . . . . . . . . . . 45011.7.1 Responses of Bridge and Train Car . . . . 45011.7.2 Maximum YQ Ratio for Moving Trains

in Earthquake . . . . . . . . . . . . . . . . 46011.7.3 Stability Assessment of Moving Trains

in Earthquake . . . . . . . . . . . . . . . . 46011.8 Concluding Remarks . . . . . . . . . . . . . . . . . 470


Contents xiii

Appendix A Derivation of Response Function P1

in Eq. (2.55) 473

Appendix B Newmark’s β Method 477

Appendix C Vertical Frequency of Vibration ofCurved Beam 481

Appendix D Horizontal Frequency of Vibration ofCurved Beam 483

Appendix E Derivation of Residual Vibration forCurved Beam in Eq. (5.53) 485

Appendix F Beam Element and StructuralDamping Matrix 489

F.1 Equation of Motion for Beam Element . . . 489F.2 Structural Damping Matrix . . . . . . . . . 493

Appendix G Partitioned Matrices and Vector forVehicle, Eq. (9.4) 497

Appendix H Related Matrices and Vectors forCFR Element 501

Appendix I Related Matrices and Vectors for 3DVehicle Model 503

Appendix J Mass and Stiffness Matrices for Rail andBridge Elements 507

J.1 Mass and Stiffness Matrices of theCFR Element for Both Tracks . . . . . . . . 507

J.2 Mass and Stiffness Matrices of theBridge Element . . . . . . . . . . . . . . . . 508


xiv Vehicle–Bridge Interaction Dynamics

J.3 Mass and Stiffness Matrices for theLSR Element . . . . . . . . . . . . . . . . . 509

J.4 Mass and Stiffness Matrices of theRSR Element . . . . . . . . . . . . . . . . . 510

J.5 Related Matrices and Vectors for theRail Elements . . . . . . . . . . . . . . . . . 510

References 513

Subject Index 527


Preface

The commercial operation of the first high-speed (or bullet) trainin 1964 with a speed of 210 km/hr in the Japanese railways con-necting Tokyo and Osaka marked the beginning of a new era inrailway engineering. Since then, high-speed trains with speeds over200 km/hr or higher have emerged as a competitive tool for inter-city transportation in several countries including Japan, Germany,France, Italy, Spain, United Kingdom and Sweden. Such a trendcontinues to spread in different parts of the world. While Japanand many European countries have been working on expanding theirhigh-speed railway networks or improving their existing railway lines,Asian countries, such as Korea, Taiwan and China, have reached thestage of planning, constructing, or field-testing their high-speed rail-way systems. Undoubtedly, high-speed train will become a key toolfor inter-city passenger transportation, at least in the aforementionedcountries.

Partly enhanced by the rapid expansion of high-speed railway sys-tems, research on the moving load problems in general, and vehicle–bridge interactionsa in particular, has been booming in the past twodecades. Nevertheless, there is an apparent lack of a timely bookthat can adequately address most of the problems encountered inthe design of high-speed railway bridges, which for the reasons stated

aIn the literature, the term “bridge–vehicle interaction” was also used. It isrealized that such a term was used by those who place more emphasis on the bridgethan on the moving vehicles. In this text, we prefer to use the term “vehicle–bridge interaction”, since we place equal weights on the dynamic behavior of thebridge and moving vehicles.

xv


xvi Vehicle–Bridge Interaction Dynamics

below, are different from those encountered in traditional railway orhighway bridges. This book is intended to fill such a gap. It has beendeveloped as a result of the research works conducted by the authorsand their co-workers. In preparing this book, special attention waspaid to the problems that may be encountered by engineers in prac-tice, with clear physical meanings given for each of the phenomenainvolved. It is hoped that the book in the present form can serveas a most updated source of reference for engineers and researchersworking in high-speed railways, and possibly to those working in thebroad area of railway or bridge engineering.

One problem encountered in high-speed railways is the impactand vibration of bridges caused by the moving trains. This problemis substantially different from that encountered in highway bridgesfor the following reasons. First, the loads induced by a moving trainon the bridge are repetitive in nature, as characterized by the se-quentially moving wheel loads, implying that certain frequencies ofexcitation will be imposed on the bridge during the passage of atrain. In contrast, the loads implied by a highway traffic are randomin nature, when expressed in terms of the wheel loads and wheeldistance. Second, high-speed trains can travel at a speed muchhigher than the vehicles moving on highways, making it possiblefor the excitation frequencies to coincide with the vibration frequen-cies of the bridge, resulting in the so-called resonance phenomenon.Whenever the condition of resonance is reached, the bridge responsewill be continuously amplified as there are more wheel loads passingthe bridge. Such a phenomenon can hardly be observed in high-way bridges. Third, the mass ratio of the vehicles to the bridge isgenerally larger for railways than for highways, due to the fact thata train consists of a number of cars in connection and the railwaybridge deck is relatively narrow, it carries no more than two tracks inmost cases. In contrast, a highway bridge deck may be so wide thatit can afford four or more lanes of running vehicles in each of thetwo directions. For this reason, the interaction between the movingvehicles and bridge appears to be much stronger for railways thanfor highways. Finally, concerning the maneuverability of the trainin motion, the riding comfort or vehicle response is an issue that


Preface xvii

should be taken into account in the design of high-speed railways.Moreover, the response of a moving vehicle is more sensitive to thevehicle–bridge interaction (VBI) compared with that of the bridge.However, the analysis of the dynamic behavior of a VBI system isnot straightforward as there are two subsystems, i.e., the movingvehicles and the bridge, interacting with each other through the con-tact forces existing between the wheels and rails surface, which, inessence, is a nonlinear, coupled and time-dependent problem.

This book intends to give a broad and systematic coverage of thevibration problems encountered in high-speed railway bridges, withparticular emphasis placed on the interaction between the movingvehicles and supporting bridge. In general, the book is divided intotwo parts, with Part I dedicated to the moving load problems andPart II to the interaction dynamics problems. These two parts canalso be distinguished by the fact that the moving load problems (i.e.,those treated in Part I) can generally be solved by analytical means,for which closed form solutions are possible, while the interaction dy-namics problems (i.e., those treated in Part II) can only be solved bynumerical means, say, using the vehicle–bridge interaction elementsderived.

Starting with a general review of the related previous worksin Chapter 1, an analytical formulation was presented for simply-supported beams subjected to a sequence of moving loads inChapter 2, from which the phenomena of resonance and cancellationwere identified, along with the optimal design criteria established forbridges. The closed-form solution presented for simple beams allowsus to identify the key parameters involved. Conventionally, elasticbearings are installed at the supports of bridge girders for isolatingthe earthquake forces transmitted from the ground to the superstruc-ture. However, such devices may adversely result in amplificationof the response of the bridge during the passage of a train. Theproblem of elastically supported beams subjected to moving loadshas received little attention in the literature, which was studied byan analytical approach in Chapter 3. The envelope impact formulaspresented in Chapter 3 can be used as a useful aid for preliminary de-signs. Moreover, the mechanism for the occurrence of resonance and


xviii Vehicle–Bridge Interaction Dynamics

cancellation was thoroughly investigated in Chapter 4, with whichthe measured results obtained from the field test for two adjacentbridges was interpreted with clear physical meanings.

The dynamic behavior of a horizontally-curved beam subjected toa series of moving masses was formulated and studied in Chapter 5.This problem was not well-treated before, due to the overlook of thecentrifugal forces induced by masses moving over a circular path,which are functions of both the speed of the moving masses andradius of the curved beam. In Chapter 5, a complete theory waspresented for the vertical and horizontal vibrations of a horizontally-curved beam under the excitation of the gravitational and centrifugalforces, respectively, that are induced by the moving masses. Partic-ular emphasis was placed on the impact effect caused by the movingmasses on the radial response of the curved beam.

One feature of the book is the derivation of a number of efficientVBI elements by condensing the vehicle’s degrees of freedom to thoseof the bridge in contact, based on the concept of dynamic condensa-tion in Chapter 6. These elements can be used to simulate problemswith bridges and moving vehicles of various complexities. The VBIelement presented in Chapter 6 was extended in Chapter 7 to includethe pitching motion of the moving vehicle. Using the VBI elementsderived, the dynamic properties of the vehicles and bridge, as well asrail irregularities, can be duly taken into account, while the dynamicresponse of the moving vehicle can be solved in addition to the bridgeresponse.

Another way to analyze the VBI dynamics is to treat the movingvehicles and bridge as two separate systems, which interact with eachother through the contact forces. By solving for the contact forcesfrom the vehicles equations, one can treat them as external forcesacting on the bridge, which can then be solved using conventionalfinite element procedures. Such a concept was utilized in developingthe VBI element in Chapter 8, which was then extended in Chapter 9to include the effect of rails with profile irregularities that form partof a railway track in the two-dimensional sense. Because of its ver-satility, the VBI element derived, based on the concept of contactforces, can be used in the simulation of various three-dimensional


Preface xix

vehicle-rails-bridge systems considering, for instance, the crossingof two trains on a bridge, the risk of derailment of a moving train(Chapter 10), and the stability of trains moving over bridges simul-taneously shaken by earthquake (Chapter 11).

The authors wish to express their sincerest gratitude to their greatteacher in civil engineering and education, Dr. Chao-Chung Yu, theformer dean of the College of Engineering, National Taiwan Uni-versity (NTU) (1972–1979) and the former President of the NTU(1981–1984), for his strong influence and continuous advice throughtheir careers of development, both as students and teachers. His ex-perience as a teacher, researcher, educationist, and in some sense asan engineer, has always been a source of inspiration to all the youngfellows under his instruction or working with him.

A large portion of the research results presented in this book hasbeen sponsored through a series of research projects granted by theNational Science Council of the Republic of China on subjects relatedto vehicle–bridge interactions, as well as on bridge dynamics. Thesenior author has been the principal investigator of all these projects.Without such a continuous support, it would be difficult to maintainsuch a large research group at the NTU working on different aspectsof the VBI problem, ranging from the vibration of substructure andsuperstructure of railway bridges to wave propagations in soils andnearby buildings; the latter forms an independent subject that re-quires further research, which was not covered in this book. Besides,we are also grateful to the China Engineering Consultants, Inc., fortheir continuous support to our research group, especially throughthe 1st Structural Department previously led by Senior Vice Presi-dent Mr. Dyi-Wei Chang. Some research results presented in thisbook have been made possible through such a support.

This book was prepared as part of the results of the researchgroup led by the senior author at the National Taiwan University.Many of the graduate students have contributed directly or indirectlyto the success of this work, including Chia-Hung Chang, Chon-MinWu, Chin-Lu Lin, Bing-Houng Lin, Lin-Ching Hsu, Shyh-Rong Kuo,Hsiao-Hui Hung, Chern-Hwa Chen, Jiann-Tsair Chang, Cheng-WeiLin, and Kuo-Wei Chang. The assistance from the administrative


xx Vehicle–Bridge Interaction Dynamics

staff of the College of Engineering, NTU, especially Ms. Hong-HuaChang, during the preparation of this book is greatly appreciated.Finally, a book can never be completed without the continuous sup-port, and expectation, from the families of the authors, colleagues,friends, and the society in which they live in.

Y. B. YangJ. D. YauY. S. Wu

Taipei, Taiwan, Republic of China


Acknowledgments

Parts of the materials presented in this book have been revised fromthe papers published by the authors and their co-workers in a num-ber of technical journals. Efforts have been undertaken to update,digest and rewrite the materials acquired from each source, suchthat a unified and progressive style of presentation can be achievedthroughout the book. In particular, the authors like to thank thecopyright holders for permission to use the materials contained inthe following papers:

Wu, Y. S. and Yang, Y. B. (2003). “Steady-state response and rid-ing comfort of trains moving over a series of simply supportedbridges,” Eng. Struct., 25(2), 251–265. Reproduced with per-mission from Elsevier.

Wu, Y. S., Yang, Y. B., and Yau, J. D. (2001). “Three-dimensionalanalysis of train-rail-bridge interaction problems,” Vehicle Sys-tem Dyn., 36(1), 1–35. c©Swets & Zeitlinger.

Yang, Y. B., Chang, C. H., and Yau, J. D. (1999). “An element foranalysing vehicle–bridge systems considering vehicle’s pitchingeffect,” Int. J. Numer. Meth. Eng., 46, 1031–1047. c©JohnWiley & Sons Limited, reproduced with permission.

Yang, Y. B., Lin, C. L., Yau, J. D., and Chang, D. W. (2004). “Mech-anism of resonance and cancellation for train-induced vibrationson bridges with elastic bearings,” J. Sound & Vibr., 269(1–2),345–360. Reproduced with permission from Elsevier.

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xxii Vehicle–Bridge Interaction Dynamics

Yang, Y. B. and Yau, J. D. (1997). “Vehicle–bridge interaction el-ement for dynamic analysis,” J. Struct. Eng., ASCE, 123(11),1512–1518 (Errata: 124(4), 479). Reproduced by permission ofASCE.

Yang, Y. B., Yau, J. D., and Hsu, L. C. (1997b). “Vibration ofsimple beams due to trains moving at high speeds,” Eng. Struct.,19(11), 936–944. Reproduced with permission from Elsevier.

Yang, Y. B., Wu, C. M., and Yau, J. D. (2001). “Dynamic response ofa horizontally curved beam subjected to vertical and horizontalmoving loads,” J. Sound & Vibr., 242(3), 519–537. Reproducedwith permission from Elsevier.

Yang, Y. B. and Wu, Y. S. (2001). “A versatile element for analysingvehicle–bridge interaction response,” Eng. Struct., 23, 452–469.Reproduced with permission from Elsevier.

Yang, Y. B. and Wu, Y. S. (2002). “Dynamic stability of trainsmoving over bridges shaken by earthquakes,” J. Sound & Vibr.,258(1), 65–94. Reproduced with permission from Elsevier.

Yau, J. D., Wu, Y. S., and Yang, Y. B. (2001). “Impact responseof bridges with elastic bearings to moving loads,” J. Sound &Vibr., 248(1), 9–30. Reproduced with permission from Elsevier.

Yau, J. D., Yang, Y. B., and Kuo, S. R. (1999). “Impact responseof high speed rail bridges and riding comfort of rail cars,”Eng. Struct., 21(9), 836–844. Reproduced with permission fromElsevier.


List of Symbols

The following is a list of symbols used throughout this book. Allthe symbols are defined at the place where they first appear in thetext. Matrices, column and row vectors are enclosed by [ ], ,and 〈〉, respectively. A quantity occurring at time t and t + ∆t

are denoted with subscript t and t + ∆t, respectively. A dot placedover a quantity is used to denote the derivative of the quantity withrespect to time t. And a prime attached to a quantity is used todenote the derivative of the quantity with respect to coordinate x.Only quantities that are not confined to local use are listed below.

A cross-sectional area of beama acceleration of vehiclea0 ∼ a7 coefficients as defined in Eq. (B.4)B(fi) weighting factor, Eq. (9.48)BYQ bogie-side lateral to vertical force ratiob0 ∼ b7 coefficients as defined in Eq. (8.8)[C] damping matrix of structure[Cb] damping matrix of bridge free of any vehicle

actionsCn nth modal damping coefficient[C0] damping matrix of railway bridge free of any

vehicle actions[cb] damping matrix of bridge element[cbi] damping matrix of eith element of bridge[cc] contact matrix as defined in Eq. (8.14)

xxiii


xxiv Vehicle–Bridge Interaction Dynamics

[c∗cij ] damping matrix caused by linking action of carbody

[cd] rail damping matrix due to interaction with thebridge

ce external damping coefficientci internal damping coefficient[cii], [cjj] damping matrices for element i and j, Eqs. (7.25)

and (7.26)[cij ], [cji] damping matrices related to pitching actions for

element i and j, Eqs. (7.25) and (7.26)[cr] damping matrix of rail element[crl] damping matrix of LSR element[crr] damping matrix of RSR elementct material damping coefficient for translational

motion of trackc∗t material damping coefficient for torsional motion

of track[cuu] a partitioned damping matrix of vehicle[cuw] a partitioned damping matrix of vehiclecv damping coefficient of suspension unit[cv] damping matrix of vehicle[cwu] a partitioned damping matrix of vehicle[cww] a partitioned damping matrix of vehicleD determinant of matrixD structural displacement vectord length of train cardb displacement vector of bridgedbi nodal displacement vector of eith element of

bridgedn

b natural deformations of bridgedr

b rigid displacements of bridgedc displacement vector of contact points of bridge or

railde displacement vector of car bodydf displacement vector of front bogie


List of Symbols xxv

dr displacement vector of rail elementdr displacement vector of rear bogie, Chapter 10drl displacement vector of LSR elementdrr displacement vector of RSR elementdu displacement vector of upper part of car bodydv displacement vector of vehicle, dv = 〈du

dw〉Tdw displacement vector of wheel part of car bodydwi displacement vector for ith wheeldwb wheelbase of each wheelsetE modulus of elasticityF (t) load functionF external nodal forces of structureFb external nodal forces of bridgeFb effective resistant force vector of structureFk(v, t) generalized forcing functionfA nodal loads of rail element of Track Af t

A total equivalent nodal forces of element underearthquake

fb frequency of vibration of bridge unitfb external nodal forces of bridge elementfbci vector of consistent nodal forces for ith contact

forcefbi vector of external forces for eith element of bridgefc contact forcefc vector of contact forcesf∗

c general vector of contact forcesfc1 ∼ fc4 contact forcesfe external force components excluding contact

forcesfh horizontal moving loadfr nodal loads of rail elementfrci equivalent nodal forces caused by ith vertical

contact forcefrl nodal loads of LSR element


xxvi Vehicle–Bridge Interaction Dynamics

frr nodal loads of RSR elementfrot rotational moment experienced by car bodyfs resistant forces of sprung mass unitfue external forces acting on upper part of vehiclefv gravitational load of moving vehicle, fv = −mvg

fv force vector of vehiclefver vertical force experienced by car bodyfwe external forces acting on wheel part of vehicleG shear modulusg acceleration of gravityH unit step functionHi ith horizontal contact forceh vertical distance between deck and torsional

center of cross sectionhci lateral displacement of ith contact pointI impact factorI moment of inertia (used together with E)IM impact factor for bending momentIp polar moment of inertia of beamIu impact factor for deflectionIV impact factor for end shear forceIv rotatory inertia of car bodyIy, Iz moments of inertia of beam about y and z axesJ torsional constantK stiffness of elastic bearings[K] stiffness matrix of structure[Kb] stiffness matrix of bridge structure[Kb] effective stiffness matrix of structure[Keff ] effective stiffness matrix[K0] stiffness matrix of railway bridge free of any

vehicle actionskB stiffness of ballast[kb] stiffness matrix of bridge element[kb] stiffness matrix of condensed system[kbi] stiffness matrix of eith element of bridge


List of Symbols xxvii

[kc] contact matrix as defined in Eq. (8.14)[k∗

cij ] stiffness matrix caused by linking action of carbody

[kii], [kjj] stiffness matrices for element i and j, Eqs. (7.25)and (7.26)

[kij ], [kji] stiffness matrices related to pitching actions forelement i and j, Eqs. (7.25) and (7.26)

[kr] stiffness matrix of rail element[krl] stiffness matrix of LSR element[krr] stiffness matrix of RSR element[ks] rail stiffness matrix due to interaction with bridge[kuu] a partitioned stiffness matrix of vehicle[kuw] a partitioned stiffness matrix of vehiclekv stiffness of suspension unit[kv] stiffness matrix of vehicle[kwu] a partitioned stiffness matrix of vehicle[kww] a partitioned stiffness matrix of vehicleL span length or characteristic length of bridgeLc distance between two wheel assemblies of train

carLd length equal to car length minus the axle distanceLr length of irregularityl length of beam element[l] transformation matrix, Chapters 8 and 9la half of axle length of wheelsetM car mass lumped at each loadM(x, t) bending moment[M ] mass matrix of structure[Mb] mass matrix of bridge structureMc mass of car bodyMn nth modal massMt mass of bogieMv half of the mass of car bodyMw mass of vehicle[M0] mass matrix of railway bridge free of any vehicle

actions


xxviii Vehicle–Bridge Interaction Dynamics

m mass per unit length[mb] mass matrix of bridge element[mbi] mass matrix of eith element of bridge[mc] contact matrix as defined in Eq. (8.14)[m∗

cij] mass matrix caused by linking action of car body[mii], [mjj] mass matrices for element i and j, Eqs. (7.25) and

(7.26)[mr] mass matrix of rail element[mrl] mass matrix of LSR element[mrr] mass matrix of RSR element[muu] a partitioned mass matrix of vehicle[muw] a partitioned mass matrix of vehiclemv mass of moving vehicle[mv] mass matrix of vehiclemw mass of wheel assembly[mwu] a partitioned mass matrix of vehicle[mww] a partitioned mass matrix of vehicleN total number of moving loadsNc interpolation vector for beam displacement

evaluated at contact point xc, Nc = N(xc)Nh

ci interpolation vector of bridge element evaluatedfor ith horizontal contact force

Nvci interpolation vector of bridge element evaluated

for ith vertical contact forceNmin minimum number of bridge units used in analysis〈Nu〉 interpolation vector for axial displacement〈Nv〉 interpolation vector for vertical displacementnA number of train cars moving on Track AnB number of train cars moving on Track Bnb number of CFR elements considered within

minimal bridge segmentnv number of vehicles comprising the trainP unified load function as defined in Eq. (5.45)P applied loads of structurePb external loads of bridge


List of Symbols xxix

P ∗c equivalent contact forces of structure

PD axle load decrement ratioPeff effective load vectorPn(x, t) function as defined in Eq. (2.50)PQ axle decrement ratioP1(v, t), P2(v, t) response functions as defined in Eq. (3.15)P1(v, t) response function as defined in Eq. (2.54)p load magnitude, p = −(Mv + mw)g, Chapter 6pb nodal loads of bridge elementpc load vector as defined in Eq. (8.15)(pc, pk) unit axial interaction forces between rail and

bridge elementsp∗ci equivalent loads as defined in Eq. (8.24)pv load vector of sprung mass model, pvT = 〈p, 0〉pw loads induced by wheelspx1p, pz1p particular solutions for in-plane vibrations of

curved beampx1h, pz1h homogeneous solutions for in-plane vibrations of

curved beampy1p, pθ1p particular solutions for out-of-plane vibrations of

curved beampy1h, pθ1h homogeneous solutions for out-of-plane vibra-

tions of curved beamQ∗

ct equivalent contact forces of structureQ1(v, t), Q2(v, t) response functions as defined in Eq. (3.21)qc load vector as defined in Eq. (8.15)(qc, qk) unit vertical interaction forces between rail and

bridge elementsq∗ci equivalent loads as defined in Eq. (8.24)qn(t) nth generalized coordinateqs internal resistant forces of sprung mass unitqu equivalent nodal loads for upper part of car

bodyquc equivalent contact forces for upper part of car

body


xxx Vehicle–Bridge Interaction Dynamics

qx1 first generalized coordinate for axial displacementux

qyi ith generalized coordinate for displacement uy

qzi ith generalized coordinate for radial displacementuz

qθi ith generalized coordinate for angle of twistθx

R radius of curvature of curved beamRd maximum dynamic responseRs maximum static responseRsM maximum bending moment under static loadsRsu maximum static deflectionRsV maximum static shear forcer(x) profile of rail irregularityr vector of irregularity profile at contact pointsrc elevation of rail irregularity at contact point xc

(rc, rk) unit vertical interaction forces between rail andbridge elements

rh(x) deviation in lateral alignment of two railsri track profile evaluated at ith contact pointrv1(x), rv2(x) vertical deviations of two railsr0 amplitude of irregularitiesr0 nominal radius of wheel, Chapter 10S speed parameterSc speed parameter for cancellationSh1 speed parameter for the horizontal vibration of

curved beamSn nth speed parameterSr speed parameter for resonanceSv1 speed parameter for vertical vibration of curved

beamSYQ single wheel lateral to vertical force ratioS(Ω) power spectral density (PSD) function for track

irregularityt time


List of Symbols xxxi

tc time lag between two sets of moving loads(= Lc/v)

tend ending time for analysistj arriving time of jth load on beamtN arriving time of Nth load on beamUb total displacements of bridgeUN,1 forced vibration caused by Nth moving load,

Eq. (5.51)UN,2 residual vibration caused by N − 1 moving loads,

Eq. (5.52)Uj(t, v, L) load configuration as defined in Eq. (2.34)Uj(x, t) residual vibration caused by jth moving load,

Eq. (5.50)U(x, t) unified displacement function as defined in

Eq. (5.45)U displacements of structureUb displacements of bridge structureub axial displacement of bridge element, Chapter 9ub nodal displacements of beam elementug support displacements of bridgeu(x, t) deflection of beam at section x and time t

ui, uj vertical displacements of elements i and j

ur axial displacement of rail elementux, uy, uz cross-sectional displacements of curved beam

along three axesV (x, t) shear forceVi vertical force acting at ith contact point of bridgev vehicle speedvb vertical displacement of bridge element,

Chapter 9vci vertical displacement of ith contact point of

bridgevr vehicle speed at resonancevr vertical displacement of rail element, Chapter 9vu displacement of car body mass


xxxii Vehicle–Bridge Interaction Dynamics

vw displacement of wheel massvwi displacement of ith wheelv0 initial velocity of vehicleW half weight of car body, W = 0.5 MvgW static weight of each wheelset, Chapter 10Wz Sperling’s ride indexWzi comfort indexx beam axisxbs0 position for first wheel to start acceleration or

brakingxbsf position for first wheel to stop acceleration or

brakingxc position of contact pointxci, xcj position of contact point for elements i and j

xend ending position of first wheelx0 reference distance in Eq. (6.38)Ylim limit on lateral track forcey a cross-sectional axisy1, y2 coordinates of the two masses of sprung mass

model, Chapter 6y nodal vector of sprung mass model, yT =

〈y1, y2〉, Chapter 6y displacements of car body as rigid beam, yT =

〈yv θv〉, Chapter 7YQ wheelset lateral to vertical force ratioyv vertical displacement of car body as rigid beamz a cross-sectional axisβ coefficient related to variation of acceleration[Γ] constraint matrixγ coefficient related to numerical dampingγ0 wavelength of corrugation∆du upper-part vehicle displacement increments∆st maximum static deflection of beam with hinge

supports∆t time increment


List of Symbols xxxiii

∆y1 displacement increment of wheel mass∆y displacement increments of sprung mass unit∆Ub displacement increments of bridge∆ub displacement increments of bridge elementδ delta functionθb rotation about x axis of bridge elementθv rotation of car body as rigid beamθx, θy, θz rotations of a cross section of curved beam about

three axesκ stiffness ratio of beam to elastic springs[λ] transformation matrix, Eq. (10.14)λr wavelength of track irregularityλu horizontal characteristic number of beam-

Winkler foundationλv vertical characteristic number of beam-Winkler

foundation systemµi coefficient of friction for ith wheelξ damping coefficientξn damping coefficient of nth modeρ density of beamϕ subtended angle of curved beam, ϕ = L/R

ϕu rotation of car body as rigid bar, Chapter 8φn nth vibration modeφn nth vibration mode of structureΨ(t) unified amplitude function as defined in

Eq. (5.45)[Ψuu] equivalent matrix for upper part of car body,

Eq. (8.10)[Ψwu] matrix as defined in Eq. (8.16a)Ω exciting frequency implied by the moving loadΩ spatial frequency of track irregularity, Chapters 9

and 10ω frequency of vibrationωd damped frequencyωdn damped frequency of vibration of nth mode


xxxiv Vehicle–Bridge Interaction Dynamics

ωh1 fundamental frequency for horizontal plane ofcurved beam

ωn frequency of vibration of nth modeωv vertical vibration frequency of car bodyωv1 fundament frequency of vertical vibration of

curved beamω0 frequency of vibration for beam with hinge

supportsωθ rotational vibration frequency of car body


Chapter 1

Introduction

The interaction between a bridge and the vehicles moving over thebridge is a coupled, nonlinear dynamic problem. Conventionally,most research has been focused on the dynamic or impact responseof the bridge, but not of the moving vehicles. For the cases whereonly the bridge response is desired, the moving vehicles have fre-quently been approximated to the extreme as a number of movingloads. However, whenever the responses of both the bridge and mov-ing vehicles are desired, as encountered in the design of high-speedrailways, models that can adequately account for the dynamic prop-erties of the moving vehicles should be adopted. In this chapter, thekey factors involved in the dynamic interaction between the bridgeand moving vehicles will be discussed, along with procedures for solv-ing the vehicle–bridge interaction problems. The materials presentedin this chapter have been revised from the review paper by Yang andYau (1998) with supplement of the relevant literature published re-cently.

1.1. Major Considerations

The dynamic interaction between a bridge and the moving vehiclesrepresents a special discipline within the broad area of structural dy-namics. The vehicles considered may be those constituting the trafficflow of a highway bridge, in general, or those that form a connectedline of railroad cars, in particular. From the theoretical point ofview, the two subsystems, i.e., the bridge and moving vehicles, can

1


2 Vehicle–Bridge Interaction Dynamics

be simulated as two elastic structures, of which each is characterizedby some frequencies of vibration. The two subsystems interact witheach other through the contact forces, i.e., the forces induced at thecontact points between the wheels and rails surface (of the railwaybridge) or pavement surface (of the highway bridge). A problemsuch as this is nonlinear and time-dependent due to the fact that thecontact forces may move from time to time, while their magnitudesdo not remain constant, as a result of the relative movement of thetwo subsystems. The way by which the two subsystems interact witheach other is determined primarily by the inherent frequencies of thetwo subsystems and the driving frequency of the moving vehicles. Inthis book, we prefer to use the term vehicle–bridge interaction (VBI)to refer to the interaction between the two subsystems. The vehi-cle considered in this book is a general term, which can be a car, atruck, a tractor-trailer, or a railroad car that forms part of the train.The term bridge is also a general one. It can be a simply-supportedbeam, a multi-span continuous beam, or a bridge of any types used inhighways and railways, with or with no account of the effects of sur-face pavement (for highways) or rails and ballast (for railways). Theconsideration of the VBI is necessary if the vehicle response, in ad-dition to the bridge response, is desired. In the design of high-speedrailway bridges, for instance, the maximum vertical and/or lateralaccelerations of the moving vehicles are used as indicators for evalu-ating the riding comfort of passengers carried by the train. Besides,the vertical and lateral contact forces of the wheels of railroad carswith the rails represent a kind of information central to assessmentof the risk of derailment for moving trains, especially in the presenceof earthquake shaking.

In many cases, especially when the vehicle to bridge mass ratio issmall, the elastic and inertial effects of the vehicles may be ignoredand much simpler models can be adopted for the vehicles. One typ-ical example is the simulation of a moving vehicle over a bridge as asingle moving load, which has been conventionally referred to as themoving load model (Fig. 1.1). Since the interaction between the twosubsystems has been ignored, the moving load model is good onlyfor computing the response of the larger subsystem, i.e., the bridge,


Introduction 3

Fig. 1.1. Moving load model.

but not of the smaller subsystem, i.e., the vehicle. In this book, themoving load problem can be regarded as a special case of the moregeneral formulation that considers the various dynamic properties ofthe moving vehicles.

The objective of this book is to establish some efficient methodswithin the framework of finite element methods for solving the dy-namic response of the VBI systems. The formulation of these meth-ods will be kept as general as possible, so that they can be appliedto most conceivable problems. However, in deriving the fundamentaltheories using the analytical approaches or in conducting the para-metric studies to illustrate the various dynamic effects involved, moreemphasis will be placed on the problems encountered in the designof high-speed railway bridges, so as to reflect the public concern overthe safety and the riding comfort of high-speed trains. It is believedthat the methodologies established herein can be applied to solvingsimilar problems encountered in traditional railways and mass rapidtransit systems.

From the point of view of structural dynamics, a railway bridgeis different from a highway bridge in that the sources of excitationcaused by the moving vehicles are different for the two cases. Forexample, the vehicles moving over a highway bridge are random innature. The vehicles constituting the highway traffic may vary interms of the axle weight, axle interval, moving speed, and even theheadway. However, a train moving over a railway bridge can gener-ally be regarded as a sequence of identical vehicles in connection, plusone or two locomotives. Conventionally, a train has been simplifiedas a sequence of moving masses, or in the extreme case as a sequence



of concentrated loads, of regular intervals. Because of the repeti-tive character of the wheel or bogie loads, a moving train usuallycontains some inherent frequencies, plus an excitation frequency as-sociated with the moving speed. If any of these frequencies coincideswith any of the frequencies of vibration of the bridge, the so-calledresonance phenomenon will be induced on the bridge by the movingtrain, in the sense that the response will be continuously built up, asthere are more railroad cars passing the bridge. Under the conditionof resonance, great amplification in the bridge responses, as well asin the vehicles response, can be expected, which is likely to affectthe life span of the bridge and the riding quality of running vehicles.It is advisable that the phenomenon of resonance be circumventedfrom the onset in the design of railway bridges.

Research on the dynamic response of bridges caused by the vehic-ular movement dates back to the mid-nineteenth century, followingprimarily the works of Willis (1849) and Stokes (1849) in investi-gating the collapse of the Chaster Rail Bridge in England in 1847,the first case for the collapse of a railway bridge in history. In thesepioneer works, the effect of inertia of the beam was ignored, and thevehicle is modeled as a concentrated moving mass traveling at con-stant speeds. Although for this particular case, an exact solution canbe obtained, its applicability remains rather limited due to the omis-sion of the inertial effect of the beam. Nevertheless, the contributionof Stokes and Willis is considered historical, since they are amongthe first to bring the problem of vehicle impacts to the design desksof bridge engineers.

In the past two decades, the amount of research conducted on thevibration of bridges under moving vehicles has been increasing at arate much faster than ever, partly due to the successful operationof high-speed railways in Japan and some European countries. Itis difficult, if not impossible, to have a complete count of all theworks conducted by previous researchers on this subject. For thedays when hand calculations and slide rules play the most importantrole in design offices, i.e., before the advent of digital computers inthe 1940s, investigations on bridge dynamics were concerned mainlywith the development of analytical or approximate solutions for some


Introduction 5

simple, fundamental problems. Researchers of this period who werefrequently cited in the literature include Timoshenko (1922), Jeffcott(1929) and Lowan (1935). The work by Inglis (1934) contains anearly general treatment on the dynamics of railway bridges, whichalso lays the foundation for the following development.

The advent of digital computers, later followed by workstations,has enabled researchers to adopt more realistic bridge and vehiclemodels in analysis. The general texts by Timoshenko and Young(1955) and Biggs (1964) on structural dynamics contain some par-tial treatment on the moving load problems. Other texts that shouldbe mentioned include the one by Fryba (1972) in analyzing the vi-bration of structures under moving loads, and those by Garg andDukkipati (1984) and Fryba (1996) in dealing with the vibration ofrailway bridges. Starting from 1975, literature reviews were con-ducted by Ting and co-workers from time to time to update the re-lated researches on vehicle–guideway interactions (Ting et al., 1975;Genin and Ting, 1979; Ting and Genin, 1980; Ting and Yener, 1983;Taheri et al., 1990). Nowadays, very powerful numerical methods, es-pecially those based on the finite element methods, can be employedto analyze the dynamic behavior of bridges and moving vehicles, withvirtually no limit placed on the level of complexity of the models usedfor the two subsystems. It should be noted that most of the worksmentioned above were concerned primarily with the vibration of thebridge or supporting structure, but not of the moving vehicles.

1.2. Vehicle Models

By neglecting the inertia effect of the vehicle and considering a vehi-cle as a moving load or pulsating force, Timoshenko (1922) derivedan enormous number of approximate solutions to the problem of sim-ple beams under moving loads. Similar models were adopted by Ayreet al. (1950) and Ayre and Jacobsen (1950) in studying the dynamicresponses of a two-span beam, and later by Vellozzi (1967) in study-ing the vibration of suspension bridges. The moving load model wasalso adopted by Chen (1978) in analyzing the dynamic response ofcontinuous beams. Research on the vibration of bridges traveled by



moving loads is abundant. It is only possible to cite a few of themost related ones, for instance, the works by Tan and Shore (1968a),Fryba (1972), Fertis (1973), Sridharan and Mallik (1979), Wu andDai (1987), Weaver et al. (1990), Galdos et al. (1993), Gbadeyanand Oni (1995), Wang (1997), Zheng et al. (1998), Rao (2000), Chenand Li (2000), and Dugush and Eisenberger (2002), among others.

The moving load model is the simplest model that can be con-ceived, which has been frequently adopted by researchers in studyingthe vehicle-induced bridge vibrations. With this model, the essentialdynamic characteristics of the bridge caused by the moving actionof the vehicle can be captured with a sufficient degree of accuracy.However, the effect of interaction between the bridge and the movingvehicle was just ignored. For this reason, the moving load model isgood only for the case where the mass of the vehicle is small relativeto that of the bridge, and only when the vehicle response is not ofinterest.

For cases where the inertia of the vehicle cannot be regarded assmall, a moving mass model (Fig. 1.2) should be adopted instead.The inertial effects of both the beam and the moving vehicle werestudied as early as in 1929 by Jeffcott (1929) by the method of suc-cessive approximations. The investigations along this line were latercarried out by a number of researchers. Stanisic and Hardin (1969)determined the response of a simple beam under an arbitrary num-ber of moving masses by employing the Fourier series expansion. Bythe use of Green’s function, algorithms for dealing with the movingmass problem has been studied by Ting et al. (1974) and Sadikuand Leipholz (1987). For a simple beam carrying a single movingmass, an exact, closed form solution was derived by Stanisic (1985)by means of expansion of the eigenfunctions in a series. The same

Fig. 1.2. Moving mass model.


Introduction 7

moving mass model was adopted by Akin and Mofid (1989) in theirstudy of the dynamic response of beams with various boundary con-ditions using an analytical–numerical approach.

One drawback with the moving mass model is that it excludesconsideration of the bouncing action of the moving mass relative tothe bridge. Such an effect is expected to be significant in the presenceof rail irregularities or pavement roughness, or for vehicles movingat rather high speeds. Occasionally, it may be necessary to considerthe separation and recontact of the moving vehicle with the bridgefor some very bad road conditions, in which the bouncing action ofthe vehicles plays a decisive role in the seperation–recontact process.

The vehicle model can still be enhanced through considerationof the elastic and damping effects of the suspension systems. Thesimplest model in this case is a moving mass supported by a spring-dashpot unit, the so-called sprung mass model (Fig. 1.3). Biggs(1964) presented a semi-analytical solution to the problem of a sim-ple beam traversed by a sprung mass. By using the series expansiontechnique, Pesterev et al. (2001) examined the response of an elas-tic continuum to multiple moving oscillations. Later, Pesterev et al.(2003) studied in depth the asymptotics of the solution of the mov-ing oscillator problem and found that in the limiting case the movingoscillator problem and the moving mass problem for a simply sup-ported beam are equivalent in terms of the beam displacements, butnot in terms of the beam stresses. Also, it was shown that for smallvalues of spring stiffness, the moving oscillator problem is equiva-lent to the moving load problem. In the book by Fryba (1972), acomprehensive treatment was given for the various vehicle models,

Fig. 1.3. Sprung mass model.



i.e., the moving load, moving mass, and moving sprung mass, con-cerning primarily the dynamic response of the structure traveled byvehicles. The analytical solutions as well as numerical solutions forsome problems were presented in this book.

Because of the emergence of high-performance computers and theadvance in computation technologies, it becomes feasible to have amore realistic modeling of the dynamic properties of the various com-ponents constituting a moving vehicle. Previously, the elastic effectof the tires and suspension mechanisms has been modeled by springs,the damping effect of the tires, suspension systems, and air-cushionby dashpots (Tan and Shore, 1968b; Genin et al., 1975; Blejwas et al.,1979; Genin and Chung, 1979; Humar and Kashif, 1993; Green andCebon, 1994), and the energy dissipating effect of the interleaf mech-anism by frictional devices (Veletsos and Huang, 1970; Chatterjeeet al., 1994; Tan et al., 1998). Using such techniques, a multiple-axletruck or tractor-trailer can be represented as a number of discretemasses each supported by a set of spring and dashpot or frictionaldevice. In the study by Yang et al. (1999), a railroad car was simu-lated as a rigid beam supported by two sets of spring-dashpot uniteach resting on a wheel mass. Such a model enables us to considerthe pitching effect of the car body.

To represent the various dynamic properties of railway freightcars, vehicle models that contain dozens of degrees of freedom(DOFs) have been devised and used by Chu et al. (1986), Wanget al. (1991), Xia et al. (2000), and Zhang et al. (2001a). In or-der to study the train–rails–bridge interaction, a train composed ofa sequence of identical cars was considered by Wu et al. (2001), inwhich each car is assumed to consist of a car body, assumed to berigid, resting on the front and rear bogies, each of which in turn issupported by two wheelsets. A total of 5 DOFs was assigned to thecar body and also to each bogie, to account for the vertical, lateral,rolling, yawing, and pitching motions. In contrast, only three DOFsare assigned to each wheelset, which relate to the vertical, lateraland rolling motions.

Although the use of a more sophisticated vehicle model can makethe simulation more realistic, it does create certain computation


Introduction 9

problems. For instance, in the simulation of bridges subjected toa series of railroad cars or highway vehicles that appear as a randomflow (Yang et al., 1996), divergence or slow convergence may occurin the process of iteration searching for a large number of contactforces at the wheels/rails or wheels/girder contact points in a step-by-step time-history analysis. The other concern here is that usingsimplified models can help identify the key parameters dominatingthe dynamic response of the bridge, which is beneficial for the devel-oping of rational formulas for use in the design codes (Humar andKashif, 1993).

1.3. Bridge Models

A beam that is simply-supported at both ends is the most popularstructure that has ever been adopted in the study of vehicle-inducedvibrations. Except for the research works that rely exclusively onanalytical approaches, there is basically no restriction on the typeof structures considered for the VBI problems, as the structures canalways be represented by finite elements of various forms; the onlydifference being that a simpler bridge model requires less preparationand computation efforts.

In the past, various types of bridges have been considered in studyof the vehicle-induced vibrations, which include the truss bridges(Chu et al., 1979; Wiriyachai et al., 1982), multispan uniform ornonuniform bridges (Wu and Dai, 1987; Yang et al., 1995; Kou andDeWolf, 1997; Cheung et al., 1999; Marchesiello et al., 1999), girderor multigirder bridges (Chu et al., 1986; Hwang and Nowak, 1991;Huang et al., 1993; Cai et al., 1994), continuous beams (Wu andDai, 1987; Yang et al., 1995), curved girder bridges (Tan and Shore,1968a,b; Galdos et al., 1993; Chang, 1997; Yang et al., 2001), guide-ways (Genin et al., 1975), steel plate girder bridges (Kawatani andKim, 2001), and arch bridges (Chatterjee and Datta, 1995; Ju andLin, 2003). The impact factor of horizontally-curved box bridges wasstudied by Galdos et al. (1993) and Senthilvasan et al. (2002). Thedynamic response of a flat plate under the moving load was studiedby Wu et al. (1987).



The dynamic response of cable-stayed bridges to moving vehi-cles has been studied by a number of researchers. By simulatingthe cable-stayed bridge as a beam resting on an elastic foundation,Meisenholder and Weidlinger (1974) proposed an approach for mod-eling the dynamic effects of cable-stayed guideways subjected to tracklevitated vehicles moving at high speeds. The effect of road surfaceroughness was considered by Wang and Huang (1992) in studyingthe cable-stayed bridge vibrations. By using an approximate bridgemodel, taking into account the nonlinear effect of cables, the dynamicresponse of cable-stayed bridges under moving loads was analyzed byYang and Fonder (1998). In the review paper by Diana et al. (2000)for the railway runability of long-span cable supported bridges, itwas noted that the impact effect of cable-stayed bridges is more sen-sitive than that of suspension bridges. Recently, Au et al. (2001a,b)investigated the impact effects of cable-stayed bridges under railwaytraffic using various vehicle models, and concluded that the movingforce and moving mass models significantly underestimate the im-pact effects and the effects of random road surface roughness on theimpact response of the bridge deck are more significant at sectionsclose to the bridge towers. Guo and Xu (2001) studied the interactionbetween a cable-stayed bridge and a tractor-trailer moving over thebridge by a fully computerized approach. Recently, a hybrid tunedmass damper system composed of several subsystems was proposedby Yau and Yang (2004) for suppressing the multiple resonant peaksof cable-stayed bridges that may be excited by high-speed trains.

The vibration of suspension bridges under the vehicular move-ment was investigated by Chatterjee et al. (1994) with the torsionalvibration taken into account. The dynamic interaction between along suspension bridge, which has a main span length of 1377 m,and the running train was shown to be insignificant by Xia et al.(2000). The same suspension bridge was later studied by Xu et al.(2003), considering that there are high winds acting on the bridge,but not directly on the running train; the latter being protected fromexposure to the high wind. Their results indicated that the wind-induced vibration on the bridge is detrimental to the running safetyof the train and also to the riding comfort of passengers.


Introduction 11

Another concern in simulation of the bridge response has been theinclusion of road surface roughness or rail irregularities. It has beenreported that road surface or pavement roughness can significantlyaffect the impact response of bridges (Paultre et al., 1992). However,the elevation of roughness or surface profile depends primarily onthe workmanship involved in the construction of pavement or railtracks and on how they are maintained, which, though random innature, may contain some inherent frequencies. In most cases, thesurface roughness or rail irregularities, which is three-dimensional innature, is often approximated by a two-dimensional profile. As forthe railways, it is realized that the profiles of irregularities on thetwo rails of a track may be different.

The road surface roughness was considered by Gupta (1980) byrepresenting the elevation of road surface by a sine function. To ac-count for its random nature, the road profile can be modeled as a sta-tionary Gaussian random process and generated using certain powerspectral density functions. Methods similar to this have been widelyadopted by researchers in studying the vehicle-induced bridge vibra-tions (Inbanathan and Wieland, 1987; Coussy et al., 1989, Hwangand Nowak, 1991; Chatterjee et al., 1994; Chang and Lee, 1994;Henchi et al., 1998; Pan and Li, 2002). The power spectral densityfunctions developed by Dodds and Robson (1973) have been modi-fied and used by Wang and Huang (1992) and Huang et al. (1993)in their analyses. The work by Marcondes et al. (1991) is of in-terest in that the power spectral density functions used to computethe road elevation have been determined by using the data collectedfrom a field measurement, with distinction made for three differentcategories of pavement. Such an approach was adopted by Yang andLin (1995) in the study of simple and continuous beams traveled byvehicles moving at different speeds.

As far as railway bridges are concerned, track irregularities mayoccur as a result of initial installation errors, degradation of supportmaterials, and dislocation of track joints. Four geometric parameterscan be used to quantitatively describe the rail irregularities, i.e.,the vertical profile, cross level, alignment, and gauge (Wiriyachaiet al., 1982; Chu et al., 1986; Wang et al., 1991). From the point of



structural dynamics, it is the wavelengths or frequencies implied bythe rail irregularities that are crucial to the dynamic behavior of theVBI system. The frequencies implied by the surface roughness of abridge plays a role similar to that of the bridge frequencies, in thatresonance may occur on the bridge and traversing vehicles, if any ofthe excitation or vehicle frequencies coincides with, or are close to,any of the frequencies implied by the surface roughness.

1.4. Railway Bridges and Vehicles

Most of the research works cited above consider only a single orvery small number of vehicular loads. In contrast, comparativelyfew works have been conducted on the dynamic response of bridgestructures under the action of a sequence of moving loads with regu-lar intervals, to simulate the effect of a connected line of train loads(Fig. 1.4). Bolotin (1964) studied a beam subjected to an infinite se-quence of equal loads with uniform interval d and constant speed v.In his study, the period d/v of the moving loads has been identifiedas a key parameter. For the same problem, Fryba (1972) concludedthat the response of the forced steady-state vibration will attain itsmaximum when the time intervals between two successive movingloads are equal to some periods of vibration of the beam in free vi-bration or to an integer multiple thereof. Kurihara and Shimogo(1978a,b) investigated the vibration and stability problems of a sim-ple beam subjected to a series of discrete moving loads. The dynamicresponse of a girder or truss bridge during the passage of a series ofrailway vehicles was studied by Chu et al. (1979). By the transfermatrix method, Wu and Dai (1987) studied the response of multispannonuniform beams subjected to two sets of identical loads moving inthe same or opposite directions. Savin (2001) derived an analyticalexpression of the dynamic amplification factor and response spec-trum for beams with various boundary conditions under successivemoving loads.

Partly enhanced by the successful operation of high-speed rail-ways worldwide, the dynamic response of railway bridges is receivingmuch more attention from researchers than ever. Matsuura (1976)


Introduction 13

Fig. 1.4. Railway vehicles in series.

studied the dynamic behavior of various bridge girders used in theShinkansen system. With the rail irregularities represented by powerspectral densities, the impact responses of various railway bridgeswere investigated by Wiriyachai et al. (1982) and Chu et al. (1986).Following a brief review of the state-of-the-art methodologies for sim-ulating the train–bridge interactions, Diana and Cheli (1989) stud-ied the dynamic behavior of a train running over a long span bridge.By modeling a vehicle as a moving force or sprung mass, Cai et al.(1994) investigated the dynamic characteristics of single- and two-span beams subjected to vehicles moving at high speeds.

With the advancement in locomotive and control technologies,railway trains that have a design speed of 350 km/h or higher arenot uncommon nowadays. In the literature, a maximum speed of515.3 km/h has been reported during a field test (Delfosse, 1991).As far as high-speed trains are concerned, one needs to consider notonly the vibration amplitudes of the bridge, but also the riding com-fort of passengers carried by the trains, which can be assessed fromthe vertical or lateral accelerations of the moving vehicles (Dianaand Cheli, 1989; Yau et al., 1999). Due to the relatively stringentrequirements imposed on the allowable deflection of the bridge andon the riding comfort of moving vehicles, the design of high-speedrailway bridges is generally governed by the conditions of serviceabil-ity, rather than by strength and yielding, as learned from the designpractices in Taiwan.

Recently, the dynamic response of a typical bridge under the pas-sage of various commercial high-speed trains was studied by Hsu



(1996), Yau (1996), Chang (1997), and Wu (2000). In particular, theeffect of column stiffness on the dynamic response of bridges traveledby high-speed trains was studied by Hsu (1996), and the effects ofballast and elastic bearings by Yau (1996). It was demonstrated byYau et al. (2001) that the insertion of elastic bearings at the sup-ports of bridge girders for the purpose of isolating the earthquakeforces may adversely amplify the dynamic response of the beam tomoving train loads. Museros et al. (2002) investigated the influenceof sleepers and ballast layers, as well as train–bridge interactions, onthe response of short high-speed railway bridges. They concludedthat inclusion of these factors can result in smaller maximum dis-placements and accelerations on the bridge, compared with thoseobtained using barely the moving loads model. The mechanism in-volved in the phenomena of resonance and cancellation for elasticallysupported beams was further explored in the study by Yang et al.(2004).

Other related effects that have been investigated include the tor-sional vibration of bridges caused by vehicles moving along one ofthe two tracks on a bridge, the crossing of two vehicles moving inopposite directions, and the mass ratio of the railway vehicles to thebridge (Hsu, 1996). A parametric study was carried out by Shen(1996) on a number of factors affecting the dynamic response of thebridge, in which both the modal superposition method and finite el-ement method were employed. In the study by Wu et al. (2001), abridge containing two railway tracks was considered, with which twotrains are allowed to move over the bridge in opposite directions.Such a vehicle–rails–bridge interaction model was adopted by Wu(2000) and Yang and Wu (2002) in evaluating the risk of derailmentfor trains traveling over a bridge and simultaneously subjected to anearthquake excitation, and further by Wu and Yang (2003) in assess-ing the steady-state response and riding comfort of trains movingover a series of simply supported beams.

Based on an analytical approach, closed form solution has beenobtained by Yang et al. (1997b) for the response of simple beamssubjected to the passage of a high-speed train modeled as a sequenceof moving loads with regular nonuniform intervals (Fig. 1.5), in which


Introduction 15

Fig. 1.5. Train load model.

the conditions for the phenomena of resonance and cancellation tooccur have been identified. Based on these conditions, optimal designcriteria that are effective for suppressing the resonant response ofthe VBI systems have been proposed. The same problem was laterexamined by Li and Su (1999) by using an alternative analyticalapproach, with similar findings obtained.

1.5. Methods of Solution

In studying the dynamic response of a VBI system, two sets of equa-tions of motion can be written, one for the bridge and the other forthe vehicles. It is the interaction or contact forces existing at the con-tact points of the two subsystems that make the two sets of equationscoupled. As the contact points move from time to time, the systemmatrices are generally time-dependent, which must be updated andfactorized at each time step. To solve these two sets of equations,procedures of an iterative nature have often been used (Hwang andNowak, 1991; Green and Cebon, 1994; Yang and Fonder, 1996; Del-gado and dos Santos, 1997). One way to do this is to start by assum-ing some values of displacements for the contact points, with whichthe contact forces can be solved from the vehicle equations. Next, bysubstituting the contact forces into the bridge equations, improvedvalues of displacements for the contact points can be solved. The ad-vantage of the iterative procedures is that the responses of both thevehicles and the bridge at any instant can be simultaneously made



available. However, the convergence rate of iteration is likely to below, when dealing with the more realistic case of bridges traveledby a large number of vehicles, whether in a random traffic flow, asencountered in highways, or in a connected line, as encountered inrailways, for there exists twice the number of contact points if eachvehicle is assumed to consist of two sets of wheel assemblies.

In the literature, Lagrange’s equation with multipliers and con-straint equations has also been applied to the analysis of VBI systems(Blejwas et al., 1979). As it is well known, the use of Lagrange mul-tipliers will increase the total number of unknowns and, therefore,the effort of computation. Theoretically speaking, a third approach isstill possible, namely, by eliminating the interaction or contact forcesfrom the original two sets of equations, one can form a new set ofcoupled equations for the entire VBI system. However, if the conden-sation procedure is performed on the structure level, the symmetryand other advantageous properties of the dynamic matrices associ-ated with each subsystem will be destroyed (Yang and Lin, 1995).

Perhaps, one of the most efficient approaches for solving the VBIequations is to perform the condensation technique on the elementlevel. Garg and Dukkipati (1984) used the Guyan (1965) reduc-tion scheme to condense the DOFs of the vehicles to those of thebridge. Recently, Yang and Lin (1995) used the dynamic condensa-tion method to eliminate all the DOFs associated with each vehicleon the element level. Such approaches are good if only the responseof the bridge (the larger subsystem) is desired. They may not yieldaccurate solutions for the response of the moving vehicles (the smallersubsystem), due to the approximations adopted in relating the vehi-cle (slave) DOFs to the bridge (master) DOFs.

Other methods that have been employed in solving the second-order differential equations of motion of the VBI problems include:(1) the direct integration methods, such as Newmark’s β method(1959) (Inbanathan and Wieland, 1987; Yang and Lin, 1995),Wilson’s θ method (Sridharan and Mallik, 1979), and fourth-orderRunge–Kutta method (Chu et al., 1986); (2) the modal superpositionmethod (Blejwas et al., 1979; Wu and Dai, 1987; Galdos et al., 1993;Cai et al., 1994), along with various integration schemes; and (3) the


Introduction 17

Fourier transformation method (Green and Cebon, 1994; Chang andLee, 1994).

One essential feature with the VBI problem is that the numberof vehicles acting on the bridge is time-dependent. The more thenumber of vehicles simultaneously acting on the bridge, the higheris the level for the vehicles to interact with the bridge. To overcomethe dependency of the system matrices on the wheel load positions,i.e., the contact point positions, one feasible approach is to eliminatethe DOFs of the vehicles not in direct contact with the bridge onthe element level by the method of dynamic condensation (Yang andLin, 1995). This will result in a VBI element that takes into accountall the coupling effects. The following is a summary of the procedurepresented by Yang and Yau (1997) for deriving the VBI element.

Consider a beam simulated by a number of elements traversed bya train, of which each railroad car is idealized as two lumped masses,each supported by a spring-dashpot unit, as shown in Fig. 1.6. For

(a)

(b)

Fig. 1.6. Train–bridge system: (a) general model and (b) sprung mass model.



Fig. 1.7. Vehicle–bridge interaction element.

the present purposes, an interaction element is defined such that itconsists of a beam element and a car-body mass and wheel massconnected by a suspension (spring-dashpot) unit directly acting overthe beam element (Fig. 1.7). For the parts of the beam that are notdirectly under the action of the railroad cars, they can be modeled byconventional beam elements. However, for the remaining parts thatare in direct contact with the wheel loads, the interaction elementshave to be used instead.

With reference to the interaction element shown in Fig. 1.7, twosets of equations of motion can be written, one for the beam elementand the other for the sprung mass unit. By Newmark’s single-step fi-nite difference formulas, the sprung mass equation can be discretizedin time domain, from which the vehicle DOFs can be solved. Further,by the method of dynamic condensation, the sprung mass DOFs canbe condensed to the associated DOFs of the beam element in con-tact. This will result in a VBI element with the effect of interactionfully taken into account. Since the VBI element possesses the samenumber of DOFs as the parent element, while retaining the prop-erties of symmetry and bandedness in element matrices, it can bedirectly assembled with the conventional beam elements to form the


Introduction 19

structure equations. Such an element is particularly suitable for mod-eling bridges under a series of moving sprung masses with constantor varying intervals. It has the advantage that the response of thesprung mass can be recovered at any time step of the time-historyanalysis, which serves as a measure of the passengers’ riding comfort.Using the VBI element, various dynamic properties of the beam andvehicles can be considered in the formulation, including the rail ir-regularities, ballast stiffness, damping of the beam, and stiffness anddamping of the suspension units.

Recently, a more versatile approach for dealing with the VBI prob-lems was proposed by Yang and Wu (2001). This method hinges oncomputation of the contact forces from the vehicles equations, interms of the contact displacements. Before this can be done, thevehicle equations should first be discretized in time domain, say, us-ing finite-difference equations of the Newmark type. The contactforces solved from the vehicles can then be treated as external loadsand transformed as consistent nodal loads onto the bridge structure.With the bridge equations discretized in time domain, the bridgedisplacements can be solved as well. Such a procedure has beendemonstrated to be quite flexible for treating vehicles of various com-plexities that appear in a sequence, in which both the vertical andhorizontal contact forces are involved.

1.6. Impact Factor and Speed Parameter

In design practice, the dynamic response of a bridge has been con-sidered indirectly by increasing the forces and stresses caused by thestatic live loads by an impact factor, defined as the ratio of the max-imum dynamic to the maximum static response of the bridge underthe same load minus one. One typical definition for the impact factorI is (Yang et al., 1995):

I =Rd(x) − Rs(x)

Rs(x), (1.1)

where Rd(x) and Rs(x) denote respectively the maximum dynamicand static responses of the bridge calculated at the cross section x of



the bridge of interest. The responses that may be considered for abeam include the deflection, bending moment, and shear force. Thedefinition given in Eq. (1.1) is more rational and computationallymore convenient than the dynamic increment factor (DIF) suggestedby the AASHTO (Guide, 1980; Galdos et al., 1993), or the dynamicamplification factor (DAF) discussed in Paultre et al. (1992) andZhang et al. (2001b), since both the maximum dynamic and staticresponses are calculated at the same cross section of the bridge. Suchan advantage will become obvious when dealing with moving loadsthat appear as a sequence, such as those of a train, or as a randomflow, such as those encountered in highways, or in treating problemsinvolving the resonance response, in the sense that the response ofthe bridge at the same cross section will be continuously amplified,as there are more loads passing the bridge. Care must be takento distinguish the maximum impact factor from the maximum totalresponse calculated for a beam. Occasionally, unreasonably largeimpact factors may be computed for a beam at some points due to thefact that the static responses, i.e., the denominator of Eq. (1.1), arevery small. For this reason, the impact factor computed or measuredfor a bridge should not be regarded as the only criterion in the designof bridges.

It is well known that a number of factors may affect the im-pact factor of a bridge under the excitation of moving vehicularloads, for instance, the dynamic properties of the vehicle, the dy-namic properties of the bridge, the vehicle speed, and the pavementroughness. Many bridge codes, including the American Associationof State Highway and Transportation Officials (AASHTO) Specifica-tions (Standard, 1989) and the Ontario Code (Ontario, 1983), haverelated the impact factor to a single parameter of the bridge, such asthe span length or frequency of vibration, and have applied the sameimpact factor to all responses of the bridge including the deflection,shear force, and bending moment. According to the AASHTO Spec-ifications, the impact factor I is related to the span length L of thebridge as

I =50

L + 125≤ 0.3 (1.2)


Introduction 21

when the bridge span length L is expressed in ft, and as

I =15.24

L + 38.1≤ 0.3 (1.3)

when L is expressed in m. Formulas such as the preceding ones havebeen established several decades ago, based on limited field mea-surements, which are valid for the particular types of vehicles andbridges available in those days, if one realizes that modern trucksused are much heavier than those used half a century ago. The pre-ceding formulas may be convenient for practical design, but are nottheoretically sound at least for two reasons. First, the formulas areinconsistent in physical units, if one notes that the impact factor I

itself is a nondimensional quantity, while the span length has somephysical units. Second, the use of span length as the control pa-rameter is not representative of the physical property of the bridgeconcerning the vehicle–bridge interactions. This is especially true forcontinuous beams, for which there exist more than one span lengths,none of which can be directly related to the modal vibration shape.Based on the evidence of more extensive theoretical analyses andfield measurements, it was reported that impact factors calculatedaccording to current design codes may significantly underestimatethe bridge response in many cases (O’Connor and Pritchard, 1984;Inbanathan and Wieland, 1987; Galdos et al., 1993; Huang et al.,1993; Chang and Lee, 1994).

By denoting the velocity of the moving vehicle as v and the char-acteristic length of the beam as L, the exciting frequency of themoving vehicle can be expressed as πv/L. The speed parameter S

that is particularly useful for expressing the dynamic response of theVBI system is defined as the ratio of the exciting frequency of themoving vehicle to the fundamental frequency ω of the beam, that is,

S =πv

ωL(1.4)

which is dimensionless. In the study by Yang et al. (1995), it wasdemonstrated that for VBI systems with a speed parameter S lessthan 0.5, the impact factor can be related to the speed parameterfor the deflection, shear force, and bending moment of simple beams



by linear formulas. Moreover, these formulas can be multiplied bysome reduction coefficients to yield formulas for continuous beams.One particular point here is that the characteristic length, ratherthan the span length, should be used for the beam in defining thespeed parameter. The former relates to the span length of a simplebeam or the distance between two adjacent inflection points of thefirst mode of vibration of a continuous beam. It is with the use ofthe characteristic length in Eq. (1.4) that simple impact formulascan be established for both the simple and continuous beams (Yanget al., 1997a). In the study by Pan and Li (2002), it was shown thatthe maximum displacement, velocity, and acceleration responses ofthe supporting structure appear to be almost linear to the speedparameter, which is consistent with the findings of Yang et al. (1995).

The speed parameter is an important parameter in the study ofmoving load problems, see, for instance, Tan and Shore (1968a,b),Veletsos and Huang (1970), Warburton (1976), Kurihara and Shi-mogo (1978b), Wang et al. (1992), Humar and Kashif (1993), Caiet al. (1994), and Chatterjee et al. (1994), among others. It wasdemonstrated that by plotting the response of the bridge against thespeed parameter, rather than the span length or frequency of vibra-tion that make up the parameter, generally more compact resultscan be obtained (Yang et al., 1995). It was also noted by Paultreet al. (1992) that the DAF generally increases with the speed pa-rameter. Noteworthy is the fact that the nondimensional characterof the speed parameter enables us to extend the range of applicationof the computed or measured results to bridges beyond those coveredin the study. Nevertheless, such a property was not fully recognizedby a substantial portion of researchers working on vehicle-inducedvibrations.

1.7. Concluding Remarks

A brief review of previous researches on the dynamic interaction ofthe bridge and moving vehicles was presented in this chapter. Vehiclemodels of increasing complexities, including the moving load, movingmass, sprung mass models, and more sophisticated ones, have been


Introduction 23

discussed. The factors that need to be considered in analyzing theresponse of the VBI systems include the dynamic properties and driv-ing frequencies of the moving vehicles, and the dynamic propertiesand surface roughness of the bridge. Even though vehicle models ofhigher complexities, e.g., those consisting of dozens of DOFs, can beemployed in studying the VBI problems nowadays, the use of simpli-fied vehicle and bridge models is helpful, since it allows us to identifythe key parameters dominating the dynamics of the VBI systems.

The impact factor adopted herein is computed based on quantitiesrelated to the same cross section of the beam that is of interest. Itcan be conveniently applied to cases involving a series of movingloads. The impact formulas provided by most current design codesare not consistent in physical units and lack a solid theoretical basis,of which the application should not be extended to bridges traveledby vehicles at high speeds. A more rational approach is to relatethe impact factor, which is nondimensional, to the speed parameter,which is also nondimensional, defined as the ratio of the drivingfrequency of the moving vehicles to the vibration frequency of thebridge.

The VBI problem is a complicated one in that the contact pointsof the vehicles with the bridge move from time to time. Variousmethods exist for solving this problem. However, the most effectiveone appears to be the one based on condensation of the noncontactDOFs of the vehicle to the beam element in contact. The VBI el-ement so derived can be applied to solving a great variety of VBIproblems, by which the dynamic response of the moving vehicles,in addition to that for the bridge, can be obtained. Other factorsthat require further studies for high-speed railway bridges includethe braking and acceleration of railroad cars, the torsional vibrationof bridges caused by vehicles not moving along the centerline of thebridge girders, the crossing of two vehicles moving in opposite direc-tions, the mass ratio of the vehicles to the bridge, the stability ofrails, the risk of derailment of railroad cars under earthquake mo-tions, and the stiffness effects of the ballast, elastic bearings, andsupporting columns, among others.



Part I

Moving Load Problems

25



Chapter 2

Impact Response ofSimply-Supported Beams

The most fundamental problem that should be considered in thestudy of vehicle-induced vibrations on bridges is the dynamic re-sponse of a simply-supported beam subjected to a single movingload. This problem is important in that the solution can be given inclosed form. In this chapter, impact formulas will be derived for thedeflection, bending moment and shear force of a simple beam undera single moving load. By the principle of superposition, the solutionobtained for a single moving load will be expanded to deal with a se-ries of identical, equi-distant moving loads, by which the key parame-ters dominating the dynamic response of the beam can be identified.Furthermore, based on the conditions of resonance or cancellationfor the waves generated by a series of moving loads, optimal designcriteria for suppressing the resonant response of beam structures willbe presented. In designing a high-speed railway bridge, such criteriaare useful for the selection of the span length and cross section of agirder bridge, if the car length, axle distance and operating speed ofthe train are already decided.

2.1. Introduction

Two effects are associated with the motion of a vehicle over a bridge,i.e., the gravitational effect and the inertial effect, both related tothe mass of the vehicle. For the cases where the mass of the vehicleis small compared with that of the bridge, the vehicle can be rep-resented as a concentrated load, with the inertial effect neglected.

27



This is the so-called moving load model, the simplest case that canbe conceived of a moving vehicle. One advantage of using such amodel is that for some special structures, e.g., for simply-supportedbeams, closed-form solutions can be obtained, from which the keyparameters dominating the dynamic response of the supportingstructure can be identified. It is from this consideration that themoving load problem for simple beams deserves a special and in-depth treatment.

The dynamic response of bridges subjected to the passage of mov-ing vehicles continues to be a subject of great interest to struc-tural engineers. In early studies, a bridge has been modeled as abeam-like structure and a vehicle as a moving load or moving mass(Timoshenko, 1922; Jeffcot, 1929; Lowan, 1935; Biggs, 1964; Fryba,1972). Such a model was adopted in later studies including thoseof Warburton (1976), Stanisic (1985), Sadiku and Leipholz (1987),and Akin and Mofid (1989). In the meanwhile, more delicate vehicleand bridge models that consider the effects of multi-axle loadings,multi-lane loadings, vehicle suspension, surface roughness, etc. havebeen developed for the analysis of bridge response (Chu et al., 1986;Inbanathan and Wieland, 1987; Galdos et al., 1993; Huang et al.,1993; Humar and Kashif, 1993; Chang and Lee, 1994; Yang and Lin,1995; Yang et al., 1995). In later chapters, it will be demonstratedthat the response of a vehicle–bridge interaction (VBI) system ob-tained by using more sophisticated models for the moving vehiclesremains in essence dominated by the key parameters identified fromthe analysis based on the moving load model.

A review of the research works cited above indicates that mostof them consider only the case of a single or very small number ofmoving vehicular loads. In comparison, rather few research has beenconducted on the dynamic response of bridges under the action ofa series of moving loads, to represent the continuous action of thewheels of a moving train on the bridge. Bolotin (1964) studied abeam subjected to an infinite sequence of equal loads with uniformintervals d and constant speed v. In his study, the period d/v of themoving loads has been identified as a key parameter. For the sameproblem, Fryba (1972) concluded that the response of the forced


Impact Response of Simply-Supported Beams 29

steady-state vibration will attain its maximum when the time inter-vals between two successive moving loads are equal to some periodsof the beam in free vibration or to an integer multiple thereof. Kuri-hara and Shimogo (1978a,b) investigated the vibration and stabilityproblems of a simple beam subjected to a series of discrete loads withrandom intervals.

Several features have to be considered in the design of bridges forhigh-speed railways. First, the moving loads acting on the bridge arenot random in nature, as encountered in highway bridges, but are ofregular nonuniform intervals in general. Second, compared with thecar length of a train, which ranges normally from 18 to 26 m, the spanlength of the elevated bridges constructed as part of the railway linesin most metropolitan areas is usually not long, which may vary from10 to 40 m. Finally, because of the rather high operating speed of thetrain, e.g., with a maximum speed ranging from 250 to 350 km/hr,and because of the repetitive nature of wheel loads, it is likely thatthe condition of resonance be excited on high-speed railway bridges.

In this chapter, only bridges that can be modeled as a simplebeam will be considered, which is the most common type of bridgesused in railways due to its relative ease in construction and otherconsiderations. The problem of a simple beam subjected to a sin-gle moving load will first be investigated in Sec. 2.2. Based on theanalytical solution given in Sec. 2.2, impact factor formulas for themidpoint deflection and bending moment of the simple beam will bepresented in Secs. 2.3 and 2.4, respectively, followed by the impactformulas for the end shear force in Sec. 2.5. The solution presentedin Sec. 2.2 will be expanded to deal with the case of a simple beamunder the passage of a train in Sec. 2.6, by modeling a train as thecomposition of two subsystems of wheel loads of constant intervals,with one subsystem consisting of all the front wheel assemblies andthe other the rear assemblies. Of interest herein is the identificationof the conditions for the phenomena of resonance and cancellationto occur on the beam, along with the optimal design criteria for theVBI system. Some illustrative examples will be presented in Sec. 2.7,followed by the concluding remarks in Sec. 2.8. The materials pre-sented in Secs. 2.2–2.5 were rewritten from the paper by Lin et al.



(2000) and those in Secs. 2.6 and 2.7 were modified from the paperby Yang et al. (1997b).

2.2. Simple Beam Subjected to a Single Moving Load

A simply-supported beam traversed by a vehicle that is modeled asa concentrated load p of speed v is shown in Fig. 2.1. The followingassumptions will be adopted in this study: (1) The beam is homoge-neous and of constant cross sections, for which the Bernoulli–Eulerhypothesis of plane cross sections remain plane after deformation ap-plies; (2) only a single moving vehicle is allowed to travel on the beamat a time; (3) only the gravitational effect of the vehicle is consid-ered, while the inertia effect of the vehicle is neglected, assumed tobe small compared with that of the bridge; (4) the vehicle moves ata constant speed v; (5) the damping of the beam is of the Rayleightype; (6) the beam is initially at rest before the vehicle moves in;and (7) no consideration is made of the road surface roughness ofthe bridge.

As shown in Fig. 2.1, a simple beam is subjected to a load ofmagnitude p moving at speed v. Here, we shall use u(x, t) to denotethe deflection of the beam along the y axis at position x and timet, L the length of the beam, m the mass per unit length, ce theexternal damping coefficient, ci the internal damping coefficient, E

the modulus of elasticity, and I the moment of inertia of the beam.Based on the aforementioned assumptions, the equation of motion of

Fig. 2.1. A simply-supported beam subjected to a moving load.



the beam can be written as

mu + ceu + ciIu′′′′ + EIu′′′′ = pδ(x − vt) , 0 ≤ vt ≤ L , (2.1)

where primes (′) and dots (˙) denote differentiation with respectto coordinate x and time t, respectively, and δ is the Diracdelta function. For the beam with simple supports, the boundaryconditions are

u(0, t) = 0

u(L, t) = 0

EIu′′(0, t) = 0

EIu′′(L, t) = 0

(2.2)

and the initial conditions are

u(x, 0) = 0

u(x, 0) = 0(2.3)

as the beam is assumed to be at rest prior to the arrival of the movingvehicle.

Let φn denote the nth vibration mode of the beam that satisfiesthe boundary conditions. The deflection of the beam u(x, t) due toonly the nth mode of vibration is

u(x, t) = φn(x)qn(t) , (2.4)

where qn(t) is the generalized coordinate corresponding to the nthmode. Substituting Eq. (2.4) into Eq. (2.1), multiplying both sidesof the equation by φn, and integrating with respect to x over thelength L of the beam, one obtains

mqn(t)∫ L

0[φn(x)]2dx + qn(t)

×

ce

∫ L

0[φn(x)]2dx + ciI

∫ L

0φ〈4〉

n (x)φn(x)dx

+ EIqn(t)∫ L

0φ〈4〉

n (x)φn(x)dx = pφn(vt) , (2.5)



where it is realized that∫ L

0δ(x − a)φn(x)dx = φn(a) . (2.6)

Let us denote the vibration frequency ωn of the nth mode of thebeam as

ω2n =

EI

m

∫ L0 φ

〈4〉n (x)φn(x)dx∫ L

0 [φn(x)]2dx. (2.7)

We shall also let ce = αem, ci = αiE, and define the damping coeffi-cient ξn of the nth mode of vibration as

ξn =12

(αe

ωn+ αiωn

). (2.8)

Consequently, Eq. (2.5) reduces to

qn + 2ξnωnqn + ω2nqn =

pφn(vt)∫ L0 m[φn(x)]2dx

. (2.9)

This is exactly the equation of motion for the nth mode of vibration,in terms of the generalized coordinate qn, which is valid only whenthe acting position vt of the moving load is located within the rangeof the beam, i.e., 0 ≤ vt ≤ L. Once the moving load leaves the beam,only free oscillation remains.

For a simply-supported beam, the nth modal shape of vibration is

φn(x) = sinnπx

L(2.10)

and the frequency of vibration ωn obtained from Eq. (2.7) is

ωn =n2π2

L2

√EI

m. (2.11)

Substituting the preceding expression into Eq. (2.9) yields the equa-tion of motion for the nth mode of the simply-supported beam as

qn + 2ξnωnqn + ω2nqn =

2pmL

sinnπvt

L, (2.12)



which is uncoupled from the other modes of vibration. From thisequation, the generalized coordinate qn for the nth mode can besolved as

qn =2pL3/(EIn4π4)

(1 − S2n)2 + 4(ξnSn)2

×

(1 − S2n) sin Ωnt − 2ξnSn cos Ωnt

+ e−ξnωnt

[2ξnSn cos ωdnt +

Sn√1 − ξ2

n

(2ξ2n + S2

n − 1) sin ωdnt

],

(2.13)

where ωdn is the damped frequency of vibration of the beam,

ωdn = ωn

√1 − ξ2

n . (2.14)

Ωn is the exciting frequency implied by the moving load,

Ωn =nπv

L(2.15)

and Sn is a nondimensional speed parameter defined as the ratio ofthe frequency of excitation of the moving load to the nth frequencyof vibration of the beam, i.e.,

Sn =Ωn

ωn=

nπv

ωnL. (2.16)

Consequently, the total displacement u(x, t) of the beam caused byall the vibration modes can be summed as follows:

u(x, t) =∞∑

n=1

2pL3/(EIn4π4)(1 − S2

n)2 + (2ξnSn)2

×

(1 − S2n) sin Ωnt − 2ξnSn cos Ωnt + e−ξnωnt

×[2ξnSn cos ωdnt +

Sn√1 − ξ2

n

(2ξ2n + S2

n − 1) sin ωdnt

]

× sinnπx

L. (2.17)



This is exactly the displacement of the beam caused by a single mov-ing load with taking into account the effect of damping. In Eq. (2.17),the terms with Ωnt represent the forced vibration of the bridge in-duced by the moving load, and the terms with ωdnt are the freevibration, which will eventually be damped out. Again, this equa-tion applies only when the acting position vt of the moving load islocated within the range of the beam.

For a wide class of moving load problems encountered in practice,the effect of damping on the bridge is so small, due to the rather shortacting time of the moving loads that it can be ignored completely.This is especially true, if one is interested in the response of the bridgeof the first few cycles. By neglecting the effect of damping, the totaldisplacement u(x, t) of the beam as given in Eq. (2.17) reduces to

u(x, t) =2pL3

EIπ4

∞∑n=1

1n4

sinnπx

L

(sin Ωnt − Sn sin ωnt

1 − S2n

). (2.18)

This is exactly the deflection of the simple beam at section x causedby the moving load p acting at position vt, with the effect of dampingneglected. Correspondingly, the bending moment M(x, t) causedby the moving load p on the beam can be computed as M(x, t) =−EIu′′(x, t), or

M(x, t) =2pL

π2

∞∑n=1

1n2

sinnπx

L

(sin Ωnt − Sn sinωnt

1 − S2n

), (2.19)

and the shear force is V (x, t) = EIu′′′(x, t) or

V (x, t) =2pπ

∞∑n=1

1n

cos(nπx

L

)(sinΩnt − Sn sin ωnt

1 − S2n

). (2.20)

Here, it should be noted that the solutions obtained above for theproblem considered are not new in the literature, see, for instance,Biggs (1964) and Warburton (1976). However, most of the previousresearchers have not proceeded further to derive impact formulasfrom these solutions, which are more useful to practicing engineers.

For the present purposes, let us consider a simple beam of lengthL = 20 m, per unit mass m = 3000 kg/m, flexural rigidity EI = 106



N-m2 and a damping coefficient ξ of 2.5% for all vibration modes,subjected to a moving load p = 6 kN of speed v = 27.8 m/s(100 km/hr). The displacements for the midpoint of the beam ob-tained from Eqs. (2.17) and (2.18) for the damped and undampedcases have been compared in Fig. 2.2. As can be seen, the effectof damping on the response of the beam during the action of themoving load is rather small. For this reason, the effect of dampinghas often been neglected in research concerning the vehicle-inducedvibrations on bridges.

To illustrate the effect of multi-modes of vibration, for the threespeeds of S1 = 0.05, 0.01 and 0.25, the deflections of the midpoint ofthe beam obtained from Eq. (2.18) for the undamped case consideringvarious numbers of vibration modes have been plotted in Fig. 2.3.As can be seen, the result obtained for the midpoint deflection byconsidering only the first mode is good enough, partly due to the factthat all the anti-symmetric modes of vibration contribute nothing tothe midpoint deflection. This gives us the impression that usingonly the first mode can yield generally good approximate solutionsfor vehicle-induced response, especially when the midpoint deflection

Fig. 2.2. The effect of damping of the beam.



Fig. 2.3. The effect of multi-modes on midpoint deflection of the beam.

of the beam is desired. Such an approximation has been previouslyadopted by a number of researchers in their analytical studies.

2.3. Impact Factor for Midpoint Displacement

In this section, the responses derived in Sec. 2.2 for a single mov-ing load will be used to derive the impact formula for the midpointdisplacement of the simply-supported beam. The results presentedin this section cover a wide range of applications, as they are allexpressed in terms of the nondimensional speed parameter. Theyalso serve as a useful reference for comparison with other results.It is realized that the impact response induced by a single movingload on the beam is generally larger than that induced by multi orcontinuously moving loads due to the suppression effect of the simul-taneous acting loads. Thus, the impact formulas presented in thischapter for a single moving load should be regarded as reasonableupper bounds for the responses considered. The other message toconvey here is that the impact factors for the deflection, bendingmoment and shear force can be quite different, and that the use of



identical impact formulas for all physical quantities, as implied bymost design codes, is not theoretically sound.

Here, we shall adopt the definition given in Eq. (1.1) for the im-pact factor I,

I =Rd(x) − Rs(x)

Rs(x), (2.21)

where Rd(x) and Rs(x) respectively denote the maximum dynamicand static response of the bridge at section x due to passage of themoving load. For a simple beam, both the maximum dynamic andstatic deflections occur at the midpoint. The following is the maxi-mum static deflection of the beam under the static load p:

Rsu

(L

2

)=

pL3

48EI. (2.22)

In contrast, the dynamic response for the midpoint deflection of thebeam can be obtained from Eq. (2.18) by setting x = L/2, that is,

u

(L

2, t

)=

2pL3

EIπ4

∞∑n=1

1n4

sinnπ

2

(sin Ωnt − Sn sin ωnt

1 − S2n

), (2.23)

where it is realized that Ωn = nπv/L and ωn = nπv/(SnL). Thepreceding equation is valid only when the acting position vt of themoving load p is located within the span of the beam. For n =2, 4, 6, . . ., the shape function sin(nπ/2) vanishes at the midpoint,as it turns out to be asymmetrical. Thus, only the modes withn = 1, 3, 5, . . ., i.e., the symmetrical modes, contribute to deflectionof the midpoint.

Correspondingly, the impact factor for the midpoint deflectionof the simple beam caused by the moving load p acting at positionvt is

Iu =96π4

∞∑

n=1,3,5...

1n4

sinnπ

2

(sinΩnt − Sn sin ωnt

1 − S2n

)− 1 , (2.24)

which is independent of the magnitude p of the moving load. As canbe seen, the contribution of higher order terms decreases by a factorn−4. It follows that the effect of higher order terms in Eq. (2.24) can



be neglected without losing accuracy. By the relation 96/π4 ∼= 1,Eq. (2.24) reduces to

Iu∼=(

sin Ω1t − S1 sin ω1t

1 − S21

)− 1 . (2.25)

The impact factors calculated for the midpoint displacement usingEqs. (2.24) and (2.25), considering only the contribution of eithermulti-modes or the first mode, have been plotted in Fig. 2.4. As canbe seen, the midpoint displacement impact response of the simplebeam is dominated by the first mode.

To illustrate the effect of the acting position vt of the moving load,different values have been assumed for vt, i.e., vt/L = 1/8, 1/4, 3/8,1/2, 5/8, 3/4, and 7/8, in calculation of the impact factor Iu forthe midpoint displacement of the simple beam in Fig. 2.5. As canbe seen, there exists an upper-bound envelope for the displacementimpact factor Iu. The maximum impact factor Iu can be regardedas proportional to the speed parameter S1 for vehicle speeds in therange S1 < 0.5, and as constant for S1 ≥ 0.5. Based on such anobservation, the following formulas can be proposed for the midpointdeflection:

Iu =

1.54S1 for S1 < 0.5 ,

0.77 for S1 ≥ 0.5 .(2.26)

In arriving at Eq. (2.26), the effect of damping of the beam has beenneglected. If the effect of damping of the beam has been considered,slightly smaller impact factors can be derived.

With regard to the impact formula presented above, several com-ments can be made here. First, the present impact formula has beenexpressed as a function of the speed parameter S1 of the moving ve-hicle, which is physically more meaningful, compared with formulasusing the frequency of vibration ω or the span length L of the beamas the key parameter, e.g., the one recommended by AASHTO inEqs. (1.2) or (1.3). Second, since the speed parameter S1, as definedin Eq. (2.16), is nondimensional, the impact formula derived hereinbased on this parameter remains valid for a wide range of simplebeams subjected to vehicles moving at various speeds. Third, for



Fig. 2.4. The effect of multi-modes on impact factor — midpoint deflection.

Fig. 2.5. The effect of loading positions — midpoint deflection.



most girder bridges with a span length no shorter than 30 m andtraversed by vehicles moving at speeds no greater than 140 km/hr,the speed parameter S1 computed is generally less than 0.5 (Yanget al., 1995). In other words, the upper bound presented for therange with S1 ≥ 0.5 should find its application for bridges withrather short span lengths and/or subjected to vehicles moving atextremely high speeds. Finally, the impact factors calculated usingthe present formula show a trend similar to, but numerically largerthan, those based on the finite element analysis (Yang et al., 1995).Such a difference can be attributed to the use of more complicatedvehicle and bridge models in the latter. For instance, in the studyby Yang et al. (1995), a five-axle tractor-trailer has been selected asthe target vehicle, and modeled as three lumped masses each restingon a spring-dashpot unit. Moreover, the bridge is represented bya number of finite elements, allowing the effects of multi-vibrationmodes to be considered. Compared with the finite element analysisresults, it is believed that the present formula is generally on theconservative side.

2.4. Impact Factor for Midpoint Bending Moment

Conventionally, the same impact factor formula has been used forthe deflection, bending moment and shear force of a bridge underthe moving loads. Such an approach has the advantage of beingconvenient, but may not be accurate enough. In this section, anindependent impact factor formula will be derived for the midpointbending moment of the simple beam. The bending moment M(x, t)acting at section x of the simple beam with zero damping due toa single moving load p of velocity v at time t has been given inEq. (2.19). The bending moment at the midpoint of the beam canbe evaluated as

M

(L

2, t

)=

2pL

π2

∞∑n=1

1n2

sinnπ

2


1 − S2n

), (2.27)

where it is realized that Ωn = nπv/L and ωn = nπv/(SnL). Cor-respondingly, the maximum bending moment of the simple beam



subjected to a static load p acting at x = L/2, i.e.,

RsM

(L

2

)=

pL

4. (2.28)

It follows that the impact factor IM for the midpoint bending mo-ment of the beam subjected to a moving load p acting at positionvt is

IM =8π2

∞∑

n=1,3,5...

1n2

sinnπ

2


1 − S2n

)− 1 . (2.29)

By comparing Eq. (2.29) with Eq. (2.24), one observes that the con-vergence rate for the bending moment impact factor IM , as repre-sented by the factor n−2, is slower than that for the displacementimpact factor Iu. Thus, the effect of higher modes of vibration onthe bending moment may not be as small as that for the midpointdisplacement. To investigate such an effect, the impact factors IM

computed for the midpoint bending moment from Eq. (2.29) usingeither a single mode or multi-modes have been plotted in Fig. 2.6,along with the finite element results. As can be seen, the effect ofhigher modes appears to be quite significant. For this reason, a totalof seven modes, i.e., with n = 7, will be included in this study incomputing the impact factor for bending moment.

To illustrate the effect of the acting position vt of the moving load,different values have been assumed for vt, i.e., vt/L = 1/8, 1/4, 3/8,1/2, 5/8, 3/4, and 7/8, in computing the impact factor IM for themidpoint bending moment using Eq. (2.29), where it is noted thatΩn = nπv/L and ωn = nπv/(SnL). As can be seen from Fig. 2.7, anupper-bound envelope exists for the impact factor IM through therange of speed parameter S1 considered, that is, for S1 < 0.36, themaximum impact factor IM is proportional to the speed parameterS1, and for S1 ≥ 0.36, it is a constant,

IM =

1.24S1 for S1 < 0.36 ,

0.45 for S1 ≥ 0.36 .(2.30)



Fig. 2.6. The effect of multi-modes on impact factor — midpoint moment.

Fig. 2.7. The effect of loading positions — midpoint moment.



In Fig. 2.7, the solutions obtained by the independent finite elementanalyses were also plotted, which appear to be well represented bythe formulas proposed. It should be noted that in deriving the pre-ceding impact factor formula, the effect of damping of the beam hasbeen ignored. Moreover, a comparison of Eq. (2.30) with Eq. (2.26)indicates that for simple beams, the impact effect caused by a mov-ing load on the bending moment is generally smaller than that onthe displacement.

2.5. Impact Factor for End Shear Force

From the shear force V expression in Eq. (2.20), one observes that theshear force V vanishes at the center point, at x = L/2, of the simplebeam, as indicated by the term cos(nπx/L). Thus, it is meaninglessto compute the impact factor IV for the shear force at the midpointof the beam. In this section, the shear impact factor IV will becomputed for a point very close to, but not at, the right end of thebeam. Note that similar result can be derived from the left end ofEq. (2.20). For a simple beam subjected to a moving load p, themaximum static shear force is RsV = p, which occurs when the loadacts on one end of the beam. Consequently, the shear impact factorIV at a point near the right end is

IV =2π

∣∣∣∣∣∞∑

n=1

1n

cos nπ

(sin Ωnt − Sn sinωn

1 − S2n

)∣∣∣∣∣− 1 , (2.31)

where it is noted that Ωn = nπv/L and ωn = nπv/(SnL). Comparedwith the impact factors Iu and IM given in Eqs. (2.24) and (2.29), thepreceding series expression for IV converges at a rate much slowerthan those for Iu and IM , as represented by the term n−1, whichindicates that the influence of higher-order terms can hardly be ne-glected in computing the impact factor for the end shear force. Forthis reason, a total of 300 terms, i.e., with n = 300, will be includedin computation of the end shear force in this section.

To illustrate the effect of the acting position vt of the moving load,different values have been assumed for vt, i.e., vt/L = 1/8, 1/4, 3/8,



Fig. 2.8. The effect of loading positions — end shear force.

1/2, 5/8, 3/4, and 7/8, in calculation of the impact factor IV for thenear-support shear force of the simple beam using Eq. (2.31). Fromthe results plotted for IV in Fig. 2.8, which have been obtained with300 terms for each of the curves shown, it is obvious that the upper-bound envelope for the impact factor IV can be very well representedby a straight line as

IV = 1.4S1 (2.32)

based on the assumption of zero damping for the beam. It is interest-ing to note that the result obtained by an independent finite elementanalysis using a more sophisticated vehicle model, also plotted inFig. 2.8, is in good agreement with the present ones.



2.6. Simple Beam Subjected to a Series ofMoving Loads

In this section, a train is modeled as the composition of two sets ofmoving loads of constant intervals, with the first set representing thewheel loads of all the front wheel assemblies and the second set therear ones. To simplify the derivation that leads to closed solutions,no consideration will be made of the variations in rail elevations, alsoknown as track irregularities, or any other initial conditions of thebeam.

2.6.1. Modeling of Wheel Loads of a Train

Figure 2.9 shows a simply-supported beam of length L traveled bya train at speed v, which consists of a number of identical cars oflength d. Only the flexural vibration of the beam will be consideredherein. This means that the train is assumed to travel along thecenterline of the beam, with no consideration made for the torsionalaction. As a first approximation, the train will be simulated as aseries of lumped loads p of constant intervals d moving at speed v,as shown in Fig. 2.10(a). The corresponding load function F (t) canbe given as

F (t) =N∑

j=1

p · Uj(t, v, L) , (2.33)

where

Uj(t, v, L) = δ[x− v(t− tj)] ·[H(t − tj) − H

(t − tj − L

v

)]. (2.34)

Here, δ denotes the Dirac delta function, x the coordinate of thebeam, H(•) a unit step function, tj the arriving time of the jth loadat the beam, tj = (j − 1)d/v, and N the total number of movingloads considered. Obviously, the action of the jth moving load isturned on by the term H(t− tj) when it enters the beam, and turnedoff by the term H(t − tj − L/v) when it leaves the beam.



Fig. 2.9. The simply-supported beam under a moving train.

(a)

(b)

Fig. 2.10. The simple beam subjected to: (a) uniform loads and (b) train loads.

The train is assumed to have N identical cars, and each car issupported by two bogies, each of which in turn is supported by twowheelsets. In this chapter, each bogie and the associated wheelsetswill be grossly referred to as a wheel assembly. Other kinds of



arrangement for the wheel assemblies that are different from the oneconsidered herein are possible. But in this section, we shall considerthe support mechanism of a train car only up to the level of wheelassembly, and we shall treat the load associated with each wheel as-sembly as a concentrated wheel load. Let Lc denote the distancebetween the two wheel assemblies of a car, and Ld the distance be-tween the rear wheel assembly of a car and the front wheel assemblyof the following car. It follows that the car length d is equal to thesum of Lc and Ld, i.e., d = Lc + Ld, and that a train can be repre-sented as a sequence of wheel loads p (assumed to be constant) withalternative intervals Lc and Ld.

For most commercial trains, the distance Lc between two wheelassemblies of a car is larger than the distance Ld between the rearwheel assembly of a car and the front wheel assembly of the fol-lowing car, i.e., Lc > Ld. For a better representation of the loadconfiguration, we shall conceive each train as the composition of twowheel load sets, with the first set representing the wheel loads ofall the front bogies and the second set the rear ones. By doing so,the distance between any two consecutive wheel loads in each set issimply d [see Fig. 2.10(b)]. It is realized, however, that a time lag oftc = Lc/v exists between the two sets of moving loads. Based on theabove considerations, the wheel load function F (t) for the train canbe modified from Eq. (2.33) as:

F (t) =N∑

j=1

p · [Uj(t, v, L) + Uj(t − tc, v, L)] , (2.35)

where p denotes the lumped load of each wheel assembly.Either the expression given in Eq. (2.33) or Eq. (2.35) considers

only the effect of moving loads, but neglects the effect of inertia ofthe moving masses and the interaction between the train cars andsupporting beam. In order to consider these two effects, the term p

in Eq. (2.35) should be replaced by the following function F (p,M, v)(Bolotin, 1964):

F (p,M, v) = p − M(u + 2vu′ + v2u′′) , (2.36)



where M denotes the mass lumped at each load or wheel position, u

is the vertical deflection of the beam, and dots and primes representdifferentiation with respect to time t and coordinate x, respectively.The physical meanings for the newly added terms in Eq. (2.36) canbe given as follows: The term Mu represents the inertial force actingalong the direction of deflection u of the beam; the term 2Mu′ is theCoriolis force relating to the rate of inclination of the beam; and theterm Mv2u′′ is the centrifugal force associated with the curvature ofthe beam induced by the mass of speed v at the position of action.

2.6.2. Method of Solution

Using the wheel load function F (p,M, v) in Eq. (2.36), the equationof motion for the beam under a moving train with the load configu-ration described above can be written as

mu+ceu+ciIu′′′′+EIu′′′′ =N∑

j=1

F (p,M, v)·[Uj(t, v, L)+Uj(t−tc, v, L)]

(2.37)subjected to the following boundary conditions:

u(0, t) = 0 ,

u(L, t) = 0 ,

EIu′′(0, t) = 0 ,

EIu′′(L, t) = 0 ,

(2.38)

and the initial conditions:

u(x, 0) = 0 ,

u(x, 0) = 0 .(2.39)

In Eq. (2.37), m denotes the mass per unit length, ce the externaldamping coefficient, ci the internal damping coefficient, E the elasticmodulus, I the moment of inertia of the beam, and tc is the time lagbetween the front and rear wheel loads of each car.

By taking into account the boundary conditions, one may expressthe vertical deflection u(x, t) for the simply-supported beam in a



series form as

u(x, t) =∞∑

n=1

qn(t) sinnπx

L, (2.40)

where qn(t) denotes the generalized coordinate of the nth mode. Bysubstituting the preceding expression for u into Eq. (2.37), multiply-ing both sides of the equation by sin(nπx/L), and integrating withrespect to length L of the beam, one can obtain the equation ofmotion for the simply-supported beam in terms of the generalizedcoordinates qn as

(1 + εm,n)qn + (2ξnωn + εc,n)qn + (ω2n + εs,n)qn = Fn(t) , (2.41)

where the damping property of the beam is assumed to be of theRayleigh type, ωn is the frequency of vibration of the nth mode, i.e.,ωn = n2π2

√EI/mL4, and ξn the damping ratio of the nth mode.

It should be noted that the equation of motion given in Eq. (2.41)for the generalized coordinates is as general as the original one inEq. (2.37), except that the conditions of hinge and roller supportsare taken into account. In Eq. (2.41), the forcing function Fn(t) forthe nth generalized coordinate is

Fn(t) =2pmL

N∑j=1

[fn(t, v, L) + fn(t − tc, v, L)] (2.42)

and the other coefficients are

εm,n =2MmL

N∑j=1

[gn(t, v, L) + gn(t − tc, v, L)] , (2.43)

εc,n =2MmL

(nπv

L

) N∑j=1

[hn(t, v, L) + hn(t − tc, v, L)] , (2.44)

εs,n = −2MmL

(nπv

L

)2N∑

j=1

[gn(t, v, L) + gn(t − tc, v, L)] . (2.45)



In Eqs. (2.42)–(2.45), the three functions fn, gn, fh are defined as

fn(t, v, L) = sinnπv(t − tj)

LH(t − tj)

+ (−1)n+1 sinnπv(t − tj − L/v)

LH

(t − tj − L

v

),

(2.46)

gn(t, v, L) =(

sinnπv(t − tj)

L

)2

H(t − tj)

−[sin

nπv(t − tj − L/v)L

]2H

(t − tj − L

v

), (2.47)

hn(t, v, L) = sin2nπv(t − tj)

LH(t − tj)

− sin2nπv(t − tj − L/v)

LH

(t − tj − L

v

). (2.48)

Here, some physical meanings can be given for the terms involvedin Eq. (2.41): εm,n represents the effect of added masses due to themoving wheel loads. Such an effect tends to elongate the period ofvibration of the beam. εc,n represents the effect of the Coriolis forcedue to the dynamic coupling between the speed of moving massesand the angular velocity of the beam. εs,n represents the effect ofthe centrifugal force, which is a function of the speed of masses andthe curvature of deflection of the beam. Such an effect tends toreduce the dynamic stiffness of the beam, especially when trains ofhigh speeds are concerned. Theoretically, it is possible that a beammay become unstable because of this effect. In practice, however, thebeams used in high-speed railways are constructed to be so stiff thatthe effects of both εc,n and εs,n on the dynamic response of the beamare much smaller than that of the added masses, as represented bythe term εm,n. For this reason, the effects of both the Coriolis forceand centrifugal force induced by the moving masses can generally beexcluded without affecting the accuracy of solution.

If only the effects of moving loads are taken into account,i.e., by neglecting all the three effects mentioned above by setting



εm,n = εc,n = εs,n = 0, the generalized coordinate qn(t) of the beamcan be evaluated from Eq. (2.41) as

qn(t) =1

mωdn

∫ t

0Fn(τ)e−ξnωn(t−τ) sin ωdn(t − τ)dτ

=2pL3

EIπ4[Pn(v, t) + Pn(v, t − tc)] , (2.49)

where ωdn is the damped frequency of vibration of the beam, i.e.,ωdn = ωn

√1 − ξ2

n, and the function Pn(v, t) can be expressed as

Pn(v, t) =1n4

N∑j=1

1(1 − S2

n)2 + 4(ξnSn)2

×[A · H(t − tj) + (−1)n+1B · H

(t − tj − L

v

)]. (2.50)

Here,

A = (1−S2n) sin Ωn(t− tj)−2ξnSn cos Ωn(t− tj)+e−ξnωn(t−tj )

·[2ξnSn cos ωdn(t− tj)+

Sn√1− ξ2

n

(2ξ2n +S2

n−1) sin ωdn(t− tj)

],

(2.51)

B = (1−S2n) sin Ωn

(t− tj − L

v

)−2ξnSn cos Ωn

(t− tj − L

v

)

+ e−ξnωn(t−tj−L/v)

·[2ξnSn cos ωdn

(t− tj − L

v

)

+Sn√1− ξ2

n

(2ξ2n +S2

n−1) sin ωdn

(t− tj − L

v

)], (2.52)

where Ωn denotes the exciting frequencies of the moving loads, Ωn =nπv/L, and the speed parameter Sn has been defined in Eq. (2.16).

As was stated previously, a train traveling over a beam can berepresented as the composition of two identical wheel load sets, with



the first set representing the wheel loads of all the front bogies andthe second set the rear ones. The terms Pn(v, t) and Pn(v, t − tc) inEq. (2.49) represent the dynamic responses excited by the two setsof wheel loads moving over the beam, in which the second set has atime lag tc behind the front set.

For a simple beam subjected to moving loads, which is basicallya transient problem with very short acting time, only the first modewill be important for determination of the deflection of the beam(Biggs, 1964). By neglecting the effect of damping and consideringonly the first mode of vibration, the dynamic response of the beamcan be derived from Eqs. (2.40) and (2.49) as

u(x, t) =2pL3

EIπ4

11 − S2

1

sinπx

L[P1(v, t) + P1(v, t − tc)] , (2.53)

where the response function P1(v, t) for the first set of wheel loads is

P1(v, t) =N∑

k=1

[sin Ω1(t − tj) − S1 sin ω1(t − tj)]H(t − tj)

+[sin Ω1

(t − tj − L

v

)− S1 sin ω1

(t − tj − L

v

)]

×H

(t − tj − L

v

). (2.54)

In Eq. (2.53), the terms P1(v, t) and P1(v, t − tc) denote the contri-bution of the front and rear wheel loads, respectively.

In this section, it is assumed that the beam has a span length L

not greater than twice the car length d, i.e., L ≤ 2d. Depending onthe bridge/car length ratio L/d, there may be two, three, or no wheelloads acting on the beam during the passage of the train, as shownin Fig. 2.11. The most severe case occurs when the front wheel loadof the (N − 1)th car has left the beam, and the front wheel load ofthe Nth car has entered the beam, namely, when the rear wheel loadof the (N − 1)th car and the front wheel load of the Nth car aresimultaneously acting on the beam, as shown in Fig. 2.11(a). Thereare two reasons for this. First, the two wheel loads, of distance Ld,



(a)

(b)

(c)

Fig. 2.11. The loading cases: (a) two wheelsets, (b) three wheelsets, and (c) nowheelset.



may induce the largest static response when passing the midpointof the beam. Second, the dynamic response of the beam has beenexcited to the utmost by the former N − 1 cars that have passed thebeam. For this case, it is known that tN < t < tN + L/v and thepreceding equation reduces to

P1(v, t)

= [sin Ω1(t − tN ) − S1 sin ω1(t − tN )]H(t − tN ) − 2S1 cosω1L

2v

·[sinω1

(t − L

2v

)+ sin ω1

(t − L

2v− tN

2

)sinω1

(tN2 − d

2v

)sin ω1d

2v

]

×H

(t − tN−1 − L

v

)(2.55)

of which the derivation has been given in Appendix A. Again, it canbe appreciated that the term containing H(t − tN ) represents thedynamic response of the beam induced by the motion of the Nth frontwheel load of the train, and the term containing H(t− tN−1−L/v) isthe free vibration caused by the former N − 1 front wheel loads thathave passed the beam. Since the two response functions P1(v, t) andP1(v, t − tc) are similar in nature, only the function P1(v, t) for thefirst set of wheel loads will be considered in the following discussion.

As a side note, whenever the car length d is greater than the spanlength L of the beam, there exist certain intervals during which thebeam is not in direct contact with the wheel loads, as was illustratedin Fig. 2.11(c). The dynamic response for the beam during theseintervals, which represents only free vibration, can be obtained fromEq. (2.55) by dropping the term containing H(t − tN ).

2.6.3. Phenomenon of Resonance

With the present moving load model for the train, it can be seenfrom Eq. (2.55) that the response of the beam reaches a maximumwhen the denominator of the second term within the brackets van-ishes, i.e., when sin(ω1d/2v) = 0 or ω1d/2v = iπ, with i = 1, 2, 3, . . ..



This is exactly the condition for resonance of the beam to occurunder repetitive loads. For the case with sin(ω1d/2v) = 0, the re-sponse function P1(v, t) in Eq. (2.55) becomes indeterminate. By therelation tN = (N −1)d/v and L’Hospital’s rule, it can be shown that

sin ω1

(t − L

2v− tN

2

)sin ω1

(tN2 − d

2v

)sin ω1d

2v

= (N − 2) sin ω1

(t − L

2v

).

(2.56)

Consequently, Eq. (2.55) reduces to

P1(v, t) = [sin Ω1(t − tN ) − S1 sin ω1(t − tN )]H(t − tN )

− 2(N − 1)S1 cosω1L

2vsinω1

(t − L

2v

)

×H

(t − tN−1 − L

v

), (2.57)

which is the response function for the case when the former N − 1wheel loads have passed the beam and the last, i.e., the Nth wheelload is acting on the beam. In Eq. (2.57), the term containing H(t−tN−1 − L/v) indicates that under the condition of resonance, theresponse of the beam will be continuously built up, as there aremore loads passing the beam.

It should be noted that the kind of resonance produced by movingloads on the beam is different from that occurring on a structure dueto excitation of a harmonic load that has a driving frequency equal tothe fundamental frequency of the structure, for which the responseof the structure may theoretically become unbounded in the absenceof damping. However, once the resonance condition is met by themoving loads, the response of the beam keeps increasing as there aremore wheel loads passing through the beam and reaches a maximumwhen the last wheel load enters the beam, if no consideration is madeof the damping effect. After all the wheel loads have passed the beam,only free vibration will remain on the beam, which eventually willbe damped out in reality.



With the condition of resonance, i.e., by letting sin(ω1d/2v) = 0,the critical car length d of the train traveling over the beam can besolved as

d = 2iπv

ω1= 2iS1L , i = 1, 2, 3, . . . . (2.58)

On the other hand, given the car length d and span length L, thespeed parameter can be found from the resonance condition as

S1 =d

2iL, i = 1, 2, 3, . . . . (2.59)

which implies that the longer the beam, the lower is the speed forresonance to occur. By letting i = 1, 2, 3, . . ., the preceding equationindicates that resonance may occur at the following speeds: S1 =0.50d/L, 0.25d/L, 0.167d/L, 0.125d/L, . . ., with diminishing values.Here, we shall call the speed 0.50d/L the primary resonant speed,and all the remaining the secondary resonant speeds. Since a trainis accelerated from zero to its full speed, it is obvious that certainresonant speeds will always be encountered by trains in their passageover a bridge. However, if one notes that all the S1 values computedabove are small compared with unity for high-speed railway bridgesand that the factor 1/(1 − S2

1) in Eq. (2.53) decreases as the speedparameter S1 decreases, the resonant responses induced by the trainmoving at the secondary speeds are generally small and can be ne-glected in practice. As will be demonstrated in the numerical study,a good design of railway bridges is to have the primary resonance, asimplied by the condition S1 = 0.50d/L, suppressed at all times, say,through adjustment of the span length or cross section of the beam,once the car length has been fixed.

2.6.4. Phenomenon of Cancellation

As can be seen from Eq. (2.55) or Eq. (2.57), whenever the conditioncos(ω1L/2v) = 0 is met, the excitation effects of all the former N −1wheel loads that have passed the beam sum to zero, that is, noresidual response will be induced by the loads that have passed thebeam. Such a condition has been referred to as the condition of



cancellation, under which the response function P1(v, t) reduces to

P1(v, t) = [sin Ω1(t − tN ) − S1 sin ω1(t − tN )]H(t − tN ) . (2.60)

This equation indicates that if the condition of cancellation is met,the response of the beam is determined solely by the last or Nthwheel load acting on the beam, as the free vibrations caused byall the former wheel loads passing the beam have been suppressed.Moreover, whenever the condition of cancellation is met, no residualresponse will be induced on the beam after the last wheel load leavesthe beam. Such a property remains valid even in the absence ofdamping.

By definition, S1 = πv/(ω1L), the speed parameter can be deter-mined for the condition of cancellation as follows:

S1 =1

2i − 1, i = 1, 2, 3, . . . (2.61)

from which the cancellation points can be computed as S1 = 1, 1/3,1/5, 1/7, . . .. As can be seen from Eq. (2.55), the condition forcancellation to occur, i.e., cos(ω1L/2v) = 0, is a condition moredecisive than that for resonance. In other words, if the conditionof cancellation is met, then the bracketed term in Eq. (2.55) justdisappears, meaning that the phenomenon of resonance is entirelysuppressed. Theoretically speaking, it is possible to select an optimalspeed for the train, aimed at providing better riding quality, suchthat the condition of cancellation is satisfied during its passage overa bridge.

2.6.5. Optimal Design Criteria

As was stated in Sec. 2.6.3, the resonance response induced by ahigh-speed train on the bridge will reach the maximum, when thespeed parameter S1 equals the primary resonant speed, i.e., 0.5d/L,as given in Eq. (2.59), rather than the secondary resonant speeds,due to the fact that the factor 1/(1 − S2

1) in Eq. (2.53) is larger forthe primary resonance than for the secondary resonance. In practice,the primary resonance can be circumvented through the selection ofproper car length d or span length L for the beam such that the



condition of cancellation is always enforced. For instance, by lettingthe first resonance speed, S1 = 0.5d/L, equal to any of the speedsthat satisfy the cancellation condition in Eq. (2.61), the span/carlength ratio L/d can be solved as follows:

L

d= i − 0.5 , i = 1, 2, 3, . . . . (2.62)

This represents exactly the optimal criterion for suppressing the res-onance response caused by repetitive, regular moving loads.

The meaning of Eq. (2.62) and its applicability to practical designshould not be underestimated. First, it is the span to car length ratioL/d that determines whether resonance will be induced by movingloads on the bridge. Second, while the integer i may take any integervalue, the value i = 2 can be easily met in practice, which impliesthat L/d = 1.5 is an optimal span/car length ratio. Thus, given thecar length d for a specific model of train, the optimal span lengthL can be found for the bridge, and vice versa. Third, the formulagiven in Eq. (2.62) has the important feature of being independent ofthe speed parameter S1 or speed v of the train. For this reason, thisformula can find a wide range of applications. In the following sectionand in many other studies conducted by the authors and co-workers,it has been confirmed that whenever the condition L/d = 1.5 is met,the first resonance response can be virtually suppressed.

2.7. Illustrative Examples

The phenomena of resonance and cancellation presented in Sec. 2.6are based primarily on the closed form solution presented inEq. (2.53), which considers only the first mode of vibration of thebeam. In this section, the accuracy of the solution presented inSec. 2.6, as well as the phenomena of resonance and cancellation, willbe numerically investigated. In particular, finite element solutionsobtained by a computer analysis program based on the vehicle–bridgeinteraction element derived in Chapter 3 will be used to generatesome reference solutions, by which the effects of all modes of vibrationof the beam are automatically taken into account.



Fig. 2.12. The most severe loading case.

2.7.1. Comparison with Finite Element Solutions

Consider a simple beam made of prestressed concrete with L = 20 m,I = 3.81 m4, E = 29.43 GPa, m = 34, 088 kg/m, for which thefirst frequency of vibration solved is ω1 = 44.75 rad/s. The trainis assumed to have N = 5 cars of identical length d = 24 m. Thetwo wheel assemblies (or bogies) of the car is separated by 18 m,i.e., Lc = 18 m and Ld = 6 m. The mass of each wheel assemblyis M = 22000 kg, corresponding to p = 215.6 kN. For the presentcase, the maximum static deflection Rs of the simple beam occurswhen two wheel loads p of interval Ld are located symmetrically onthe beam as shown in Fig. 2.12, which can be computed as (Gereand Timoshenko, 1990):

Rs =pa(3L2 − 4a2)

24EI, (2.63)

where a = (L − Ld)/2 for the present case. The maximum staticdeflection Rs is required in computation of the impact factor I inthis section based on the definition of Eq. (2.21).

For the purpose of verification, two cases are considered,particularly to demonstrate the phenomena of resonance and can-cellation. In the first case, the speed parameter S1 is selected sothat the resonance condition given in Eq. (2.59) is met. By settingi = 5, S1 = d/(2iL) = 0.12, the resonance speed is found to bev = 34 m/s = 122.4 km/h. In the second case, the speed parameterS1 is selected to meet the condition of cancellation given in Eq. (2.61).



By setting i = 6, S1 = 1/(2i − 1) = 1/11, the speed of cancellationis found as v = 26 m/s = 93.6 km/h. By simulating the train as aset of concentrated wheel loads, the midpoint responses of the beam(undamped) subjected to the action of the wheel loads moving atthe above two speeds have been plotted in Fig. 2.13, along with thefinite element solutions, which have been obtained by discretizingthe beam into 16 elements. As can be seen, for the two cases consid-ered, the present solution, which was obtained by considering onlythe first mode of vibration of the beam, agrees very well with thefinite element result. Moreover, for a train with v = 34 m/s, themidpoint response of the beam tends to increases steadily as thereare more loads passing the beam, which is indicative of the resonancephenomenon. For a train with v = 26 m/s, the response of the beamappears merely as a periodic function, with no amplification effectobserved during the passage of wheel loads on the bridge. Of inter-est is the fact that as long as all the wheel loads depart from thebeam, no residual response remains on the beam, which is a typicalcancellation phenomenon.

Fig. 2.13. The cases of verification.



For the present case of d/L = 1.2, the points of resonance can befound from Eq. (2.59) as follows: S1 = 0.60, 0.30, 0.20, 0.15, 0.12,0.10, . . ., and the points of cancellation from Eq. (2.61) as follows:S1 = 1.00, 0.33, 0.20, 0.143, 0.111, 0.091, . . .. Correspondingly, thespeeds of resonance for the train are v = 615.4, 307.7, 205.1, 153.9,123.1, 102.6 km/h, . . ., and the speeds of cancellation are v = 1025.6,338.4, 205.1, 146.5, 114.0, 93.0 km/h, . . .. The impact factor Iu anddisplacement u computed for the midpoint of the beam have beenplotted against the speed parameter S1 in Figs. 2.14 and 2.15, re-spectively. From these figures, the following observations can bemade: First, the present result agrees very well with the finite ele-ment solution that considers the contribution of all modes of vibra-tion, indicating that the effect of higher modes is rather minor for thepresent problem. Second, the resonance points with S1 = 0.60, 0.15,0.12 can generally be observed, while the other resonance points aremerely suppressed as they are coincident with or close to the pointsof cancellation. Third, the first resonance point (S1 = 0.6) should

Fig. 2.14. Impact factor versus speed parameter S1.



Fig. 2.15. Response amplitude u versus speed parameter S1.

be avoided in practical design for its relatively large magnitude ofresponse. Fourth, if the operation speed of the train can be con-trolled such that it falls in the range with 0.18 ≤ S1 < 0.33, or with184.6 km/h ≤ v < 338.4 km/h, minimal resonant responses will beinduced on the beam. Note that the point with S1 = 0.18 is impliedby another cancellation condition cos(ω1Lc/2v) = 0 not mentionedin the text. Finally, the results presented in Figs. 2.14 and 2.15 arenondimensional, which can be applied to simple beams of other crosssections, as long as the car length d and span length L remain un-changed. For instance, for a beam with a smaller cross section, say,with I = 1.75 m4, m = 33, 144 kg/m, and ω1 = 30.76 rad/s, the firstresonant speed becomes v = S1ω1L/π = 117.5 m/s = 423.0 km/h.

2.7.2. Effects of Moving Masses and Damping

All the data adopted in this example are identical to those of theprevious one in Sec. 2.7.1, except that a damping ratio of ξ = 2.5%



Fig. 2.16. The effects of damping and moving masses.

is considered for the beam. All the results presented in Fig. 2.16have been obtained by directly integrating the generalized equationof motion in Eq. (2.41) using Newmark’s β method (see Appendix B)with or without considering the effect of moving masses. As can beseen, the added mass effect of the moving masses tends to elongatethe period of vibration of the beam, which is the main cause forshifting of the resonance speed to smaller S1 values. On the otherhand, the inclusion of 2.5% of damping has resulted in significantreduction of the resonant responses. Nevertheless, the general impactcharacteristics of the beam, including the speeds for occurrence ofresonance and cancellation, can still be observed using the movingload model.

2.7.3. Effect of Span to Car Length Ratio

In order to demonstrate the effect of span to car length ratio onthe impact response, a number of simple beams made of prestressed



concrete (with E = 29.43 GPa) and with different span lengths L

are considered herein, of which the unit mass and first frequency ofvibration of each beam are specified as follows: m = 30 + 0.2L t/mand ω1 = 900/L rad/s, which represent the interpolation formula forthe seven simple beams of span lengths 20, 25, 30, 35, 40, 45, and50 m studied by T. Y. Lin Taiwan (1993). Zero damping is assumedfor all the beams.

Let each of the beams be traveled by the train model consideredpreviously in Sec. 2.7.1. The impact factor Iu computed for the mid-point displacement of the beam has been plotted with respect to thespeed parameter S1 and span/car length ratio L/d in Fig. 2.17(a),along with the contour lines in Fig. 2.17(b). An important trend re-vealed by these figures is that the shorter the span length of a beam,the larger the impact factor for the displacement of the beam. Mean-while, the resonance speed S1 shifts to smaller values in response tothe increase in span length, meaning that a long beam can be moreeasily excited to resonance by moving loads than a short beam. How-ever, from the response amplitude given in Fig. 2.18(a) and contourlines in Fig. 2.18(b), it can be observed that the maximum responsesfor long beams are not necessarily smaller than those of short beams.As such, the effect of resonance on long beams has to be consideredin design as well. Another observation from the figures is that onlythe first resonance responses are of engineering significance, whilethe other resonance responses can generally be neglected in practice.

Noteworthy is the fact that when the span/car length ratio L/d

equals 1.5 or 0.5, virtually no resonance response will be induced onthe beams under the action of a sequence of continuously movingloads, as can be seen from the contour lines in Figs. 2.17(b) and2.18(b). In fact, these conditions have been identified to be the opti-mal conditions presented in Eq. (2.62), with i set equal to 2 or 1, bywhich the first resonance has been suppressed through enforcementof the condition of cancellation. Another merit with the three-phaseplots in Figs. 2.17 and 2.18 is that they have all been presented ina nondimensional form, which can be applied virtually to all othersimple beams subject to the action of the particular train modelconsidered.



(a)

(b)

Fig. 2.17. The effect of span/car length on impact factor: (a) I − S1 −L/d plotand (b) contour lines.



(a)

(b)

Fig. 2.18. The effect of span/car length on displacement: (a) u − S1 −L/d plotand (b) contour lines.




This chapter consists primarily of two parts. In the first part, i.e.,in Secs. 2.2–2.5, the impact response of simple beams subjected toa single moving load was analytically studied. The results indicatethat there exist upper-bound envelopes for the midpoint deflections,bending moments, and near-support shear forces, which can all berelated to the nondimensional speed parameter S1, defined as theratio of the exciting frequency of the moving load to the fundamentalfrequency of the beam. For the case with S1 < 0.5, which is normallythe case encountered in practice, all the envelopes for the impactfactors are linearly proportional to the speed parameter S1, and forthe case with S1 ≥ 0.5, all the envelopes can be taken as constant.Although the impact factor formulas established herein for a singlemoving load may be larger than those for multiple moving loads, theyserve as conservative and useful upper bounds for the latter cases.

In the second part, i.e., in Secs. 2.6 and 2.7, the moving loadproblem is extended to deal with the case of train loads, in the sensethat a train is modeled as the composition of two sets of wheel loadsof constant intervals, each to account for the front and rear wheel as-semblies. The following are the conclusions: (1) The first resonance,as indicated by S1 = 0.5d/L, represents the most critical conditionand should always be avoided in practical design. (2) Once the con-dition of cancellation is met, the resonance peak can be effectivelysuppressed, which forms the basis for the optimal design criteria. (3)The inertia effect of the moving vehicles tends to elongate the periodof vibration of the beam, resulting in smaller resonance speeds. (4)Inclusion of damping of the beam can significantly reduce the peakresponses. (5) The shorter the span length of a beam, the larger theimpact factor for the midpoint deflection of the beam. (6) Whenthe span to car length ratio L/d equals 1.5, virtually no resonanceresponse will be induced on the beam, as the first resonance has beensuppressed.



Chapter 3

Impact Response of Railway Bridgeswith Elastic Bearings

In earthquake-prone regions, elastic bearings are often inserted atthe supports of bridge girders to reduce the earthquake forces trans-mitted upward from the ground. While they are effective for such apurpose, they may equally prevent the bridge vibrations induced bymoving vehicles from transmitting downward to the ground, therebyresulting in accumulated or amplified response on the bridge. Thelatter problem seems not well addressed in the literature. In thischapter, the dynamic response of an elastically-supported beam tomoving train loads will be studied using an analytical approach. Thepresent results indicate that the dynamic response of the beam at res-onance remains generally constant, if the effect of damping is takeninto account, and the installation of elastic bearings at the supportsof the beam to reduce the earthquake forces may adversely amplifythe dynamic response of the beam to moving train loads. Envelopeimpact formulas are derived for the deflection of the beam with lightdamping, which serve as a useful, preliminary design aid to railwayengineers.

3.1. Introduction

To prevent the damage of bridges from severe earthquakes, elas-tic bearings are often used as base isolators in bridge engineering.Conventionally, they are installed at the supports of bridge gird-ers to serve as filters for isolating the vibration energy transmittedfrom the ground. However, as they are effective for shielding the

69



upward-transmitting forces, they may be equally effective in pre-venting the vehicle-induced vibrations of the bridge girder from dis-sipating to the ground, thereby resulting in accumulated, amplifiedresponse on the bridge when subjected to a connected line of movingloads, as is the case encountered in railways. In this chapter, an ana-lytical approach is presented for investigating the dynamic responseof bridges with elastic bearings to high-speed trains. Particular em-phasis is placed on the resonant response that may be induced byhigh-speed trains.

The dynamic response of bridge structures to moving loads athigh speeds is a problem of great concern in the design of high-speedrailway bridges. As can be seen from the review presented in Chap-ter 1, a large number of analytical investigations have been carriedout by assuming the bridges to be simply-supported, and by mod-eling the vehicles as moving loads or moving masses. In Chapter 2,based on the paper by Yang et al. (1997b), we have presented aclosed-form solution for the dynamic response of simply-supportedbeams to a series of moving loads at high speeds, in which both thephenomena of resonance and cancellation have been investigated.By considering the effect of damping, Li and Su (1999) investigatedthe fundamental characteristics and dominant factors for the reso-nant vibration of a girder bridge under high-speed trains. Using thedynamic stiffness approach and damped Timoshenko beam theory,Chen and Li (2000) investigated the dynamic response of elevatedhigh-speed railways considered by the Bureau of Taiwan High SpeedRail in the preliminary stage of design.

On the other hand, based on the finite element methods, more so-phisticated models have been devised to study the dynamic behaviorof vehicle–bridge interaction problems by researchers. To resolvethe coupling effect between the bridge and moving vehicles, methodsthat are of iterative nature have been employed (Hwang and Nowak,1991; Green and Cebon, 1994; Yang and Fonder, 1996). By the con-cept of dynamic condensation, Yang and Yau (1997) and Yang et al.(1999) developed the vehicle–bridge interaction (VBI) elements forthe dynamic analysis of railway bridges subjected to moving trains,which will be described in Chapters 6 and 7. Cheung et al. (1999)


Impact Response of Railway Bridges with Elastic Bearings 71

used the modified beam vibration functions to investigate the re-sponse of multi-span nonuniform bridges under moving vehicles andtrains. Ichikawa et al. (2000) used the modal analysis method tostudy the dynamic behavior of the continuous beam subjected to amoving mass. They found that the inertial effect of the moving masshas greater influence on the second and successive spans than on thefirst span. By using the ultimate vertical accelerations of the railwaybridge deck to evaluate the operability of high-speed trains, and bycomparing the theoretical results with the experiments carried outon high-speed railway lines in Europe, Fryba (2001) proposed someinteroperability formulas for quick assessment of the suitability ofrailway bridges for high-speed trains. For a given train configuration,Savin (2001) derived analytically a pseudo-acceleration spectrum forpredicting the maximum accelerations of weakly-damped beams withvarious boundary conditions.

To the knowledge of the authors, very few studies have been con-ducted on the dynamic response of elastically-supported beams toa series of moving loads. The objective of this chapter is to an-alytically investigate the dynamic behavior of elastically-supportedbeams subjected to moving loads in the high-speed range. Basedon the analytical results, envelope impact formulas that take intoaccount the effect of damping will be proposed for the deflection ofthe beam. The accuracy of such a formula will be demonstrated inthe numerical examples through comparison with the finite elementsolutions. The materials presented in this chapter are based largelyon the work by Yau et al. (2001).

3.2. Equation of Motion

As shown in Fig. 3.1, a beam supported by two elastic bearings ofstiffness K at the two ends is considered. The beam is assumed tobe of length L and uniform cross sections. The train moving overthe beam at speed v is modeled as a sequence of equidistant movingloads. The interval between two adjacent moving loads is d and theweight of each moving load is p. The equation of motion for the



Fig. 3.1. Elastically-supported beam subjected to uniform moving loads.

beam traveled by the moving loads can be written as

mu + ceu + ciu′′′′ + EIu′′′′

= p

N∑k=1

δ[x − v(t − tk)] ×[H(t − tk) − H

(t − tk − L

v

)], (3.1)

where a prime denotes derivative with respect to coordinate x, anoverdot denotes derivative with respect to time t, m = the mass perunit length of the beam, u(x, t) = vertical displacement, ce = externaldamping coefficient, ci = internal damping coefficient, E = elasticmodulus, I = moment of inertia of the beam, δ(x) = Dirac’s deltafunction, H(t) = unit step function, N = total number of movingloads, and tk = (k − 1)d/v = arriving time of the kth load at thebeam. Correspondingly, the boundary conditions of the beam are

EIu′′(0, t) = 0 ,

EIu′′(L, t) = 0 ,

EIu′′′(0, t) = −Ku(0, t) ,

EIu′′′(L, t) = Ku(L, t) ,

(3.2)

and the initial conditions are

u(x, 0) = 0 ,

u(x, 0) = 0 ,(3.3)

assuming that the beam is initially at rest.



3.3. Fundamental Frequency of the Beam

To analyze the dynamic response of the elastically-supported beamto a sequence of moving loads, the vibration shape of the beam will beapproximated by the combination of a flexural sine mode and a rigiddisplacement mode, as shown in Fig. 3.2. Thus, the displacementu(x, t) of the elastically-supported beam can be expressed as

u(x, t) = q(t)φ(x) ∼= q(t) × sin(πx/L) + κ

1 + κ, (3.4)

where q(t) denotes the generalized coordinate of the vibration shape,φ(x) the assumed shape function, and κ(= EIπ3/KL3) is the stiff-ness ratio of the beam to the elastic bearings. In particular, the termκ in the numerator of Eq. (3.4) denotes the rigid displacement mode.For the special case with κ = 0, which implies hinge supports, the

(a)

(b)

(c)

Fig. 3.2. The concept for modal vibration shape of elastically-supported beam.



assumed mode shape of vibration, φ(x), reduces to the first flexuralmode. As can be seen, higher modes other than the first flexural andrigid modes have been excluded from Eq. (3.4), which is justifiedfor the moving load problems, because of the transient nature of thebeam in response to the moving loads. This is particularly true ifonly the midpoint displacement of the beam is desired.

By Rayleigh’s method, the fundamental frequency ω of theelastically-supported beam can be computed as

ω2 =

∫ L0 EI[φ′′(x)]2dx + K[φ(0)]2 + [φ(L)]2∫ L

0 m[φ(x)]2dx

∼= ω20(1 + 4κ/π)

1 + 8κ/π + 2κ2, (3.5)

where ω0 = (π/L)2√

EI/m = the fundamental frequency of thecorresponding beam with hinge supports. When κ equals 0, thefundamental frequency ω of the elastically-supported beam equalsthat of the corresponding simple beam. On the other hand, if κ

approaches infinity, the fundamental frequency ω reduces to zero,meaning that the beam is unsupported. To verify the accuracy ofthe approximate fundamental frequency given by Eq. (3.5), the exactsolution will be computed from the following frequency equation forthe elastically-supported beam (Gorman, 1975):

b2

(cos kL − 1

cosh kL

)+ 2b(sin kL − tanh kL cos kL)

− 2 sin kL tanh kL = 0 , (3.6)

where kL = π√

ω/ω0 and b = EIk2. The fundamental frequen-cies computed from Eq. (3.5), which are approximate, have beencompared with the exact ones obtained from Eq. (3.6) for differentstiffness ratios in Fig. 3.3. As can be seen, the approximate frequen-cies agree excellently with the exact ones, implying that the use ofthe approximate mode shape for the elastically-supported beam, asgiven in Eq. (3.4), is acceptable.



Fig. 3.3. The comparison of fundamental frequencies.

3.4. Dynamic Response Analysis

By substituting the expression for the displacement u(x, t) inEq. (3.4) into Eq. (3.1), multiplying both sides of the equation bythe shape function φ(x), and then integrating with respect to thebeam axis x over the length L, one obtains the equation of motionin terms of the generalized coordinate q(t) as

q(t)+2ξωq(t)+ω2q(t) =2p(1 + κ)

mL(1 + 8κ/π + 2κ2)

N∑k=1

Fk(v, t) , (3.7)

where ξ is the modal damping ratio and Fk(v, t) is the generalizedforcing function,

Fk(v, t) = [κ + sin Ω(t − tk)]H(t − tk)

+[−κ + sin Ω

(t − tk − L

v

)]H

(t − tk − L

v

). (3.8)

Here, Ω(= πv/L) is the driving frequency, as implied by the movingloads.



First, consider the case when only a single moving load is crossingthe bridge. The equation of motion, Eq. (3.7), becomes

q(t) + 2ξωq(t) + ω2q(t) =2p(1 + κ)

mL(1 + 8κ/π + 2κ2)

(sin

πvt

L+ κ

).

(3.9)

By Duhamel’s integral, the generalized coordinate q(t) can be solvedfrom Eq. (3.9) as

q(t) =[∆st(1 + κ)]/[1 + 4κ/π]

(1 − S2)2 + (2ξS)2

(1 − S2) sin Ωt − 2ξS cos Ωt

+ e−ξωt

[2ξS cos ωdt − S(1 − S2 − 2ξ2)√

1 − ξ2sin ωdt

]

+∆st(1 + κ)1 + 4κ/π

× κ1 − e−ξωt[cos ωdt + ξ√

1 − ξ2 sin ωdt] ,

(3.10)

where ∆st = 2pL3/π4EI ≈ pL3/48EI = the maximum static deflec-tion of the corresponding simple beam, S = Ω/ω = the speed pa-rameter, and ωd = ω

√1 − ξ2 = the damped frequency. It should be

noted that the speed parameter S represents the ratio of the drivingfrequency to the frequency of the beam. For most of the vehicle–bridge problems encountered in practice, the speed parameter S isless than 0.3.

In this chapter, only elastically-supported beams with light damp-ing (ξ < 0.05) are considered, which implies that terms involving ξ2,ξS, and ξκ are so small that they can be neglected. As a result, theresponse in Eq. (3.10) reduces to

q(t) ∼= ∆st(1 + κ)1 + 4κ/π

[1

1 − S2G1(v, t) + κG2(t)

], 0 ≤ vt ≤ L ,

(3.11)



where the functions G1(v, t) and G2(t) are

G1(v, t) = sin Ωt − Se−ξωt sin ωt ,

G2(t) = 1 − e−ξωt cos ωt .(3.12)

Consider the case when a series of moving loads of constant intervalsd are crossing the bridge. The dynamic response can be extendedfrom Eq. (3.11) as follows:

q(t) ∼= ∆st(1 + κ)1 + 4κ/π

1

1 − S2

N∑k=1

[G1(v, t − tk)H(t − tk)

+ G1

(v, t − tk − L

v

)H

(t − tk − L

v

)]

+ κ

N∑k=1

[G2(t − tk)H(t − tk)

−G2(t − tk − L/v)H(t − tk − L/v)]

, (3.13)

where tk = (k − 1)d/v denotes the arriving time of the kth loadon the bridge, the unit step function H(t − tk) is used to repre-sent the direct action of the kth moving load on the beam, whilethe function H(t − tk − L/v) is the residual action of the kth mov-ing load. It is easy to see that for the undamped case, ξ = 0, asκ = 0, the preceding expression for the generalized coordinate re-duces to that for the simply-supported beam, as the one implied byEq. (2.53).

3.5. Phenomena of Resonance and Cancellation

In this chapter, the span length L of the beam is assumed to be nogreater than twice the interval d between two consecutive movingloads, i.e., L 2d which is the case implied by the most high-speed



railway constructions. In order to derive the conditions for the phe-nomena of resonance and cancellation to occur under the action ofa series of moving loads of constant intervals, we shall neglect theeffect of damping on the beam in this section. For this special case,ξ = 0, the generalized coordinate of the response of the beam, asgiven in Eq. (3.13), reduces to

q(t) =∆st(1 + κ)1 + 4κ/π

[P1(v, t)1 − S2

+ κP2(v, t)]

, (3.14)

where

P1(v, t) =N∑

k=1

[sin Ω(t − tk) − S sinω(t − tk)]H(t − tk)

+[sin Ω

(t − tk − L

v

)− S sin ω

(t − tk − L

v

)]

×H

(t − tk − L

v

), (3.15a)

P2(v, t) =N∑

k=1

[1 − cos ω(t − tk)]H(t − tk)

−[1 − cos ω

(t − tk − L

v

)]H

(t − tk − L

v

). (3.15b)

In Eqs. (3.14) and (3.15), the function P1(v, t) indicates the contri-bution of the flexural vibration mode of the simple beam, which isidentical to the function P1(v, t) given in Eq. (2.54) for the simply-supported beam, and P2(v, t) the rigid displacement mode of theelastic bearings. The beam will be excited to its utmost, in the sensethat the response will reach the maximum, when the last (i.e., theNth) moving load enters the beam. For this case, tN < t < tN +L/v,



the function P1(v, t) in Eq. (3.15a) can be simplified as

P1(v, t) = [sin Ω(t − tN ) − S sin ω(t − tN )]H(t − tN )

− 2S cosωL

2v×[

sinω

(t − L

2v

)+ sin ω

(t − tN + L/v

2

)

× sinω

2

(tN − d

v

)/sin(

ωd

2v

)]H

(t − tN−1 − L

v

)

(3.16)

by following the procedure presented in Sec. 2.6.2 leading toEq. (2.55). In the meantime, the function P2(v, t) in Eq. (3.15b)reduces to

P2(v, t) = [1 − cos ω(t − tN )]H(t − tN )

+

N−1∑k=1

[cos ω

(t − tk − L

v

)− cos ω(t − tk)

]

×H

(t − tN−1 − L

v

). (3.17)

Through introduction of the following relations:

cos ω

(t − tk − L

v

)− cos(t − tk)

= 2 sin(

ωL

2v

)sin[ω

(t − tk − L

2v

)],

(3.18)N−1∑k=1

sinω

(t − tk − L

2v

)

= sin ω

(t − L

2v

)+ sinω

(t − tN + L/v

2

)sin ω

2 (tN − dv )

sin(ωd2v )

,



the function P2(v, t) in Eq. (3.17) can be rearranged as

P2(v, t) = [1 − cos ω(t − tN )]H(t − tN )

+ 2 sinωL

2v×[

sin ω

(t − L

2v

)+ sin ω

(t − tN + L/v

2

)

× sinω

2

(tN − d

v

)/sin(

ωd

2v

)]H

(t − tN−1 − L

v

).

(3.19)

By using Eqs. (3.14), (3.16), and (3.19), the dynamic response forthe midpoint of the elastically-supported beam can be obtained fromEq. (3.4) as

u

(L

2, t

)=

∆st(1 + κ)1 + 4κ/π

×[Q1(v, t)H(t − tN ) + Q2(v, t)H

(t − tN−1 − L

v

)],

(3.20)

where the dynamic response factors Q1(v, t) and Q2(v, t) are

Q1(v, t) =sinΩ(t − tN ) − S sin ω(t − tN )

1 − S2

+ κ[1 − cos ω(t − tN )] , (3.21a)

Q2(v, t) = 2[κ sin

( π

2S

)− S

cos(π/2S)1 − S2

]

×[

sinω

(t − L

2v

)+ sin ω

(t − tN + L/v

2

)

× sin((ω/2))(tN − (d/v))sin(ωd/2v)

]. (3.21b)

It should be noted that the relation ωL/2v = π/2S has been utilizedin arriving at Eq. (3.21). From Eq. (3.20), it can be seen that the



term Q1(v, t) is associated with H(t−tN ), which represents the forcedresponse induced by the Nth moving load directly acting on thebeam, and the term Q2(v, t) is associated with H(t − tN−1 − L/v),which represents the residual response induced by the N − 1 movingloads that have passed the beam.

From Eq. (3.21b), it can be seen that the response reaches a max-imum when sin(ωd/2v) = 0. This is exactly the condition for res-onance to occur. Correspondingly, the resonant speed is denotedas vr. Note that the resonance speed vr as implied by the condi-tion sin(ωd/2v) = 0 is independent of the stiffness ratio κ. By theL’Hospital rule, the dynamic response factor Q2(vr, t) for resonancein Eq. (3.21b) can be manipulated to yield

Q2(vr, t) = 2(N − 1)[κ sin

(π

2Sr

)− Sr

cos(π/2Sr)1 − S2

r

]

× sin ω

(t − L

2vr

), (3.22)

where Sr is the speed parameter for resonance to occur. The preced-ing equation indicates that under the condition of resonance, largerresponse will be induced on the beam if there are more loads pass-ing the beam, as implied by (N − 1). The other observation fromEq. (3.22) is that when the signs of sin(π/2Sr) and cos(π/2Sr) aredifferent, i.e., when the vibration phases of the elastic bearings andthe beam are the same, the response factor Q2(vr, t) attains its maxi-mum, meaning that the beam response will be amplified. In contrast,when the elastic bearings and beam are out of phase, less severe re-sponse will be induced on the beam.

On the other hand, as can be observed from Eq. (3.21b), wheneverthe following condition is met, that is,

κ sin( π

2S

)− S

cos(π/2S)1 − S2

= 0 , (3.23)

the residual response caused by the previous N − 1 moving loadsthat have passed the beam disappears. Because of this, the conditionin Eq. (3.23) has been referred to as the condition of cancellation.Under this condition, the midpoint dynamic response of the beam in



Eq. (3.20) becomes

u

(L

2, t

)=

∆st(1 + κ)1 + 4κ/π

×

sinΩ(t − tN ) − Sc sinω(t − tN )1 − S2

c

+ κ[1 − cos ω(t − tN )]

, (3.24)

where Sc denotes the speed parameter for cancellation to occur. Ascan be seen, whenever the condition of cancellation is met, the re-sponse of the beam is determined solely by the Nth moving load,while the effect of the moving loads that have passed the beam isfully suppressed.

3.6. Effect of Structural Damping

Consider the case when the resonance condition implied byEq. (3.21b) is met, i.e., sin(ωd/2vr) = 0, or ωd/vr = 2nπ, withn = 1, 2, 3 . . ., and when the Nth moving load is acting on the beamat time t = tE + tN , that is, t = tE + (N − 1)d/vr, where tE denotesthe time for the load to reach the position shown in Fig. 3.4. Thenω(t−tk) = ω(tE +tN −tk) = ωtE +(N−k)ωd/vr = ωtE +2nπ(N−k)and the following relations can be derived:

sinΩ(t − tk) + sin Ω(

t − tk − L

vr

)= 0 ,

sin ω(t − tk) = sin ωtE ,

sinω

(t − tk − L

vr

)= sin ω

(tE − L

vr

),

cos ω(t − tk) = cos ωtE ,

cos ω

(t − tk − L

vr

)= cos ω

(tE − L

vr

),

(3.25)



Fig. 3.4. The Nth moving load is acting at position vtE.

for 0 < t < tN . By introducing the preceding relations intoEq. (3.13), where the effect of damping of the beam has been con-sidered, the resonance response for the elastically-supported beamunder the action of the last (i.e., the Nth) moving load can be ex-pressed as

qr(tE) ∼= ∆st(1 + κ)1 + 4κ/π

×[

sin ΩtE − Sre−ξωtE sinωtE

1 − S2r

+ κ(1 − e−ξωtE cos ωtE)

]H(tE) + e−ξω(tE+tN )

N−1∑k=1

eξωtk

×[

−Sr

1 − S2r

(sin ωtE + eξπ/Sr sin ω

(tE − L

vr

))

+ κ

(eξπ/Sr cos ω

(tE − L

vr

)− cos ωtE

)]

×H

(tE − L − d

vr

). (3.26)

Furthermore, by using the approximation in expansion for the expo-nential function, i.e., exp(ξπ/Sr) ∼= 1 + ξπ/Sr for ξπ/Sr ≤ 0.3, and



the following relations for the series sum:

N−1∑k=1

eξωtk =N∑

k=1

eξωd(k−1)/vr =eξωd(N−1)/vr − 1

eξωd/vr − 1,

e−ξωtN

N−1∑k=1

eξωtk =1 − e−ξωd(N−1)/vr

eξωd/vr − 1,

(3.27)

the dynamic response in Eq. (3.26) can be further expressed as

qr(tE)∼= ∆st(1+κ)1+4κ/π

×[

sin ΩtE −Sre−ξωtE sin ωtE

1−S2r

+ κ(1−e−ξωtE cos ωtE)

]H(tE)

+

[2(

κ sinπ

2Sr− Sr

1−S2r

cosπ

2Sr

)sinω

(tE − L

2vr

)

+ξπ

Sr

(κ cos ω

(tE − L

vr

)− Sr

1−S2r

sinω

(tE − L

vr

))]e−ξωtE

× 1−e−ξ(N−1)ωd/vr

eξωd/vr −1H

(tE − L−d

vr

). (3.28)

For the special case of zero damping, i.e., by letting ξ = 0, thedynamic response in Eq. (3.28) reduces to

qr(tE) =∆st(1 + κ)1 + 4κ/π

×[

sin ΩtE − Sr sin ωtE1 − S2

r

+ κ(1 − cos ωtE)

]H(tE)

+ 2(N − 1)(

κ sinπ

2Sr− Sr

1 − S2r

cosπ

2Sr

)sin ω

(tE − L

2vr

)



×H

(tE − L − d

vr

)

=∆st(1 + κ)1 + 4κ/π

[Q1(vr, tE)H(t − tN )

+ Q2(vr, t)H(

t − tN−1 − d

vr

)], (3.29)

which is identical to the one given in Eq. (3.20) for the condition ofresonance.

For the purpose of obtaining closed-form solutions, let us assumethat there is an infinite number of moving loads crossing the beam.By letting the number of vehicles approach infinity, N → ∞, andusing the relation exp(ξωd/vr) − 1 ∼= ξωd/(SrL) for light damping,Eq. (3.28) becomes

qr(tE) ∼= ∆st(1 + κ)1 + 4κ/π

×[

sinΩtE − Sre−ξωtE sin ωtE

1 − S2r


]H(tE) + e−ξωtE

× SrL

ξπd

[2(

κ sinπ

2Sr− Sr

1 − S2r

cosπ

2Sr

)sin(

ωtE − π

2Sr

)

+ξπ

Sr

(κ cos

(ωtE − π

Sr

)− Sr

1 − S2r

sin(

ωtE − π

Sr

))]

×H

(tE − L − d

vr

). (3.30)

As can be verified from Eq. (3.30) and in the examples to follow,if the effect of damping is considered, the resonant response of the



beam subjected to an infinite number of moving loads remains moreor less constant. Furthermore, by the use of the following relations:

cos(

ωtE − π

Sr

)= cos

π

2Srcos(

ωtE − π

2Sr

)

+ sinπ

2Srsin(

ωtE − π

2Sr

),

sin(

ωtE − π

Sr

)= cos

π

2Srsin(

ωtE − π

2Sr

)

− sinπ

2Srcos(

ωtE − π

2Sr

),

(3.31)

the resonance response in Eq. (3.30) becomes

qr,max(tE) ≈ ∆st(1 + κ)1 + 4κ/π

×[

sin ΩtE − Sre−ξωtE sin ωtE

1 − S2r


]H(tE)

+L

de−ξωtE ×

[(2Sr

ξπ+ 1)

×(

κ sinπ

2Sr− Sr

1 − S2r

cosπ

2Sr

)sin(

ωtE − π

2Sr

)

+(

κ cosπ

2Sr+

Sr

1 − S2r

sinπ

2Sr

)cos(

ωtE − π

2Sr

)]

×H

(tE − L − d

vr

). (3.32)

Since sin(ωtE − π/2Sr) and cos(ωtE − π/2Sr) are out of phase,when the function sin(ωtE − π/2Sr) reaches the maximum, the



function cos(ωtE − π/2Sr) is at its minimum. As far as the max-imum response is concerned, the preceding expression can be ap-proximated by dropping the term containing cos(ωtE − π/2Sr) asfollows:

qr,max(tE) ≈ ∆st(1 + κ)1 + 4κ/π

×[

sin ΩtE − Sre−ξωtE sinωtE

1 − S2r


]H(tE) +

L

de−ξωtE

(2Sr

ξπ+ 1)

×(

κ sinπ

2Sr− Sr

1 − S2r

cosπ

2Sr

)sin(

ωtE − π

2Sr

)

×H

(tE − L − d

vr

), (3.33)

where it is recognized that 2Sr/ξπ + 1 > 1. At this point, we havederived the maximum response for the elastically-supported beam inEq. (3.33) considering the effect of damping.

3.7. Envelope Formula for Resonance Response

In this section, envelope formulas will be derived for the elastically-supported beam based on the maximum response presented inEq. (3.33) assuming that the beam is lightly-damped and is sub-jected to an infinite number of moving loads. When the conditionof resonance is met, while the condition of cancellation as given inEq. (3.23) is not, the dynamic response of the beam is dominated bythe term containing H(tE − (L− d)/vr) in Eq. (3.33). Moreover, thefunction sin(ωtE − π/2Sr) reaches its maximum when

ωtE =π

2+

π

2Sr=

π(1 + Sr)2Sr

. (3.34)



Substituting Eq. (3.34) into Eq. (3.33) and noting that exp(−ξωtE) ∼=1 for light damping, one obtains

qr,max(tE) ≈ ∆st(1 + κ)1 + 4κ/π

×[

κ +cos(Srπ/2)

1 − S2r

+(

κ sinπ

2Sr− Sr

1 − S2r

cosπ

2Sr

)e−ξπ(1+Sr)/2Sr

]H(tE)

+L

d

(2Sr

ξπ+ 1)(

κ sinπ

2Sr− Sr

1 − S2r

cosπ

2Sr

)

× e−ξπ(1+Sr)/2SrH

(tE − L − d

vr

). (3.35)

Accordingly, the absolute maximum response for the beam is

qr,max ≈ ∆st(1 + κ)1 + 4κ/π

×[

κ +| cos(Srπ/2)|

1 − S2r

]

+∣∣∣∣κ sin

π

2Sr− Sr

1 − S2r

cosπ

2Sr

∣∣∣∣×[L

d

(2Sr

ξπ+ 1)

+ 1]

e−ξπ(1+Sr)/2Sr

. (3.36)

For the case of light damping considered in this study, ξπ/Sr < 0.3,the following relations may be adopted:

cos(Srπ/2)1 − S2

r

∼= 1 ,

1 − S2r∼= 1 ,

[L

d

(2Sr

ξπ+ 1)

+ 1]

e−ξπ(1+Sr)/2Sr ∼= L

d× 2Sr

ξπ.

(3.37)



It follows that the maximum response in Eq. (3.36) reduces to

qr,max ≈ ∆st(1 + κ)1 + 4κ/π

×(

1 + κ +L

d

2Sr

ξπ

∣∣∣∣κ sinπ

2Sr− Sr cos

π

2Sr

∣∣∣∣)

,

(3.38)

which is applicable for the case where the resonance condition is met,but the cancellation condition is not.

On the other hand, when both the resonance condition, as im-plied by sin(ωd/2v) = 0, and the cancellation condition, as given inEq. (3.23), are satisfied, i.e., when Sr = Sc, the response in Eq. (3.33)becomes

qr(tE) ≈ ∆st(1 + κ)1 + 4κ/π

×[

sin ΩtE − Sce−ξωtE sin ωtE

1 − S2c


]H(tE) . (3.39)

Here, due to the damping effect and consideration of operating speedsin the range 0 < Sr < 0.3, the forced vibration term sin ΩtE and theconstant κ will dominate the response. By letting sin ΩtE = 1, orωtE = π/(2Sr), Eq. (3.39) becomes

qr(tE) ≈ ∆st(1 + κ)1 + 4κ/π

×[

11 − S2

c

+ κ − e−ξπ/2Sc

×(

Sc sin(π/2Sc)1 − S2

c

− κ cosπ

2Sc

)]H(tE) , (3.40)

which can further be expressed as follows:

qr,max ≈ ∆st(1 + κ)1 + 4κ/π

×[κ +

11 − S2

c

+(

1 − ξπ

2Sc

)√S2

c + κ2

],

(3.41)

if the relations for approximations are used: exp(−ξπ/2Sc) ∼= 1 −ξπ/2Sc and Sc/(1 − S2

c ) ∼= Sc. These formulae are valid for the case



when both the condition of resonance and the condition of cancella-tion are satisfied.

3.8. Impact Factor and Envelope Impact Formulas

The impact factor defined in Eq. (1.1) is adopted for the deflectionof an elastically-supported beam subjected to the moving loads, i.e.,

I =Rd(x) − Rs(x)

Rs(x), (3.42)

where Rd(x) and Rs(x), respectively, denote the maximum dynamicand static responses of the beam at position x due to the action ofthe moving loads. The following is the maximum static deflection foran elastically-supported beam subjected to a lumped load p at themidpoint:

Rs(x) =∆st(1 + κ)2

1 + 4κ/π, (3.43)

where ∆st = 2pL3/π4EI ∼= pL3/48EI = the maximum midpointstatic deflection of the corresponding beam with hinge supports. Byusing Eqs. (3.38), (3.41) and (3.42), the deflection impact formulafor the elastically-supported beam subjected to a sequence of movingloads can be expressed as:

I ≈ 1(1 + κ)

[1 + κ +

L

d

2Sr

ξπ

∣∣∣∣κ sin(

π

2Sr

)− Sr cos

(π

2Sr

)∣∣∣∣]− 1

≈ L

(1 + κ)d2Sr

ξπ

∣∣∣∣κ sin(

π

2Sr

)− Sr cos

(π

2Sr

)∣∣∣∣ , (3.44)

for the case of resonance, and

I ≈ 1(1 + κ)

κ +

11 − S2

c

+(

Sc − ξπ

2

)√1 +(

κ

Sc

)2− 1

≈ 1(1 + κ)

S2

c

1 − S2c

+(

Sc − ξπ

2

)√1 +(

κ

Sc

)2 , (3.45)



for the case when both the conditions of resonance and cancellationare met. Because of the impact factors for S other than Sr or Sc aresmaller than those under the resonance or cancellation conditions,it is possible to construct an envelope formula by connecting all thepeak values computed from Eqs. (3.44) or (3.45) for all the resonancepoints in a piecewise manner, which serves as an upper bound for theimpact factor of elastically-supported beams subjected to a sequenceof moving loads at various speeds.

3.9. Numerical Examples

3.9.1. Phenomenon of Resonance

As shown in Table 3.1, two bridges supported by elastic bearingsare considered in this example. The train moving over the bridge isassumed to have N = 8 cars, each of length d = 25 m. The weightof each car is represented by a moving load of p = 300 kN. Differentvalues of resonant speeds computed for the two beams have beenlisted in Table 3.2, together with those for the corresponding beamswith hinge supports. Evidently, most of the resonance speeds canbe encountered by modern high-speed trains. The responses of thebeam traversed by the moving loads have been plotted in Figs. 3.5–3.7, along with those obtained by the finite element method to be

Table 3.1. Properties of bridges.

L (m) m (t/m) EI (kN-m2) κ ω0 (rad/s) ω (rad/s)

23 30 1.4 × 108 0.24 40.3 35.127 32 2.0 × 108 0.2 33.9 29.9

Table 3.2. Resonant speeds.

Resonant speed L = 23 m L = 27 m L = 27 mvr = (ωd/2π)/n (n = 2) (n = 2) (n = 4)

Simple beam v0 80 m/s (= 288 km/h) 67 m/s (= 242 km/h) 34 m/s (= 122 km/h)[Sr = d/2nL] [Sr = 0.272] [Sr = 0.231] (Sr = 0.116)

Elast. supp. beam v 70 m/s (= 252 km/h) 59 m/s (= 214 km/h) 30 m/s (= 108 km/h)[Sr = d/2nL] [Sr = 0.272] [Sr = 0.231] (Sr = 0.116)



Fig. 3.5. The time history responses at resonant speed S = d/4L for 23 m beam.





presented in Chapter 6. As can be seen, all the solutions obtainedby the present approach agree excellently with the finite elementsolutions. For the case when the motions of the simple beam andelastic bearings are in phase, the elastic bearings inserted at thesupports can significantly increase the dynamic response of the bridgeto the moving loads, as can be seen from Figs. 3.5 and 3.7. Such aphenomenon is harmful to the riding comfort or maneuverability ofthe train. However, for the case when the motions of the simple beamand elastic bearings are out of phase, as indicated in Fig. 3.6, thedynamic response of the beam is reduced through the introductionof the elastic bearings.

3.9.2. Effect of Structural Damping

To investigate the effect of damping on the resonant response of anelastically-supported beam due to an infinite series of moving loads,30 moving loads are considered in this example. A damping ratio of2% is assumed for the beam. As can be seen from Figs. 3.8–3.11,



Fig. 3.8. The comparison of time history responses at resonant speed S = d/4Lfor 23 m simple beam.

Fig. 3.9. The comparison of time history responses at resonant speed S = d/4Lfor 23 m elastically-supported beam.



Fig. 3.10. The comparison of time history responses at resonant speed S = d/4Lfor 27 m simple beam.

Fig. 3.11. The comparison of time history responses at resonant speed S = d/4Lfor 27 m elastically-supported beam.



due to the presence of damping, the vibration of the beam remainsrather bounded, behaving in a steady state manner, even when theresonance condition is met. This is very different from the undampedcase, in which the response amplitude tends to grow continuously, asthere are more loads passing the beam under the resonance condition.

3.9.3. Envelope Impact Formula

From the condition of resonance, sin(ωd/2vr) = 0, the resonancespeed can be solved: vr = ωd/2nπ or Sr = d/2nL, with n = 1, 2, 3 . . ..It is possible to connect all the peak values computed from the enve-lope impact formulas in Eqs. (3.44) and (3.45) for all the resonancepoints in a piecewise manner, as shown in Figs. 3.12–3.15. In thesefigures, both the elastically-supported beams and simple beams areconsidered, assuming two different values of damping ratios, ξ = 0.02and 0.04. The impact factors I computed using the more accurateEq. (3.33) in Sec. 3.6 are also plotted in the figures. The figuresshown here are known as the I − S plots, i.e., impact factor versusspeed parameter plots. As can be seen, the envelope impact formulasshow a trend in good consistency with the more accurate I −S plotsfor the two values of damping ratios throughout the entire range ofspeed parameters considered.

A comparison of Fig. 3.12 with Fig. 3.13 for the 23 m beams is thatinstallation of elastic bearings at the supports tends to drasticallyincrease the impact response of the beam for the entire speed rangeconsidered. On the other hand, by comparing Fig. 3.14 with Fig. 3.15for the 27 m beams, one observes that the installation of elasticbearings may amplify the impact response only over the low-speedrange (S < 0.125), but may suppress the response for the high-speedrange (S > 0.125). To give the readers an overview of the effect ofelastic bearings on the bridge response, two three-dimensional I−S−L/d plots for the impact factor of the midpoint deflection of the beamwere given in Figs. 3.16 and 3.17 for two values of stiffness ratios,κ = 0.15 and 0.25. A comparison of the two figures indicates that theuse of less rigid elastic bearings, as implied by a larger stiffness ratioκ, tends to increase the impact response of the elastically-supportedbeam.



Fig. 3.12. The I − S plot for 23 m simple beam.

Fig. 3.13. The I − S plot for 23 m elastically-supported beam.



Fig. 3.14. The I − S plot for 27 m simple beam.

Fig. 3.15. The I − S plot for 27 m elastically-supported beam.



Fig. 3.16. The I − S − L/d diagram for ξ = 0.02, κ = 0.15.

Fig. 3.17. The I − S − L/d diagram for ξ = 0.02, κ = 0.25.




In this chapter, the impact response of elastically-supported beamssubjected to a sequence of moving loads has been investigated by ananalytical approach. Light damping is assumed for the beams con-sidered. Both the conditions for resonance and cancellation to occurhave been identified. Unlike the case for a beam with zero damping,whose resonance response tends to grow continuously as there aremore moving loads passing the beam, the resonance response for adamped beam remains more or less constant, regardless of the num-ber of moving loads that have moved over the beam. For the case ofan infinite number of moving loads, envelope impact formulas havebeen derived for the elastically-supported beam with light damping.It is concluded that the installation of elastic bearings may generallyincrease the response of the beam under most resonance conditions.The more flexible the elastic bearings, the larger is the peak responseof the beam.

In the following chapter, the mechanisms for occurrence of theresonance and cancellation phenomena on an elastically-supportedbeam will be unveiled. In particular, it will be shown that the can-cellation condition is a condition that is more decisive than the reso-nance condition. In case there occurs the cancellation condition, allthe resonance response will just be suppressed.


Chapter 4

Mechanism of Resonanceand Cancellation for

Elastically-Supported Beams

This chapter can be regarded as a supplement to the preceding chap-ter on elastically-supported beams. It is aimed at unveiling the mech-anism underlying the phenomena of resonance and cancellation onelastically-supported beams induced by passing trains, which is veryuseful for the mitigation of train-induced vibrations. Basically thesame approach as in the preceding chapter was followed. The train ismodeled as a sequence of equi-distant moving loads. The vibrationshape of the elastically-supported beam is approximated as the com-position of a flexural sine mode and a rigid body mode. The analysisindicates that resonances of much higher peaks can be excited onbeams with elastic supports by trains moving at much lower speeds,compared with those for beams with rigid supports. On the otherhand, the speed for cancellation to occur is generally independentof the support stiffness. Of particular interest is that cancellation isa phenomenon more decisive than resonance. Whenever the condi-tion of cancellation is met, the dynamic response of the beam willbe suppressed, whether the beam be under resonance or not. Theanalytical results presented herein were verified by a field test on twoelastically-supported bridges located at existing railway lines.

4.1. Introduction

Taiwan is located at one of the most active seismic zones in the Pa-cific Rim. In order to prevent the bridge structures from damages orcollapse under severe earthquakes, various protective measures have

101



been adopted by structural engineers. Elastic bearings represent akind of devices commonly installed at the supports of bridge gird-ers for isolating the earthquake forces transmitted from the ground.While they are effective for isolating the ground-borne seismic forces,they can equally prevent the vehicle-induced vibrations from trans-mission or dissipation to the supports and then to the ground orsoils. This is certainly one disadvantage with the use of elastic bear-ings. For high-speed railway bridges installed with elastic bearings,it is likely that the huge amount of vibration energy brought by thehigh-speed train be accumulated in the bridge, which may result inamplified or resonant vibrations on the bridge at some critical speeds.The repetitive nature of resonant vibrations may cause early fatigueproblems on the associated railway tracks, as well as deterioration inthe riding quality of passengers, while increasing the cost of mainte-nance for the railway lines.

Due to the regular, repetitive nature of the wheel loads that con-stitute a train, both the phenomena of resonance and cancellationmay be induced on the bridge by the train moving at high speeds.The resonance phenomenon relates to the continuous built-up of thefree-vibration response on the bridge as there are more loads passingby. In contrast, the cancellation phenomenon implies that the wavesassociated with the free-vibration responses of the bridge generatedby the sequential moving loads cancel out each other. If the reso-nance condition can be reached by a train within its range of speedof operation, then some detrimental effects can be expected on thetrack system, as well as on the moving train itself. Such a problemwill be aggravated when the factor of elastic bearings is taken intoaccount.

Previously, rather few research works have been conducted on thedynamic response of elastically-supported beams to moving loads.Based on the work of Yau et al. (2001), envelope formulas were pre-sented in the preceding chapter for elastically-supported beams withlight damping subjected to moving loads. Recently, Lin (2001) inves-tigated the vibrations of railway bridges installed with elastic bear-ings, together with measures for vibration reduction. In this chap-ter, focus is placed on the physical interpretation of the mechanism


Resonance and Cancellation for Elastically-Supported Beams 103

involved in the phenomena of resonance and cancellation of the dy-namic response of elastically-supported beams to moving loads. Thekey factors affecting the dynamic response of bridges will be investi-gated, with comments made concerning the suitability of using elasticbearings as seismic isolation devices for railway bridges. The resultsobtained from a field test that serve to verify the theory are alsopresented. This chapter has been rewritten mainly from the paperby Yang et al. (2004).

4.2. Formulation of the Theory

The bridge model adopted is the one shown in Fig. 4.1, in which abeam supported by two identical elastic bearings is considered. Aswas done in Chapter 3, the following assumptions will be adopted inthe derivation of a closed-form solution for the elastically-supportedbeam under the moving loads: (1) The vertical stiffness of each elasticbearing is K. The mass of the spring is negligible compared withthat of the bridge. (2) The train is modeled as a sequence of equally-spaced moving loads, with the inertial effect neglected. (3) For beamssubjected to moving loads, which is basically a transient problemwith very short acting time, only the fundamental mode of vibrationof the bridge needs to be considered, while the higher modes can beneglected without losing accuracy (Biggs, 1964). (4) The dampingof the beam can be neglected, also due to the transient nature of themoving loads over the beam.

4.2.1. Assumed Modal Shape of Vibration

By the concept of modal superposition, the deflection u(x, t) of theelastically-supported beam can be expressed as

u(x, t) =∑

φn(x)qn(t) , (4.1)

where qn(t) denotes the generalized coordinate and φn(x) the shapefunction of the nth vibration mode. As was stated above, only thefundamental mode of vibration will be considered in analyzing the vi-brational response of the beam, as it is essentially a transient problem



K

m , E I , L

K

KK

flexible beam

rigid beam

Fig. 4.1. The model beam as a superposition of simple and rigid beams.

with very short acting time. If a mathematically exact approach isemployed to find the first modal shape of vibration for the elastically-supported beam, the final form of the modal shape will be rathercomplex, rendering it impossible to obtain simple closed-form solu-tions. As the first priority herein is to derive a closed-form solution,by which the mechanism behind the key phenomena can be inter-preted, the vibration shape of the elastically-supported beam will beapproximated as the superposition of the first modal shape of the flex-ural deflection of the beam with simple supports and the first modalshape of a rigid beam supported by two elastic springs, as indicatedin Fig. 4.1, that is,

φ(x) = sinπx

L+ κ , (4.2)

where κ = EIπ3/KL3 denotes the ratio of the flexural rigidity EI ofthe beam to the stiffness K of the elastic bearing, and L the lengthof the beam. As can be seen, a higher stiffiness ratio κ means asofter elastic spring and a zero stiffness ratio means the special caseof simple supports. The shape function φ(x) in Eq. (4.2) differs fromthat used in Chapter 3, i.e., the one implied by Eq. (3.4) by a factor1/(1+κ), which is acceptable, since it is known that shape functionsdo not have absolute magnitudes.



4.2.2. Single Moving Load

For an elastically-supported beam subjected to a single moving loadp of speed v, the equation of motion for the deflection u(x, t) of thebeam is

mu + EIu′′′′ = pδ(x − vt) for 0 ≤ vt ≤ L , (4.3)

where m is the mass per unit length and EI the flexural rigidity ofthe beam. The boundary conditions are

EIu′′(0, t) = 0 ,

EIu′′(L, t) = 0 ,

EIu′′′(0, t) = −Ku(0, t) ,

EIu′′′(L, t) = Ku(L, t) .

(4.4)

By multiplying both sides of Eq. (4.3) by the shape function φ(x) inEq. (4.2) and integrating with respect to the length L of the beam,one obtains

q + ω2q =2pmL

(1 +

8κπ

+ 2κ2

)−1(sin

πvt

L+ κ

), (4.5)

where the frequency of vibration ω is

ω = ω0

√π + 4κ

π + 8κ + 2πκ2, (4.6)

which is a function of the stiffness ratio κ. Here, ω0 indicates thefrequency of vibration of the associated beam with simple supports,

ω0 =(π

L

)2√

EI

m. (4.7)

It has been demonstrated that the fundamental frequency ω of vi-bration solved using the present approximate shape function φ(x),as given in Eq. (4.6), is in excellent agreement with the exact one,as indicated in Chapter 3.



The generalized coordinate q(t) can be solved from Eq. (4.5), to-gether with zero initial conditions, as

q(t) =2pL3

EIπ4

(1 +

4κπ

)−1 [(sin Ωt − S sin ωt

1 − S2

)+ κ(1 − cos ωt)

],

(4.8)

where Ω denotes the driving frequency implied by the moving load,

Ω =πv

L, (4.9)

and S is a speed parameter defined as the ratio of the driving fre-quency Ω to the frequency ω of vibration of the bridge, i.e.,

S =Ωω

=πv

ωL. (4.10)

In Eq. (4.8), the term containing the stiffness ratio κ represents theeffect of the elastic supports. For the special case of simple sup-ports, i.e., with κ = 0, the present solution reduces to that given inChapter 2 for simply-supported beams.

4.2.3. A Series of Moving Loads

As shown in Fig. 4.2, consider now that the elastically-supportedbeam is subjected to a series of concentrated loads p of equal intervalsd moving at speed v, as a representation of the loading action of atrain consisting of N cars of length d. The equation of motion forthe beam should now be modified as

mu + EIu′′′′ = p

N∑k=1

δ[x − v(t − tk)]

·[H(t − tk) − H

(t − tk − L

v

)], (4.11)

where δ(x) denotes the Dirac delta function, H(x) the unit stepfunction, tk = (k − 1)d/v the arriving time of the kth load at thebeam, and N is the total number of moving loads. The term H(t −tk) indicates the arrival of the kth load at the beam and the term



Fig. 4.2. The elastically-supported beam subjected to equi-distant moving loads.

H(t− tk −L/v) the departure of the same load from the beam. Theboundary conditions given in Eq. (4.4) remain valid.

Based on the hypothesis of linear theories, the deflection of thebeam induced by a sequence of moving loads can be obtained as thesuperposition of the deflection induced by each of the moving loads, ifthe time lag of each moving load is taken into account. Consequently,the generalized deflection q(t) of the beam for the present case canbe obtained as a generalization of Eq. (4.8) as

q(t) =2pL3

EIπ4

(1 +

4κπ

)−1

[Q1(t) +Q2(t)] , (4.12)

where Q1(t) represents the contribution caused by the flexural vibra-tion of the beam with simple supports, and Q2(t) the rigid displace-ment of the elastic bearings, namely,

Q1(t) =1

1− S2N∑

k=1

[sinΩ(t− tk)− S sinω(t− tk)]H(t− tk)

+[sinΩ

(t− tk − L

v

)− S sinω

(t− tk − L

v

)]

×H(t− tk − L

v

), (4.13a)



Q2(t) = κ

N∑k=1

[1 − cos ω(t − tk)]H(t − tk)

−[1 − cos ω

(t − tk − L

v

)]H

(t − tk − L

v

), (4.13b)

where Ω denotes the driving frequency implied by the moving loadsand ω the frequency of vibration of the elastically-supported beam.For the special case of κ = 0, the solution given in Eq. (4.12) reducesto that given in Chapter 2 for a beam with simple supports.

4.3. Conditions of Resonance and Cancellation

The generalized deflection of the elastically-supported beam given inEq. (4.12) consists of two parts, that is, the forced vibration caused bythe moving loads that are directly acting on the beam, as indicatedby the terms containing the driving frequency Ω, and the residual orfree vibration caused by the moving loads that have passed the beam,as indicated by the terms containing the bridge frequency ω. Whenall the moving loads have passed the beam, the forced vibration partterminates immediately. However, the free vibration part, which isof sinusoidal form, continues to be functional until the vibration iseventually damped out.

Both the phenomena of resonance and cancellation relate to thefree vibrations induced by the moving loads. When a moving loadhas passed the beam, waves of the sinusoidal form will be induced onthe beam. If the time lag of the wave components induced by eachmoving load equals a multiple of the period 2π/ω, then superpositionof all the wave components will result in amplified responses. This isthe so-called phenomenon of resonance. On the contrary, if the timelag equals an odd multiple of half of the period, the wave compo-nents induced by the sequentially moving loads will just cancel out,indicating that the phenomenon of cancellation has occurred.

Whether the phenomena of resonance or cancellation will occuror not depends only on the free vibration part of the motion. In

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order to interpret the two phenomena using the analytical solution,let us consider the critical case when the (N − 1)th moving load hasleft the beam and the Nth load has entered the beam, that is, whentN < t ≤ tN +∆t. Such a case is considered critical, since the beam isexcited to the utmost. From Eq. (4.12), one can obtain the followingfor such a case:

q(t) =2pL3

EIπ4

(1 +

4κπ

)−1

[A(t)H(t − tN ) + B(t)H(t − tN − ∆t)] ,

(4.14)

where the dynamic response factors A(t) and B(t) are

A(t) =1

1 − S2[sin Ω(t − tN ) − S sin ω(t − tN )]

+ κ[1 − cos ω(t − tN )] , (4.15a)

B(t) =( −2S

1 − S2cos

ωL

2v+ 2κ sin

ωL

2v

)×[

sin ω

(t − L

2v

)

+sinω(t − (L/2v) − (tN/2)) · sin ω((tN/2) − (d/2v))

sin(ωd/2v)

].

(4.15b)

In Eq. (4.14), the term A(t)H(t − tN ) indicates the forced vibrationof the beam caused by the Nth moving load, and the term B(t)H(t−tN − ∆t) the sum of all the free vibrations caused by the previousN − 1 moving loads that have already passed the beam.

Some physical interpretations can be given using Eq. (4.15b).First of all, if the denominator within the brackets vanishes, i.e.,when sin(ωd/2v) = 0, the response of the beam reaches a peak. ByL’Hospital’s rule, the second term within the brackets of the dynamicresponse factor B(t) under the resonance condition becomes

sin ω(t − (L/2v) − (tN/2)) · sin ω((tN/2) − (d/2v))sin(ωd/2v)

= (N − 2) sin ω

(t − L

2v

). (4.16)



Accordingly, the dynamic response factor B(t) can be written as

B(t) =( −S

1 − S2cos

ωL

2v+ κ sin

ωL

2v

)× 2(N − 1) sin ω

(t − L

2v

).

(4.17)

As can be seen, larger amplitude for the response can be expectedas there are more loads passing the beam, indicated by the term2(N − 1). Such a phenomenon is similar to that observed for beamswith simple supports, as discussed in Chapter 2. Correspondingto sin(ωd/2v) = 0, the condition of resonance is ωd/2v = iπ withi = 1, 2, 3, . . ., or v = ωd/(2iπ), which can also be written in termsof the speed parameter S based on the definition of Eq. (4.10) as

Sr =πv

ωL=

12i

· d

Lwith i = 1, 2, 3, . . . , (4.18)

which is the same as that for simply-supported beams. As can beseen either from Eq. (4.18) or Fig. 4.3, the resonant speed parameterSr is a function of the ratio d/L of the car length to the bridgelength. For trains of the commercially available models of whichthe car length d is known, the resonant speed parameter Sr, which isdimensionless, is generally small for bridges of practical length L. Anobservation from Eq. (4.18) is that the longer the span length L of abeam, the easier is for the resonance phenomenon to occur. It is truethat the resonant speed parameter Sr computed from Eq. (4.18) is thesame for both the elastically-supported and simply-supported beams.However, because the fundamental frequency of the former is muchlower than that of the latter, the resonant speed, i.e., vr = ωLSr/π,for vehicles moving over an elastically-supported beam will be muchlower than that over a simply-supported beam. Besides, it should benoted that according to Eq. (4.14), the amplitude of the resonanceresponse also depends on the stiffness ratio κ of the elastic supports.

On the other hand, by setting the parenthesized term inEq. (4.15b) equal to zero, all the residual free vibrations caused bythe previous N − 1 moving loads just cancel out. This is exactly the



Fig. 4.3. The Sr − d/L plot for resonance.

condition of cancellation for the elastically-supported beam:

Y (S) =−S

1 − S2cos

ωL

2v+ κ sin

ωL

2v= 0 . (4.19)

For simply-supported beams, κ = 0, the preceding condition reducesto cos(ωL/2v) = 0, or ωL/2v = π(2i− 1)/2 with i = 1, 2, 3, . . ., fromwhich the speed parameter S can be determined as

Sc =1

2i − 1with i = 1, 2, 3, . . . , (4.20)

which is valid only for the simply-supported beam, i.e., for the casewith κ = 0. The cancellation speed for the elastically-supportedbeam cannot be presented explicitly, since Eq. (4.19) is an implicitfunction. The solution computed from Y (S) = 0 in Eq. (4.19)for the cancellation speed parameter Sc has been plotted with re-spect to the stiffness ratio κ in Fig. 4.4. As can be seen, the can-cellation speed parameter Sc increases slightly as the stiffness ra-tio κ increases. For example, for the case with i = 2, we haveSc/Sc(κ=0) ≈ 1 + 0.5κ. However, since the fundamental frequency



Fig. 4.4. The Sc − κ plot for cancellation.

ω of the elastically-supported beam also decreases as the stiffness ra-tio κ increases, i.e., ω/ω0 ≈ 1− 0.5κ. It turns out that the differencebetween the real cancellation speed vc, computed as vc = ScωL/π,for vehicles moving over an elastically-supported beam and that overa beam with simple supports is generally small.

4.4. Mechanism of Resonance and Cancellation

As was stated before, the impact factor I as defined in Eq. (1.1) hasoften been used to account for the dynamic amplification effect onthe bridge due to the passage of moving vehicles through increase ofthe design forces and stresses. In order to unveil the mechanism un-derlying the phenomena of resonance and cancellation of the bridgeresponses in relation to the effect of elastic supports, two bridges, B1and B2, will be considered, of which the key properties have beenlisted in Table 4.1. The train is simulated as eight moving loads ofequal weight p = 220 kN spaced at an interval d = 25 m. By chang-ing the vertical stiffness of the elastic bearings, say, allowing it to



Table 4.1. The properties of the bridges used in analysis.

L (m) m (t/m) EI (kN-m2) ω0 (rad/s)

B1 bridge 23 30 1.4 × 108 40.3B2 bridge 27 32 2.0 × 108 33.9

Fig. 4.5. The I − κ − S plot and contour map (bridge B1).

vary in terms of the stiffness ratio κ from 0 to 0.4, and computingthe impact factor I for the midpoint deflection of the beam due tothe loads moving at different speeds, one can establish the I − κ−S

plots as in Figs. 4.5 and 4.6 for the two bridges.



Fig. 4.6. The I − κ − S plot and contour map (bridge B2).

As can be seen from Fig. 4.5, for bridge B1, higher stiffness ra-tios κ generally result in higher resonant peaks, indicating that theelastic bearings inserted at the bridge supports tend to amplify thebridge response. Such a phenomenon can be clearly explained usingFig. 4.7, where two impact curves were plotted each for κ = 0 (i.e.,for the beam with simple supports) and κ = 0.2. As can be seenfrom Fig. 4.7(a), the resonance phenomenon appears to be not sovisible for the simply-supported beam in the range S < 0.1, whichcan be practically ignored. However, it is amplified drastically andbecomes rather significant and non-negligible due to the installation



(a)

(b)

Fig. 4.7. The impact response of bridge B1: (a) I − S plot and (b) I − v plot.

of the elastic bearings on the bridge. In the design of high-speedrailway bridges, the detrimental effect of elastic bearings in amplify-ing the low-speed resonant responses should be seriously taken intoconsideration.

On the other hand, a comparison of Fig. 4.7(a) with Fig. 4.7(b)reveals two interesting facts. First, the real resonant speed vr for



an elastically-supported bridge is much smaller than that for thebridge with simple supports (see Fig. 4.7(b)), although the resonantspeed parameter Sr remains the same for both cases (see Fig. 4.7(a)).Second, the real cancellation speed vc seems to be close for boththe elastically- and simply-supported bridges (see Fig. 4.7(b)), al-though the corresponding speed parameter Sc is slightly larger forthe elastically-supported bridge (see Fig. 4.7(a)). Such findings areconsistent with the statements made following Eq. (4.20). In study-ing the impact response on bridges, the results are often presentedas an I −S plot for its elegance in expression and convenience in ex-tension to more general cases, as both I and S are nondimensional.However, we should not misinterpret the real physical meanings im-plied by these nondimensional parameters.

As can be seen from Fig. 4.6, for the speed parameter in therange S < 0.11, larger resonant peaks can be expected for bridgeB2 for larger stiffness ratio κ. However, the same is not true forthe speed parameter in the range S > 0.11. The reason can begiven as follows. First, for bridge B2, the car/bridge length ratio isd/L = 25/27 = 0.926. By drawing a vertical line at d/L = 0.926, onecan obtain from the resonance plot (i.e., Fig. 4.3) several intersectionsof which the ordinates (for Sr) represent the points of resonance, asindicated in Fig. 4.8(a). Because the resonance condition in terms ofspeed parameter is independent of the stiffness ratio κ, one can drawa resonance plot as shown in Fig. 4.8(b) for bridge B2, in whicheach horizontal line represents one of the ordinates for resonance(i.e., passes through one of the intersections) shown in Fig. 4.8(a).Finally, we can superimpose the resonance plot of Fig. 4.8(b) withthe cancellation plot of Fig. 4.4 to obtain the resonance/cancellationplot as shown in Fig. 4.8(c), which contains all the information weneed for explaining the dynamic response of bridge B2.

Take the resonance line S = 0.23 in the resonance/cancellationplot, i.e., Fig. 4.8(c), for example. The cancellation line below be-comes closer to this line as the stiffness ratio κ increases, whichmeans that the resonance peaks will be suppressed as the stiffnessratio κ increases. The fact that cancellation is a condition moredecisive than resonance is attributed to the fact that the dynamic



(a) (b)

(c)

Fig. 4.8. The mechanism of resonance and cancellation: (a) points of resonance,(b) resonance plot, and (c) resonance/cancellation plot.

response factor B(t) simply vanishes under the cancellation condi-tion, regardless of the presence of resonance, as can be observed fromEq. (4.15b). For the same reason, we can explain why the resonantpeak at S = 0.11 first deminishes and then grows as the stiffnessratio κ increases. This is due to the fact that the resonance andcancellation lines are close in the beginning, but are getting apart



for increasing κ. After explaining the mechanism of cancellationand resonance for the train-induced response of elastically-supportedbridges, we shall proceed to verify the theory presented herein bysome field tests.

4.5. Field Measurement of Vibration ofRailway Bridges

There have been reports by train drivers on the unusually high levelsof oscillations when maneuvering the trains to traverse the bridgesover the Fongshan Creek on the Western Railway Lines in north-ern Taiwan. In order to seek for the reasons behind the problem, afield test was carried out for two adjacent bridges, A1 and A2, lo-cated at the Fongshan Creek, which were not known to be elastically-supported at the time of testing. Two locomtives of type E300 (seeFig. 4.9) linked back-to-back were used to generate the action of mov-ing loads at different speeds. Some typical dimensions of the bridgesand locomotives were shown in Fig. 4.10. Each locomotive has abogie-to-bogie distance of 9.6 m and a gross weight of 96 t, muchheavier than the normal passenger cars used. The railway gauge is1067 mm, which is typical in Taiwan. The fundamental frequen-cies of the two bridges measured from an ambient vibration test are:f = 5.17 Hz for bridge A1 and f = 5.13 Hz for bridge A2, whichrepresent the combined dynamic effect of all the components con-stituting the railway bridge, including the continuous rails, sleepers,ballast, elastic bearings, the girder and two side flanges that form across section of the U shape. By design, the two bridges are identi-cally the same, but due to degradation in material properties theyturn out to be slightly different in the material properties.

During the testing, the two locomotives connected back-to-backare allowed to travel on one side, i.e., the test side, of the two-track railway running through the tested bridges at the followingspeeds: 15, 30, 45, 60, 75, 85, and 110 km/h. The maximum accel-erations measured at the midpoint of the A1 and A2 bridges usingseismometers during the passage of the two locomotives at differentspeeds have been plotted in Fig. 4.11. The maximum accelerations



Fig. 4.9. The E300 locomtive (unit: mm).

Fig. 4.10. The schematic of tested bridges.

computed for the midpoint of the two bridges by simulating the twolocomotives as four equal-weight moving loads, but with simple sup-port conditions, using the finite element analysis program developedin Chapter 6 is also plotted.

One observation from the measured and computed resultsshown in Fig. 4.11 is that they both show the occurrence of a peak re-sponse at the speed arround 60 km/h. Such a speed can be recognizedas one of the resonance speeds, as can be verified from Eq. (4.18),



Fig. 4.11. The maximum midpoint accelerations.

that is,

Bridge A1: vr =ωd

2nπ=

fd

n

∣∣∣∣ f=5.17d=9.6n=3

= 16.55 m/s = 59.6 km/h .

Bridge A2: vr =ωd

2nπ=

fd

n

∣∣∣∣ f=5.13d=9.6n=3

= 16.42 m/s = 59.1 km/h .

However, the computed response in Fig. 4.11 appears to be muchsmaller than the measured ones for the two bridges considered, whilethe peak response around the speed of 60 km/h is not quite visi-ble. This is primarily due to the adoption of simple supports forthe bridges in the finite element analysis. We assumed the bridgesto be supported by hinges simply because we were not aware of theexistence of elastic bearings at the bridge supports. Nevertheless,the relatively high amplitudes of the measured responses for thetwo bridges, compared with the computed one, suggested that the



substantial magnification in the bridge response may be caused bythe existence of elastic bearings, which was latter known to the au-thors. Such a problem seems to have received little attention fromresearchers working on railway bridges in the past.

Based on the static deflection tests under the locomotive loads,which were conducted as part of the preliminary tests, the spring con-stants measured for the A1 and A2 bridges are K1 = 3.5×106 kN/mand K2 = 8 × 106 kN/m, respectively, assuming that the elasticbearings installed at the two ends of a bridge are the same. All thekey properties identified for the tested bridges, including the flexuralrigidity EI and stiffness ratio κ, have been listed in Table 4.2. Withthe data given in Table 4.2, the midpoint responses computed by thefinite element program for the A1 and A2 bridges using the movingloads assumption were plotted in Fig. 4.12. As can be seen, becauseof the inclusion of elastic bearings, the computed responses turn outto be generally consistent with the measured ones for the two bridges.Moreover, larger response exists for the A1 bridge simply because ithas softer support bearings.

Let us now turn to the phenomenon of cancellation. FromEq. (4.20), for the case with simple supports, the speed parameter S

for cancellation to occur is:

Sc =1

2i − 1

∣∣∣∣i=8

= 0.067 .

Correspondingly, the cancellation speeds for the two bridges are

Bridge A1: vc =ω0LSc

π= 2fLSc = 2 × 5.17 × 31.3 × 0.067

= 21.68 m/s = 78 km/h .

Bridge A2: vc =ω0LSc

π= 2fLSc = 2 × 5.13 × 31.3 × 0.067

= 21.51 m/s = 77.5 km/h 78 km/h .

This is exactly the cancellation speed for the bridge with simple sup-ports, as can be seen from the numerical solution shown in Fig. 4.11.



Table 4.2. The properties of tested bridges.

L (m) f (Hz) K (kN-m) EI (kN-m2) κ = EIπ3/KL3

A1 bridge 31.3 5.17 3.5 × 106 2.40 × 108 0.069A2 bridge 31.3 5.13 8 × 106 2.44 × 108 0.031

Fig. 4.12. The computed solutions for elastic supports.

To consider the effect of elastic bearings, the speed parameter S

for the cancellation to occur can be solved from Eq. (4.19) or

Y (S) =−Sc

1 − S2c

cosπ

2Sc+ κ sin

π

2Sc= 0 . (4.21)

By substituting the stiffness ratios κ of 0.063 and 0.031, as givenin Table 4.2, into the preceding equation, the speed parameter S

solved for the A1 and A2 bridges respectively are 0.069 and 0.068.Correspondingly, the cancellation speeds v for the two bridges are

Bridge A1: vc = 0.069 × 2fL = 0.069 × 2 × 5.17 × 31.3

= 22.3 m/s = 80.4 km/h .



Bridge A2: vc = 0.068 × 2fL = 0.068 × 2 × 5.13 × 31.3

= 21.8 m/s = 78.5 km/h .

Clearly, the above (computed) speeds are consistent with the (mea-sured) speeds for the occurrence of minimal responses for the twobridges, as shown in Fig. 4.12, which is an indication of the reliabil-ity of the present theory concerning the occurrence of resonance andcancellation.


In this chapter, the mechansims underlying the resonance and can-cellation phenomenona of elastically-supported bridges caused by asequence of equi-distant moving loads have been analytically stud-ied. A field test on two adjacent bridges traveled by two back-to-backconnected locomotives was also conducted to confirm the phenom-ena of resonance and cancellation identified. The conclusions drawnfrom this chapter are: (1) The resonance condition in terms of thespeed parameter S is the same for the beam with both the elas-tic and simple supports. Since an elastically-supported beam has alower frequency of vibration, it therefore has a lower resonant speedv, meaning that it can be more easily excited than a beam with sim-ple supports. (2) The speed parameter for the cancellation conditionto occur increases slightly as the stiffness ratio increases. However,since the frequency of vibration is slightly smaller for an elastically-supported beam, it turns out that the real cancellation speed foran elastically-supported beams remains close to that for the simply-supported beam. (3) Whenever the cancellation speed comes closeto or coincides with the resonance speed, the phenomenon of reso-nance will be suppressed, meaning that the cancellation condition ismore decisive than the resonance condition. (4) Once a resonancecondition is reached for an elastically-supported beam, much largerpeak responses will be induced on the beam, compared with those ofthe simply-supported beam.



There is no doubt that elastic bearings are effective devices forisolating the earthquake forces transmitted from the ground to thesuperstructure. However, the installation of these devices can alsoprevent the transmission or dissipation of vehicle-induced forces fromthe superstructure to the ground. Thus, the huge amount of vibra-tion energy brought by a train may be accumulated and amplified onthe bridge during its passage. Such a fact should not be overlookedin the design of railway bridges, especially of those to be traveled byhigh-speed trains, since it is harmful not only for the riding comfortof the passing trains, but also for the maintenance of track struc-tures, as the repetitive occurrence of high-amplitude resonant peaksmay cause fatigue problems on related components.


Chapter 5

Curved Beams Subjected to Verticaland Horizontal Moving Loads

Analytic solutions are derived in this chapter for a horizontal curvedbeam subjected to vertical and horizontal moving loads. The hori-zontal moving loads may be regarded as the centrifugal forces gen-erated by vehicles moving along a curved path, which were rarelystudied by previous researchers. Unlike the vertical moving loads,a horizontal moving load is not constant, but is proportional to thesquare of the speed of the moving vehicle. The case of a single mov-ing load will be studied first, which will then be extended to thecase of a series of equidistant moving loads. By superposition ofthe waves generated by consecutively moving loads on the curvedbeam, the conditions for the resonance and cancellation phenomenato occur will be derived. As compared with the approaches that arebased fully on numerical simulation, the present analytical approachhas the advantage of providing clear physical insight into the variousphenomena induced by moving vehicles, thereby enabling us to graspthe key parameters involved in the curved-beam dynamics.

5.1. Introduction

The vehicle-induced vibration on bridges has been a subject of in-terest for more than one and half centuries. The pioneer works onthis subject include those of Willis (1849) and Stokes (1849) follow-ing the collapse of the Chaster Rail Bridge in England in 1847. Theproblem of simple beams under moving vehicular loads was studiedby Timoshenko (1922) neglecting the inertia effect of the vehicle.

125



In addition to the inertia of the beam, the inertia of the vehiclewas included by Jeffcott (1929) in his study, followed by Stanisicand Hardin (1969) and Ting et al. (1974). By taking into accountthe suspension properties of the vehicle, the sprung mass model wasadopted in the studies by Tan and Shore (1968a,b), Veletsos andHuang (1970), Blejwas et al. (1979), etc. The book by Fryba (1972)represents an early treatise on this subject, in which vehicle models ofvarious complexities were studied. Partly enhanced by the operationof high-speed railways worldwide, research on the vehicle-induced vi-brations of bridges continued to grow in the past two decades. Theproblems that have been studied include the implementation of morerealistic vehicle models (Chu et al., 1986; Hwang and Nowak, 1991)and bridge models (Galdos et al., 1993; Chatterjee et al., 1994), theinclusion of surface roughness (Hwang and Nowak, 1991; Inbanathanand Wieland, 1987), or the derivation of various solution schemes,including the vehicle–bridge interaction (VBI) elements that will bepresented in later chapters. Some partial reviews of the research onvehicle-induced vibrations of the bridge are available in Diana andCheli (1989), Paultre et al. (1992), and Yang and Yau (1998). In arecent book by Fryba (1996) on the dynamics of railway bridges, 231references have been cited.

Previously, a majority of research conducted on vehicle-inducedvibrations has been aimed at straight beams. While some researchhas been conducted for horizontally-curved beams under movingloads (Tan and Shore, 1986a,b; Genin et al., 1982), concern was gen-erally placed on the vertical or out-of-plane vibration of the curvedbeam. To the knowledge of the authors, the radial or in-plane vibra-tion of curved beams under the action of centrifugal forces, generatedby masses moving along circular paths, has rarely been studied. Justas a vertical moving load will cause some impact effect on the verti-cal vibration of a straight beam, a centrifugal force generated by amass moving over a horizontally-curved beam will also induce certainimpact effect on the radial response of the beam. The objective ofthis chapter is to establish a general theory for treating the vibra-tion of a horizontally-curved beam subjected to a series of movingmasses, of which will be simulated as a pair of gravitational force


Curved Beams Subjected to Vertical and Horizontal Moving Loads 127

and centrifugal force. The conditions for superposition of the wavesgenerated by consecutively moving loads to result in the phenomenaof resonance and cancellation on the curved beam will be identified.The reliability of the present theory will be demonstrated in the ex-emplary studies. The materials presented in this chapter have beenrevised mainly from the paper by Yang et al. (2001).

5.2. Governing Differential Equations

Consider the horizontally-curved beam shown in Fig. 5.1, in whichϕ denotes the subtended angle, R the radius of curvature, and L

the length of the beam. For the present purposes, a right-handedcoordinate system is chosen, of which the y and z axes coincide withthe principal axes of the cross section, and the x axis is tangentto the centroidal axis of the beam. Let ux, uy and uz denote thedisplacements of the centroid of each cross section of the curved beamalong the three axes, and θx, θy and θz the rotations about the three

Fig. 5.1. The coordinates of curved beam.



axes. All the deformations are assumed to be small so that the lineartheory applies. The curved beam is assumed to be made of constant,bi-symmetric cross sections with negligible warping resistance.

The following are the linear differential equations for ahorizontally-curved beam of solid cross sections (Yang and Kuo,1987, 1994):

• Axial displacement:

EA

(u′′

x +u′

z

R

)= 0 . (5.1)

• Radial displacement:

EIy

(u′′′′

z + 2u′′

z

R2+

uz

R4

)+

EA

R

(u′

x +uz

R

)= 0 . (5.2)

• Vertical displacement:

EIz

(u′′′′

y − θ′′xR

)− GJ

R

(θ′′x +

u′′y

R

)= 0 . (5.3)

• Torsional rotation:

EIz

R

(u′′

y −θx

R

)+ GJ

(θ′′x +

u′′y

R

)= 0 , (5.4)

where a prime denotes differentiation with respect to the axis x, E

and G denote the moduli of elasticity and rigidity, respectively, of thebeam, A the cross-sectional area, Iy and Iz respectively the momentsof inertia about the y and z axes, and J the torsional constant.From Eqs. (5.1)–(5.4), one can observe that the differential equationsfor the in-plane displacements, i.e., ux and uz, are independent ofthose for the out-of-plane displacement, i.e., uy and θx. Furthermore,the differential equation for the axial displacement ux and that forthe radial displacement uz are coupled, and the same is true for thevertical displacement uy and torsional rotation θx.



5.3. Curved Beam Subjected to a Single Moving Load

The two ends of the curved beam are assumed to be simply-supported, in the sense that the flexural displacements and twistingrotation of the beam are restrained at the supports, but their firstderivatives are not zero. In general, the action of the moving ve-hicle can be replaced by a vertical moving load, to simulate thegravitational effect, and a horizontal moving load, to simulate thecentrifugal effect, as shown in Fig. 5.2(a).

5.3.1. Vertical Moving Load

As the in-plane and out-of-plane behaviors of the curved beam areuncoupled, we shall consider first the vertical vibration of the curvedbeam under the action of a vertical moving load (Fig. 5.2(c)). Letmv denote the mass of the vehicle moving at speed v. The load ofthe vehicle is fv = −mvg, where g is the acceleration of gravitation.By taking into account the effect of inertia, the equations of motionfor the vertical vibration of the curved beam can be modified fromEqs. (5.3) and (5.4) as follows:

muy + EIz

(u′′′′

y − θ′′xR

)− GJ

R

(θ′′x +

u′′y

R

)= fvδ(x − vt) , (5.5a)

ρJθx +EIz

R

(u′′

y − θx

R

)+ GJ

(θ′′x +

u′′y

R

)= 0 , (5.5b)

where m denotes the mass per unit length, ρ the density of the curvedbeam and δ(x) is Dirac’s delta function. The term on the right-hand side of Eq. (5.5a) represents the effect of the vertical movingload fv, where fv = −mvg. For the present problem, the verticaldisplacement uy can be expressed as the summation of a series ofsine functions that satisfy the boundary conditions:

uy(x, t) =∞∑i=1

qyi(t) siniπx

L, (5.6)



vf

hf

O

(a)

hf

O

(b)

vf

O

(c)

Fig. 5.2. The curved beam subjected to moving loads: (a) general,(b) horizontal, and (c) vertical.



where qyi denotes the generalized coordinate for the ith mode. Theexpression for the angle of twist θx is not arbitrary, but must bedetermined from Eq. (5.4). Substituting Eq. (5.6) into Eq. (5.4) andmaking use of boundary conditions for twisting, one obtains

θx(x, t) =∞∑i=1

γiqyi(t) siniπx

L, (5.7)

where γi = −R(iπ/L)2(GJ + EIz)/[(iπ/L)2GJR2 + EIz], orequivalently,

θx(x, t) =∞∑i=1

qθi(t) siniπx

L, (5.8)

where qθi denotes the ith generalized coordinate for the angle of twistθx. For the present purposes, let us consider only the contributionof the first modes, i.e.,

uy(x, t) = qy1(t) sinπx

L, (5.9a)

θx(x, t) = qθ1(t) sinπx

L, (5.9b)

where qy1 and qθ1 denote the first generalized coordinates for uy andθx, respectively.

To solve the two differential equations in Eq. (5.5), Galerkin’smethod will be used. First, one multiplies both sides of Eq. (5.5a) bythe variation δuy and those of Eq. (5.5b) by δθx. Use the first-modeapproximations for uy and θx in Eq. (5.9). Then integrate the twodifferential equations each with respect to x from 0 to L. The resultsare as follows:

(qy1 + a1qy1 + a2qθ1)δqy1 =2fv

mLsin

πvt

Lδqy1 ,

(qθ1 + b1qθ1 + b2qy1)δqθ1 = 0 ,

(5.10)



where

a1 =1m

(π

L

)2[EIz

(π

L

)2+

GJ

R2

],

a2 =1

mR

(π

L

)2(EIz + GJ) ,

b1 = − 1ρJ

[EIz

R2+ GJ

(π

L

)2]

,

b2 = − 1ρJ

1R

(π

L

)2(EIz + GJ) .

(5.11)

Since the variations δqy1 and δqθ1 are arbitrary, the two equations inEq. (5.10) reduce to

qy1 + a1qy1 + a2qθ1 =2fv

mLsin

πvt

L,

qθ1 + b1qθ1 + b2qy1 = 0 .

(5.12)

The general solutions to the two differential equations in Eq. (5.12)are composed of two parts, i.e., the homogenous solution and partic-ular solution,

qy1 = qy1h + qy1p ,

qθ1 = qθ1h + qθ1p ,(5.13)

where the subscripts h and p respectively denote the homogeneousand particular solutions.

The homogeneous solutions can be given as follows:

qy1h = h1 sin ωv1t + h2 cos ωv1t , (5.14a)

qθ1h = k1 sin ωv1t + k2 cos ωv1t , (5.14b)

where ωv1 denotes the fundamental frequency of vibration for thevertical direction of the curved beam and h1, h2, k1, k2 are the coef-ficients to be determined from the initial conditions. By substitutingthe equations in Eq. (5.14) into Eq. (5.12) and dropping the term



containing fv, one obtains[a1 − ω2

v1 a2

b2 b1 − ω2v1

]qy1h

qθ1h

=

00

. (5.15)

By letting the determinant in Eq. (5.15) equal to zero, the funda-mental frequency ωv1 can be solved as (see Appendix C)

ωv1 =

√a1 + b1 +

√(a1 − b1)2 + 4a2b2

2. (5.16)

As for the particular solutions, the following may be assumed:

qy1p = py1 sinπvt

L, (5.17a)

qθ1p = pθ1 sinπvt

L, (5.17b)

where py1 and pθ1 denote the amplitudes of vibration. Substitutingthe equations in Eq. (5.17) into Eq. (5.10) yields

a1 −(πv

L

)2a2

b2 b1 −(πv

L

)2

py1

pθ1

=

2fv

mL0

. (5.18)

From the preceding equation, py1 can be solved,

py1 =2fv

mL

1ω2

v1

11 − S2

v1

β , (5.19)

where

Sv1 =πv

Lωv1, (5.20)

and

β =b1 − (πv/L)2

b1 + a1 − ω2v1 − (πv/L)2

. (5.21)

Here, Sv1 denotes the speed parameter for the vertical vibration ofthe curved beam, which represents the ratio of the driving frequency,πv/L, implied by the moving load to the fundamental frequency ofthe beam, ωv1.



The initial displacement and velocity of the beam are zero beforeit is subjected to the moving load. Such conditions should be obeyedby the general solution qy1 given in Eq. (5.13), or the sum of qy1h

and qy1p given in Eqs. (5.14a) and (5.17a), respectively. By theseconditions, the coefficients of the homogeneous solution can be de-termined as: h1 = −py1Sv1 and h2 = 0. It follows that the generalsolution qy1 for the vertical vibration of the curved beam is

qy1(t) = py1

(sin

πvt

L− Sv1 sin ωv1t

). (5.22)

By the expression in Eq. (5.19) and the relation fv = −mvg, thepreceding equation may be rewritten as

qy1(t) = −2mvg

mL

1ω2

v1

11 − S2

v1

βΨv1(t) , (5.23)

where the amplitude function Ψv1(t) is

Ψv1(t) = sinπvt

L− Sv1 sin ωv1t . (5.24)

Consequently, the vertical displacement of the curved beam is

uy(x, t) = −2mvg

mL

1ω2

v1

11 − S2

v1

βΨv1(t) sinπx

L. (5.25)

For the midpoint of the curved beam, x = L/2, the vertical displace-ment is

uy

(L

2, t

)=

2mvg

mL

1ω2

v1

11 − S2

v1

βΨv1(t) . (5.26)

The above solution has been obtained by considering only the firstmode of vibration of the curved beam. More accurate solutions canbe obtained through consideration of more vibration modes. In prac-tice, however, the acting time of the moving vehicles on the curvedbeam is so short that the moving load problem is by nature a tran-sient vibration problem. As a result, only the first mode of the beamwill be significantly excited. This is especially true when only themidpoint response of the beam is desired, and when the beam issubjected to a series of moving loads, as encountered in high-speedrailways. The accuracy of the present solutions will be demonstrated



in the numerical example through comparison with the finite elementresults.

5.3.2. Horizontal Moving Load

The horizontal moving load fh considered herein is the centrifugalforce generated by a vehicle of mass mv moving at speed v along ahorizontally-curved beam of radius R. By taking into account theeffect of inertia, the equations of motion for the radial vibration of acurved beam can be modified from Eqs. (5.1) and (5.2) as

mux + EA

(u′′

x +u′

z

R

)= 0 , (5.27a)

muz + EIy

(u′′′′

z + 2u′′

z

R2+

uz

R4

)= fhδ(x − vt) , (5.27b)

where the horizontal moving load fh can be related to the speedv as fh = mvv

2/R. Similarly, the radial displacement uz can beexpressed as the summation of the sine functions that satisfy theboundary conditions of the curved beam as

uz(x, t) =∞∑i=1

qzi(t) siniπx

L, (5.28)

in which qzi denotes the ith generalized coordinate for the radialdisplacement. The relation between the radial displacement uz andaxial displacement ux is not arbitrary. By substituting Eq. (5.28)into Eq. (5.1), the axial displacement ux can be solved as

ux(x, t) =∞∑i=1

−qzi(t)R

(L

iπ

)1 − cos

iπx

L− [1 − (−1)i]

x

L

. (5.29)

If only the first modes are considered for the axial and radial vibra-tions, the following may be written:

ux(x, t) = qx1(t)(

1 − cosπx

L− 2x

L

),

uz(x, t) = qz1(t) sinπx

L,

(5.30)



where qx1 and qz1 respectively denote the first generalized coordinatefor the axial and radial displacements.

Again, Galerkin’s method will be employed to solve the twosecond-order differential equations in Eq. (5.27). Namely, by mul-tiplying both sides of Eq. (5.27a) by δux and those of Eq. (5.27b)by δuz , making use of the expressions for the displacement functionsux and uz in Eq. (5.30), integrating from 0 to L, and taking thearbitrary nature of virtual displacements δqx1 and δqz1, one obtains

qx1 + a1qx1 + a2qz1 = 0 ,

qz1 + b1qz1 + b2qx1δqz1 =2fh

mLsin

πvt

L,

(5.31)

where

a1 =EAπ2

L

(4π2

− 12

)/mL

(8π2

− 56

),

a2 =EAπ

R

(4π2

− 12

)/mL

(8π2

− 56

),

b1 =EIy

m

[(π

L

)2 − 1R2

]2+

EA

mR2,

b2 =EA

mRL

(π − 8

π

).

(5.32)

As was stated, the general solutions to the two differential equa-tions in Eq. (5.31) consist of two parts,

qx1 = qx1h + qx1p , (5.33a)

qz1 = qz1h + qz1p . (5.33b)

The homogenous parts can be given as

qx1h = h1 sin ωh1t + h2 cos ωh1t ,

qz1h = k1 sin ωh1t + k2 cos ωh1t ,(5.34)



where ωh1 denotes the frequency of vibration for the horizontal planeof the curved beam and the coefficients h1, h2, k1, k2 are to bedetermined from the initial conditions. Substituting Eq. (5.34) backinto the differential equations in Eq. (5.31), and dropping the termcontaining fh yields[

a1 − ω2h1 a2

b2 b1 − ω2h1

]qx1h

qz1h

=

00

. (5.35)

By letting the determinant equal to zero, the vibration frequency ωh1

can be solved as (Appendix D)

ωh1 =

√√√√ a1 + b1 −√

(a1 − b1)2 + 4a2b2

2. (5.36)

The particular solutions for the present problem are

qx1p = px1 sinπvt

L,

qz1p = pz1 sinπvt

L.

(5.37)

Substituting the preceding expressions for qx1p and qz1p into the dif-ferential equations in Eq. (5.31) yields

a1 −(πv

L

)2a2

b2 b1 −(πv

L

)2

px1

pz1

=

0

2fh

mL

, (5.38)

from which the generalized coordinate pz1 can be solved as

pz1 =2fh

mL

1ω2

h1

11 − S2

h1

α , (5.39)

where

Sh1 =πv

Lωh1, (5.40)

and

α =a1 − (πv/L)2

a1 + b1 − ω2h1 − (πv/L)2

. (5.41)



Here, Sh1 denotes the speed parameter for the horizontal vibration ofthe curved beam, which again is defined as the ratio of the drivingfrequency, πv/L, to the fundamental frequency ωh1 of the beam.

By assumption, the zero initial conditions must be obeyed by thegeneral solution qz1, or the one given in Eq. (5.32b). From theseconditions, the coefficients of the homogeneous solution qz1h can bedetermined as: k1 = −pz1Sh1 and k2 = 0. By the relation fh =mvv

2/R and using Eq. (5.39) for pz1, the radial displacement qz1 canbe derived as

qz1(t) =2mvϕ

mπ2

S2h1

1 − S2h1

αΨh1(t) , (5.42)

where ϕ = L/R for the curved beam and the amplitude functionΨh1(t) is

Ψh1(t) = sinπvt

L− Sh1 sin ωh1t . (5.43)

As a result, the radial displacement of the curved beam is

uz(x, t) =2mvϕ

mπ2

S2h1

1 − S2h1

αΨh1(t) sinπx

L. (5.44)

Of interest is the fact that when the vehicle speed v approaches zero,Sh1 → 0, the radial displacement uz(x, t) of the curved beam also ap-proaches zero. In contrast, the vertical displacement uy(x, t) as givenin Eq. (5.25) approaches a constant under the same condition. Thiscan be realized if one notes that as the vehicle speed v reduces tozero, so does the horizontal moving load fh, as there is no centrifu-gal force. However, the vertical moving load fv remains constant,regardless of the vehicle speed.

5.4. Unified Expressions for Vertical andRadial Vibrations

The solutions for the vertical and radial displacements, uy(x, t) anduz(x, t), as given in Eqs. (5.25) and (5.44) are similar in form. Duringthe travel time L/v of the vehicle on the beam, the two equations



can be expressed in the same form as

U(x, t) = PΨ(t)H(t) sinπx

Lfor 0 ≤ t ≤ L

v, (5.45)

where H(t) is a unit step function, indicating that the functionΨ(t) is turned on at t = 0. For the vertical vibration case, thedisplacement U(x, t) should be interpreted as uy(x, t), the loadP as −(2mvg/mL)(1/ωv1)[1/(1 − S2

v1)]β, and the amplitude func-tion Ψ(t) as Ψv1(t). For the horizontal vibration case, the dis-placement U(x, t) should be interpreted as uz(x, t), the load P as(2mvϕ/mπ2)[S2

h1/(1 − S2h1)]α, and the amplitude function Ψ(t) as

Ψh1(t). For t > L/v, the beam reaches a state of free vibration, asthe moving load already leaves the beam. By the fact that L/v de-notes half of the period of the moving load over the beam, the freevibration response of the beam at this stage is

U(x, t) = P

[Ψ(t)H(t) + Ψ

(t − L

v

)H

(t − L

v

)]

× sinπx

Lfor t >

L

v. (5.46)

Here, the amplitude function Ψ(t) is

Ψ(t) = sinπvt

L− S1 sin ω1t , (5.47)

where the frequency ω1 and speed parameter S1 should be interpretedas ωv1 and Sv1 for the vertical vibration, and as ωh1 and Sh1 forthe horizontal vibration. By substitution of Eq. (5.47) for Ψ(t) andnoting that H(t) = H(t − L/v) for t > L/v, sin(πvt/L) + sin[πv(t −L/v)/L] = 0 and sin a+sin b = 2 sin[(a+ b)/2] cos[(a− b)/2], one canrearrange the free vibration response in Eq. (5.46) as

U(x, t) = −2PS1 cosω1L

2vsinω1

(t − L

2v

)H

(t − L

v

)

× sinπx

Lfor t ≥ L

v. (5.48)

This is exactly the residual response of the beam after the movingload has left the beam, based on the first mode approximation.



By letting cos(ω1L/2v) = 0 or ω1L/2v = (2i − 1)π/2, withi = 1, 2, 3, . . ., one observes that the residual response of the beamreduces to zero. Such a condition is equivalent to

Sc =1

2i − 1for i = 1, 2, 3, . . . . (5.49)

Theoretically speaking, if the condition in Eq. (5.49) is met, theresidual response of the beam simply vanishes. For this reason, thecondition in Eq. (5.49) is referred to as the condition of cancellation.

5.5. Solutions for Multi Moving Loads

Based on the assumption of small deformations, the vibration of abeam caused by a series of moving loads can be obtained as thesuperposition of the vibration induced by each of the moving loads.In such a process, care must be taken in calculating the entering timeand the departing time for each moving load from the beam and thetime lag between any two consecutive loads.

Consider N identical masses of interval d moving at constantspeed v. As shown in Fig. 5.3, each of the masses will induce avertical load fv and a centrifugal force fh. Assuming that the firstmoving load enters the beam at t = 0, the time lag for the jth mov-ing load is tj = (j−1)d/v. By setting x = L/2, the residual vibrationresponse of the midpoint of the beam caused by the jth moving loadcan be modified from Eq. (5.46) through consideration of the timelag tj as

Uj

(L

2, t

)= P

[Ψ(t − tj)H(t − tj) + Ψ

(t − tj − L

v

)

×H

(t − tj − L

v

)]for t − tj ≥ L

v. (5.50)

The most critical condition for the beam occurs when the first N −1masses have left and only the Nth mass is acting on the beam. Thetime interval for such a case is max(tN , tN−1 + L/v) < t < tN + L/v,as shown in Fig. 5.4. For this case, the midpoint response of the



Fig. 5.3. The curved beam subjected to equi-spaced moving loads.

Fig. 5.4. The critical loading case.

beam is composed of two parts. The first part relates to the forcedvibration caused by the Nth moving load, which can be obtained byletting x = L/2 and replacing t by t − tN in Eq. (5.45), i.e.,

UN,1

(L

2, t

)= PΨ(t − tN )H(t − tN ) for 0 < t − tN <

L

v.

(5.51)



The second part is simply the residual vibration caused by the N −1moving loads that passed the beam. This can be obtained by lettingx = L/2 and replacing t by t − tj in Eq. (5.48), and then summingthe responses induced by the moving loads from j = 1 to j = N −1.a

Noting that when H(t − tN−1 − L/v) = 1, it can be ascertainedH(t − tj − L/v) = 1 for j = 1, 2, . . . , N − 1. Thus, the step functionH(t − tj− − L/v) for j = 1, 2, . . . , N − 1 can be replaced by H(t −tN−1 −L/v). The following is the result for part two or the residualvibration response:

UN,2

(L

2, t

)=

N−1∑j=1

−2PS1 cos

ω1L

2vsin ω1

[(t − tj) − L

2v

]

×H

(t − tN−1 − L

v

)for t − tN−1 ≥ L

v.

(5.52)

Following the procedure presented in Appendix E, one can removethe summation sign in the preceding equation and obtain the residualvibration response of the beam as

UN,2

(L

2, t

)= −2PS1 cos

ω1L

2v

[sinω1

(t − L

2v

)+ sin ω1

(N − 2

2d

v

)

× sin ω1

(t − L

2v− N − 1

2d

v

)sin−1 ω1d

2v

]

×H

(t − tN−1 − L

v

)for t − tN−1 >

L

v.

(5.53)

Consequently, the midpoint response of the beam under the actionof the Nth moving load can be computed as the summation of UN,1

and UN,2 given in Eqs. (5.51) and (5.53), respectively.aThis is equivalent to summing the expression in Eq. (5.50) for each load thatpassed the beam.



5.6. Conditions of Resonance and Cancellation

The phenomena of resonance and cancellation relate to the free vi-bration response U2 caused by the N − 1 moving loads that havepassed the beam. From Eq. (5.53), one observes that the response ofthe beam reaches a maximum when the denominator of some termsvanishes, i.e., when sin(ω1d/2v) = 0 or when ω1d/2v = iπ, withi = 1, 2, 3, . . ., or equivalently,

Sr =12i

d

Lfor i = 1, 2, 3, . . . . (5.54)

This is exactly the condition for resonance of the beam to occur. Forthe case with sin(ω1d/2v) = 0, the expression in Eq. (5.53) becomesindeterminate. By the condition sin(ω1d/2v) = 0 and L’Hospital’srule, Eq. (5.53) can be manipulated to yield

UN,2

(L

2, t

)= −2P (N − 1) cos

ω1L

2vsin ω1

(t − L

2v

)

×H

(t − tN−1 − L

v

). (5.55)

Here, it is interesting to note that the midpoint response of the beamincreases as there are more loads passing the beam.

On the other hand, one observes from Eq. (5.53) that whenevercos(ω1L/2v) = 0, or ω1L/2v = (2i − 1)π/2, with i = 1, 2, 3, . . ., orequivalently the one given in Eq. (5.49), the free vibration responseU2(L/2, t) reduces to zero, which means that no residual responsewill be generated by the moving loads that have passed the beam.Such a condition has been referred to as the condition of cancellation.The conditions of resonance and cancellation, as given in Eqs. (5.54)and (5.49), are identical in form for both the vertical and horizontaloscillations in terms of the speed parameter Sr. However, becausethe vertical and horizontal vibration frequencies, ωv1 and ωh1, aregenerally different for the beam, the resonance or cancellation phe-nomena for the two directions do not occur at the same speed v.



5.7. Numerical Examples

The curved beam considered herein is simply-supported, of lengthL = 24 m and subtended angle ϕ = 30. The following propertiesare assumed for the beam: cross-sectional area A = 9 m2, moments ofinertia Iy = 18.75 m4, Iz = 2.43 m4, torsional constant J = 21.18 m4,elastic modulus E = 33.2 GPa, Poisson’s ratio ν = 0.2, and densityρ = 2.4 t/m3. Each vehicle has a mass of mv = 29.9 t and thedistance between two adjacent vehicles is d = 25 m. Unless notedotherwise, all the data assumed here will be used throughout this sec-tion. For comparison, finite element solutions obtained by approxi-mating the curved beam by 10 piecewise straight beam elements willalso be presented. As the straight beam element was derived fromthe straight beam theory (Yang and Kuo, 1994), the results obtainedfrom the finite element analysis are totally independent of the presentanalytical results based on the curved beam theory (Yang and Kuo,1987), as represented by Eqs. (5.1)–(5.4).

5.7.1. Comparison of Analytic with Finite Element

Solutions

Consider a single moving mass with speed v = 40 m/s, which willgenerate a gravitational force and a centrifugal force. The frequen-cies of vibration computed for the two directions of the beam usingEqs. (5.16) and (5.36) are: ωv1 = 32.10 rad/s and ωh1 = 115.4 rad/s,compared with the finite element results of ωv1 = 32.24 rad/s andωh1 = 116.61 rad/s.

The static vertical response, dynamic vertical response and dy-namic horizontal response of the midpoint of the curved beam havebeen plotted in Figs. 5.5–5.7, along with the finite element solutions.As can be seen, the present solutions agree very well with the nu-merical ones, which is a demonstration of the accuracy of the presentsolutions considering only the first mode of vibration. Noting thatthe acting time of the mass is L/v = 0.6 s, one can easily distinguishbetween the forced vibration of the beam from the residual responsein Figs. 5.5–5.7; the latter does not decay since the damping effect



Fig. 5.5. The midpoint static vertical response of curved beam (one movingmass).

Fig. 5.6. The midpoint dynamic vertical response of curved beam (one movingmass).

was ignored. On the other hand, it is observed that the verticalresponse is much larger than that of the horizontal one, indicatingthat the vertical response can be more easily excited by the loadmoving at the speed v = 40 m/s.



Fig. 5.7. The midpoint dynamic horizontal response of curved beam (one movingmass).

5.7.2. Phenomenon of Cancellation Under Single or

Multi Moving Masses

As an illustration, we shall let i = 2 in Eq. (5.49) and computethe speed of cancellation as Sc = 0.333. By the definition for S1

in Eqs. (5.20) and (5.40), along with ωv1 = 32.24 rad/s and ωh1 =116.61 rad/s, the speed of cancellation can be computed for the twodirections as: vv = 82 m/s and vh = 297 m/s. Considering a singlemass moving at these speeds, each for one direction, the time-historymidpoint response computed for the two directions were plotted inFigs. 5.8 and 5.9. As can be seen, the residual response inducedby the moving mass after it leaves the beam, i.e., after 0.29 s and0.08 s respectively for the vertical and horizontal directions, deviatesslightly from the theoretical value of zero response, due to the neglectof higher modes. The close agreement of the present solutions withthe finite element ones indicates that the effect of higher modes isgenerally small.

Consider next the case of eight moving masses and use the samespeeds of cancellation, i.e., vv = 82 m/s and vh = 297 m/s, for the



Fig. 5.8. The phenomenon of cancellation for vertical direction (one movingmass).

Fig. 5.9. The phenomenon of cancellation for horizontal direction (one movingmass).

two directions. The time-history response computed for the mid-point response for the two directions have been plotted in Figs. 5.10and 5.11. The time for all the moving loads to depart from thebeam is 2.43 s and 0.67 s for the vertical and horizontal direction,



Fig. 5.10. The phenomenon of cancellation for vertical direction (eight movingmasses).

Fig. 5.11. The phenomenon of cancellation for horizontal direction (eight movingmasses).

respectively. In the two figures, it is observed that there is a to-tal of eight peaks, each of which corresponds to one moving load.Moreover, the residual responses for both directions remain negli-gibly small, as was expected. Finally, the effect of higher modes



of vibration appears to be generally small, as the present solutionsagree very well with the finite element ones.

5.7.3. Phenomenon of Resonance Under Multi

Moving Masses

Consider also the case of eight moving masses. Using d = 25 mand L = 24 m, the resonance speed computed from Eq. (5.54) fori = 1 is S1 = 0.521. By using Eqs. (5.20) and (5.40), along withωv1 = 32.24 rad/s and ωh1 = 116.61 rad/s, the corresponding speedscomputed of the moving masses for two directions are: vv = 128 m/sand vh = 464 m/s. In Figs. 5.12 and 5.13, the time-history responseobtained for the vehicles traveling at the resonance speed for eachdirection has been plotted. The fact that the response increases forthe two directions as there are more masses passing the beam is atypical resonance phenomenon. After all the masses pass the beam,i.e., after 1.55 s and 0.43 s for the vertical and horizontal directions,respectively, the beam tends to oscillate with the largest amplitudethat has been excited, as no damping is assumed. Again, the presentsolutions match very well the finite element solutions.

Fig. 5.12. The phenomenon of resonance for vertical direction (eight movingmasses).



Fig. 5.13. The phenomenon of resonance for horizontal direction (eight movingmasses).

5.7.4. I S Plot Impact Effect Caused by

Moving Loads

By letting Rd(x) and Rs(x) respectively denote the maximum dy-namic and static deflection of the beam at position x due to theaction of the moving loads, the impact factor I for the deflection ofa beam subjected to the moving loads was defined in Eq. (1.1), i.e.,

I =Rd(x) − Rs(x)

Rs(x). (5.56)

For simple beams, both the maximum static and dynamic deflectionswill occur at the midpoint. Consider the case of eight equally-spacedmoving masses. The impact response of the vertical deflection ofthe beam versus the speed parameter of the moving masses has beenplotted in Fig. 5.14, along with the finite element solution. Theimpact response of the horizontal deflection can hardly be plotted,because of the lack of a static centrifugal force. However, if thecentrifugal force fh computed as mvv

2/R can be treated as if it wasa static force, the impact response of the horizontal deflection canbe computed as well (not shown), which appears to be quite similarto that presented in Fig. 5.14 for the vertical vibration (Wu, 1998).



Fig. 5.14. The impact factor for midpoint vertical deflection (eight movingmasses).

Fig. 5.15. The maximum response for midpoint horizontal deflection (eight mov-ing masses).

In Fig. 5.14, it is confirmed that the present solutions agree verywell with the finite element results. By substituting d = 25 m andL = 24 m into Eq. (5.54), the resonance speeds computed for thetwo directions are: S1 = 0.521, 0.260, 0.174, 0.130,. . ., which are



close to the points where the peaks occur in Fig. 5.14. On the otherhand, from Eq. (5.49) the speeds of cancellation can be computedas: S1 = 1.000, 0.333, 0.200, 0.143,. . ., which are also close to thepoints where the minimum values occur in Fig. 5.14.

As was noted, the impact response for the horizontal vibration ofthe curved beam is unreal, since there is no static centrifugal force.In Fig. 5.15, the maximum horizontal deflection for the midpoint ofthe curved beam has been plotted. As can be seen, the speeds forthe maximum and minimum values to occur are generally consistentwith those of Fig. 5.14, although they are not as clear.


In this chapter, a general theory has been presented for treating thevibration of a horizontally-curved beam subjected to either a singleor a series of moving masses, with each moving mass simulated as aset of gravitational and centrifugal forces. Unlike the gravitationalforce, the centrifugal force is not constant, but is proportional to thesquare of the speed of the moving vehicle along the curved beam.The problem has been solved in an analytical but approximate man-ner considering the contribution of the first mode of vibration. Theaccuracy of the present approach has been confirmed by an indepen-dent finite element analysis, which, by nature, considers the contri-bution of all modes of vibration of the beam. The present approachhas the advantage that it provides clear physical insights into thevarious vibration phenomena induced by vehicles, in particular, thephenomena of resonance and cancellation, while allowing us to iden-tify the key parameters involved. The solution established herein forthe horizontal vibration of curved beams subjected to the centrifugalforce is new in the literature.


Part II

Interaction Dynamics Problems

153



Chapter 6

Vehicle–Bridge Interaction ElementBased on Dynamic Condensation

In this chapter, an element that is both accurate and efficient formodeling the vehicle–bridge interactions is presented, which is suit-able for applications to high-speed railway bridges. The train ismodeled as a sequence of lumped sprung masses, with the effects ofsuspension and energy dissipation of each vehicle taken into account,and a bridge sustaining the railway track with irregular elevationsby a number of beam elements. Two sets of equations of motionare written, with the first set for the bridge element and the secondset for the moving mass, which interact with each other through thecontact forces. To resolve the problem of coupling between the twosubsystems, the sprung mass equation is first discretized using New-mark’s finite difference formulas, by which the sprung mass degreesof freedom (DOFs) are condensed to those of the bridge element incontact. The element derived is referred to as the vehicle–bridge in-teraction (VBI) element, which possesses the same number of DOFsas the parent element, while the properties of symmetry and band-edness in the element matrices are preserved.

6.1. Introduction

In the past two decades, partly due to the construction of high-speed railways worldwide, the vibration of bridges caused by thepassage of trains is becoming a subject of increasing interest. In thesurvey paper by Diana and Cheli (1989), issues related to the train–bridge interactions have been discussed and a total of 90 papers were

155



cited. By modeling a moving vehicle as a moving load, moving mass,or moving sprung mass considering the suspension and/or energydissipation mechanisms, the dynamic response of bridges inducedby moving vehicles has been studied by researchers from time totime (Timoshenko, 1922; Jeffcott, 1929; Lowan, 1935; Ayre et al.,1950; Ayre and Jacobsen, 1950; Biggs, 1964; Fryba, 1972; Chu et al.,1979; Stanisic, 1985; Sadiku and Leipholz, 1987; Chatterjee et al.,1994). More sophisticated models that consider the various dynamicproperties of vehicles or railroad cars have also been implemented inthe study of vehicle–bridge interactions by researchers (Veletsos andHuang, 1970; Garg and Dukkipati, 1984; Yang and Lin, 1995).

In studying the dynamic response of a vehicle–bridge system, twosets of equations of motion can be written, one for the supportingbridge (i.e., the stationary subsystem) and the other for each of themoving vehicles (i.e., the moving subsystem) over the bridge. It isthe interaction or contact forces existing at the contact points of thetwo subsystems that make the two sets of equations coupled andnonlinear. As the contact forces vary with respect both to time andspace, the system matrices, which are functions of the contact forces,must be updated and factorized at each time step in a time-historyanalysis. To solve the two sets of differential equations, procedures ofiterative nature are often adopted (Hwang and Nowak, 1991; Greenand Cebon, 1994; Yang and Fonder, 1996). In this case, one mayfirst start by assuming the displacements for the contact points, andthen solve the vehicle equations to obtain the contact forces. Bysubstituting the contact forces into the bridge equations, improvedvalues of displacements for the contact points can be obtained. Theadvantage of such a procedure is that the responses of both the vehi-cles and bridge are simultaneously made available at each time step.However, the rate of convergence of iteration is likely to be low, whendealing with the situation of a bridge sustaining a large number ofmoving vehicles, for there exists twice the number of contact pointsif each vehicle is modeled as the composition of two lumped sprungmasses.

Other approaches for solving the vehicle–bridge interaction prob-lems include those based on the condensation method. Garg and


Vehicle–Bridge Interaction Element Based on Dynamic Condensation 157

Dukkipati (1984) used the Guyan reduction scheme to condense thevehicle degrees of freedom (DOFs) to the associated DOFs of thebridge. Recently, Yang and Lin (1995) employed the dynamic con-densation method to eliminate the vehicle DOFs on the element level.These condensation approaches were initially proposed for solvingthe response of the bridge (i.e., the larger or master subsystem), andthey are good for this purpose. However, if the response of the mov-ing vehicles (i.e., the smaller or slave subsystem) is desired, whichserves as a reference for evaluating the riding comfort of passengers,the aforementioned two approaches cannot be relied to yield accu-rate solutions, since approximations have been made in relating thevehicle (slave) to the bridge (master) DOFs.

To resolve the dependency of system matrices on the wheel loadpositions, the condensation technique that eliminates the vehicleDOFs on the element level will be adopted in this chapter. First,two sets of equations of motion will be written, one for the bridgeelement and the other for the sprung mass lumped from the train cardirectly acting on the element. The sprung mass equation is then dis-cretized using Newmark’s finite difference formulas, and condensedto those of the bridge element in contact, which will result in theso-called vehicle–bridge interaction (VBI) element. Such an elementhas the advantage that it possesses exactly the same number of DOFsas the parent element, while the properties of symmetry and banded-ness in element matrices are preserved. The materials presented inthis chapter concerning the element formulation, i.e., in Secs. 6.2–6.7,have been revised mainly from the paper by Yang and Yau (1997),and those related to the parametric studies in Sec. 6.8 from the paperby Yau et al. (1999).

6.2. Equations of Motion for the Vehicle and Bridge

As shown in Fig. 6.1, a bridge is modeled as a beam-like structure,and the train traveling over the bridge with constant speed v is ideal-ized as a sequence of lumped sprung mass units of regular intervals.In this chapter, each sprung mass unit is used to represent either thefront or rear half of a train car, which consists of two concentrated



(a)

(b)

Fig. 6.1. The train–bridge system: (a) general model and (b) sprung massmodel.

masses, with the top one representing the mass lumped from the carbody, which is equal to half of the car body mass, and the bottomone the mass of the wheel assembly. The two masses are connectedby a set of spring and dashpot that serves to represent the vehicle’ssuspension and energy dissipation mechanism.

For the present purposes, an interaction element is defined suchthat it consists of a bridge (beam) element and the sprung massunit directly acting on it, as shown in Fig. 6.2, where the ballastwith stiffness kB and the rail irregularity with a profile r(x) are alsoindicated. The stiffness of the beam element may be computed asthe summation of the stiffness of the bridge girder, rails and trackstructures. In other words, no consideration will be made specificallyof the stiffness of the rails or track structures. For the parts of thebridge that are not directly acted upon by the moving vehicles or



Fig. 6.2. The vehicle–bridge interaction element.

sprung mass units, they are modeled by traditional bridge elements.However, for the remaining parts that are in direct contact with thevehicles, interaction elements considering the effects of the sprungmass units have to be used instead. In this study, the notation [ ] isused for a square matrix, for a column vector, and 〈〉 for a rowvector.

The vehicle model depicted in Fig. 6.2 will be grossly referred tothe sprung mass model throughout this chapter. Let the stiffnessand damping coefficients of the suspension unit be denoted by kv

and cv, respectively, the mass of the wheel assembly by mw, and themass lumped from the car body as Mv (assumed to be equal to halfof the mass of the car body). The sprung mass model is regarded asa two-node system, with one node associated with each of the twoconcentrated masses. Also, let the vertical displacements of the twonodes measured from the static equilibrium positions be denotedby the coordinates zT = 〈z1, z2〉. Corresponding to the nodaldisplacements zT are the external forces pvT = 〈p, 0〉, where p

represents the total weight of the two masses, p = −(Mv + mw)g,with g denoting the acceleration of gravity.



The equations of motion for the sprung mass model shown inFig. 6.2 can be written as follows (Fryba, 1972):[

mw 00 Mv

]z1

z2

+[

cv −cv

−cv cv

]z1

z2

+[

kv −kv

−kv kv

]z1

z2

=

p + fc

0

,

(6.1)

where fc is the interaction or contact force existing between the wheelmass and the bridge element. Let xc denote the acting position ofthe sprung mass unit (see Fig. 6.2) and Nc a vector containing thecubic Hermitian interpolation functions for the vertical displacementof the beam evaluated at the contact point xc, i.e., Nc = N(xc)(see, for instance, p. 52 of Yang and Kuo (1994) for the interpolationfunctions of a two-dimensional beam element). The contact force fc

can be expressed as

fc = kB(〈Nc〉ub + rc − z1) ≥ 0 , (6.2)

where the condition of fc ≥ 0 is imposed for the VBI system to ex-clude the possibility of separation for the wheels from the bridge,kB is the ballast stiffness, ub the nodal displacements of the beamelement, and rc the elevation of rail irregularity at the contact pointxc, which is assumed to be known throughout the element consid-ered. Note that the interpolation function vector Nc contains onlyfour nonzero entries for those DOFs of the element related to thevertical displacements, the other entries not related to the verticaldisplacements are simply set to zero.

The equations of motion for the bridge element in contact withthe sprung mass unit can be written as

[mb]ub + [cb]ub + [kb]ub = pb − Ncfc , (6.3)

where [mb], [cb], and [kb] denote the mass, damping, and stiffnessmatrices of the bridge element; and pb denotes the external nodalloads acting on the element. Regarding the bridge element as a three-dimensional solid beam element, one can assign six DOFs to each



node of the element, with three for translations and the other threefor rotations. The mass matrix [mb] and stiffness matrix [kb] for thethree-dimensional beam element have been given in Appendix F. Inthis study, Rayleigh damping is assumed for the bridge, which meansthat the damping matrix of the entire bridge can be computed as alinear combination of the mass and stiffness matrices of the structure,for which the procedure has also been described in Appendix F. Thus,the element damping matrix [cb] appearing in Eq. (6.3) is requiredonly for conceptual needs.

As can be seen from Eqs. (6.1) and (6.3), the sprung mass unitand the bridge element interact with each other through the contactforce fc, which varies as a function of both time and position. Toensure that the sprung mass unit is in full contact with the bridgeelement, the reactive force exerted by the bridge element on thesprung mass unit must be greater than zero. Whenever the contactforce fc is less than zero, the wheel mass will jump upward, meaningthat the contact condition between the sprung mass unit and thebridge is violated. In this chapter, no consideration will be made forseparation of the moving vehicles from the bridge. From the first lineof Eq. (6.1), together with the second line, the contact force fc canalso be expressed as follows:

fc = −p + mwz1 + Mv z2 . (6.4)

From Eqs. (6.3) and (6.4), it can be seen that the dynamic response ofthe beam element is affected not only by the static weight p, but alsoby the inertial effects of the moving vehicles in the vertical direction.

6.3. Element Equations in Incremental Form

The system equations as given in Eqs. (6.1)–(6.3) are coupled, non-linear, and time-dependant in nature, due to existence of the contactforce fc between the two subsystems, i.e., the moving vehicle and thesustaining bridge. In this chapter, an incremental method will bepresented for solving the system equations of motion, with iterationsperformed at each incremental step for removing the unbalancedforces. Consider a typical incremental step for the system from time



t to t+∆t. The component equations as presented in Eqs. (6.1)–(6.3)should be interpreted as those established for the deformed positionof the system at time t + ∆t. For the incremental time step, onemay let

zt+∆t = zt + ∆z , (6.5)

where ∆z denotes the displacement increments of the sprung massunit occurring during the time step considered. By using Eq. (6.2),the sprung mass equation (6.1) can be rewritten in an incrementalform as[

mw 00 Mv

]z1

z2

t+∆t

+[

cv −cv

−cv cv

]z1

z2

t+∆t

+[

kv + kB −kv

−kv kv

]∆z1

∆z2

t+∆t

=

p + kB(〈Nc〉ub + rc)

0

t+∆t

−

qs1

qs2

t

, (6.6)

where qst denotes the internal resistant forces of the sprung massunit computed at the end of the last incremental step, i.e., attime t,

qs1

qs2

t

=[

kv + kB −kv

−kv kv

]z1

z2

t

. (6.7)

It should be noted that the resistant forces qst and rail irregularitiesrc,t+∆t appearing on the right-hand side of Eq. (6.6) are known atthe beginning of the current incremental step. However, the bridgedisplacements ubt+∆t remain unknown. The vehicle equations arecoupled with the bridge equations to be shown below.

Similarly, one may write the following for the bridge displacementsat the current time step:

ubt+∆t = ubt + ∆ub , (6.8)



where ∆ub denotes the displacement increments of the bridge ele-ment during the current incremental step. By using Eqs. (6.2) and(6.8), the bridge equations of motion as given in Eq. (6.3) can berewritten in an incremental form,

[mb]ubt+∆t + [cb]ubt+∆t + [[kb] + kBNc〈Nc〉]∆ub= pbt+∆t − NckB(rc − z1)t+∆t − [[kb] + kBNc〈Nc〉]ubt ,

(6.9)

where all the terms on the right-hand side of the equal sign are eithergiven, such as the external loads pbt+∆t and irregularity functionrc,t+∆t, or have been made available at the last time step, such asthe bridge displacements ubt, except the wheel mass displacementz1,t+∆t. This again represents another source of coupling betweenthe bridge and moving vehicle.

6.4. Equivalent Stiffness Equation for Vehicles

As can be seen from Eqs. (6.6) and (6.9) for the sprung mass unitand the bridge element, the two system equations are coupled, sincethey depend on the solution of each other. In the following, thesprung mass equation (6.6) will first be transformed into an equiv-alent stiffness equation using Newmark’s single-step finite differencescheme. From the equivalent stiffness equation, the displacementsof the sprung mass unit can be solved and then substituted into thebridge equations. By doing so, the vehicle DOFs can be eliminatedand condensed to those of the bridge element in contact.

Based on Newmark’s β method with constant average accelera-tion, i.e., with β = 0.25 and γ = 0.5 for its unconditional stability(see Appendix B),

zt+∆t = zt + (1 − γ)zt + γzt+∆t∆t ,

zt+∆t = zt + zt∆t + (0.5 − β)zt

+ βzt+∆t∆t2 ,

(6.10)



from which the accelerations and velocities of the vehicle at timet + ∆t can be solved,

zt+∆t = a0∆z − a2zt − a3zt ,

zt+∆t = zt + a6zt + a7zt+∆t ,(6.11)

where the coefficients and those to appear later are defined as follows:

a0 =1

β∆t2, a1 =

γ

β∆t, a2 =

1β∆t

,

a3 =12β

− 1 , a4 =γ

β− 1 , a5 =

∆t

2

(γ

β− 2)

,

a6 = ∆t(1 − γ) , a7 = γ∆t .

(6.12)

With the relations given in Eq. (6.11) for the vehicle accelerationsand velocities, the sprung mass equation (6.6) can be manipulatedto yield an equivalent stiffness equation:

[kv + kB + a0mw + a1cv −kv − a1cv

−kv − a1cv kv + a0Mv + a1cv

]∆z1

∆z2

=

p + kBrc + kB〈Nc〉ub0

t+∆t

−(

qs1

qs2

t

+

qe1

qe2

t

),

(6.13)

where

qe1,t = −mw(a2z1 + a3z1) − cv [a4(z1 − z2) + a5(z1 − z2)] ,

qe2,t = −Mv(a2z2 + a3z2) − cv[a4(z2 − z1) + a5(z2 − z1)] .(6.14)

Here, all the terms on the right-hand side of the equal sign ofEq. (6.13) are either given or already made known at the end ofthe last incremental step, except the bridge displacements ubt+∆t.

Accordingly, the displacement increments ∆z of the sprungmass unit can be solved from Eq. (6.13) and related to the bridge



displacements ubt+∆t as∆z1

∆z2

= − 1D

kv + a0Mv + a1cv

kv + a1cv

(p + kBrc,t+∆t + kB〈Nc〉ubt+∆t)

− 1D

(qs1,t + qe1,t)a0Mv + (qs,t + qe,t)(kv + a1cv)

(qs1,t + qe1,t)(a0mw + kB) + (qs,t + qe,t)(kv + a1cv)

,

(6.15)

where D represents the determinant of the vehicle equations inEq. (6.13),

D =∣∣∣∣ kv + kB + a0mw + a1cv −kv − a1cv

−kv − a1cv kv + a0Mv + a1cv

∣∣∣∣ (6.16)

and

qe,t = (qe1 + qe2)t ,

qs,t = (qs1 + qs2)t = kBz1,t .(6.17)

Equation (6.15) represents the master–slave relation for condensingthe sprung mass DOFs to the bridge DOFs. Compared with theapproximate master–slave relation used by Yang and Lin (1995), thepresent relation in Eq. (6.15) has the advantage of being simple,accurate, and explicit. Since Eq. (6.15) has been derived throughintroduction of the finite difference formulas of the Newmark type,it possesses the same order of accuracy as that implied in solution ofthe equations for the entire vehicle–bridge interaction system, whichwill also be solved by Newmark’s β method. Because of this, theaccuracy and convergence characteristics of the VBI element to bederived below can be assured with no difficulty.

6.5. Vehicle–Bridge Interaction Element

By the fact that z1,t+∆t = z1,t + ∆z1 and using the first line ofEq. (6.15) for ∆z1, one can derive from Eq. (6.9) the condensed



equations of motion for the beam element at time t + ∆t with theinteraction effect of the sprung mass unit taken into account, that is,

[mb]ubt+∆t + [cb]ubt+∆t + [kb]∆ub= (pbt+∆t + pwt+∆t) − (fst + [kb]ubt) , (6.18)

where [kb] is the stiffness matrix for the condensed system,

[kb] = [kb]+kBa0

D[(Mv+mw)(kv+a1cv)+a0Mvmw]Nc〈Nc〉 . (6.19)

pwt+∆t denotes the load actions induced by the wheels, includingthe contributions from rail irregularities and ballast stiffness,

pwt+∆t

= −kB

[rc,t+∆t − (p + kBrc,t+∆t)

1D

(kv + a0Mv + a1cv)]Nc(6.20)

and fst denotes the resistant forces associated with the sprungmass unit,

fst = kB

[1D

(qs1,t + qe1,t)a0Mv

+ (qs,t + qe,t)(kv + a1cv) − z1,t

]Nc . (6.21)

In Eq. (6.18), the term [kb]ubt should be interpreted as the resistantforces exerted by the bridge element at the last time step.

The condensed stiffness matrix [kb] as given in Eq. (6.19) containsessentially two major parts. The [kb] matrix represents the nonin-teraction part, which originates from the bridge element itself. Theremainder in Eq. (6.19) represents the interaction part induced bythe moving sprung mass, which is a function of the loading positionxc, and thus is time-dependent, as implied by the shape functionNc. In a time-history analysis, only the second (interaction) partmust be updated at each incremental step. A convenient way is toassemble the [kb] matrices as if it was the case for the bridge with



no moving vehicles, keeping this part of stiffness matrix unchangedthroughout all the incremental steps, and to update only the second(interaction) part for each time step according to the acting positionsof the sprung mass units.

The equations as given in Eq. (6.18) represent exactly the equa-tions of motion for the vehicle–bridge interaction (VBI) element. Us-ing such an element, only one sprung mass unit is allowed to be actingat one beam element. The maximum length that is allowed for eachelement should be no greater than the minimum spacing betweenany two wheel assemblies. Such a restriction can be removed, if theforegoing procedure has been generalized to include the case of twoor more sprung mass units acting simultaneously on a single beamelement.

Consider the special case of moving loads, i.e., with the effectof inertia of the moving vehicles neglected. By letting kv = 0,mw = 0, and cv = 0, one can obtain from Eq. (6.16) the de-terminant as D = a0MvkB , from Eq. (6.7) the resistant forceqs1,t = kBz1,t, and from Eq. (6.14a) the sprung mass force qe1,t = 0.Furthermore, from Eqs. (6.19)–(6.21), the following can be obtained:[kb] = [kb], pst+∆t = pNc and fst = 0. Consequently, thecondensed equation as given in Eq. (6.18) reduces to

[mb]ubt+∆t + [cb]ubt+∆t + [kb]∆ub= pbt+∆t + pNc − [kb]ubt , (6.22)

in which the term pNc denotes the action of the moving load, ascan be expected according to the concept of consistent nodal loads.On the other hand, by setting the damping coefficient cv and themass mw of the wheel assembly equal to zero, and assigning a verylarge number to the stiffnesses of the vehicle and ballast, i.e., kv andkB , one can arrive at the moving mass model as another special case.

Since the VBI element derived above possesses exactly the samenumber of DOFs as the parent element, while preserving the proper-ties of symmetry and bandedness in element matrices, conventionalelement assembly procedure, which takes into account the condi-tions of equilibrium and compatibility of the structure at each nodal



point, through transformation of all the element equations and as-sociated variables to the same global coordinates, can be appliedto constructing the equations of motion for the entire vehicle–bridgesystem. Namely, by looping over all the VBI elements, as representedby Eq. (6.18), and the elements not in contact with the moving ve-hicles, the equations of motion for the structure can be establishedas follows:

[Mb]Ubt+∆t + [Cb]Ubt+∆t + [Kb]∆Ubt+∆t

= Pbt+∆t − Fbt , (6.23)

where Ubt+∆t denotes the total displacements of the bridge at timet + ∆t, and ∆Ub the displacement increments of the bridge fromtime t to t + ∆t,

Ubt+∆t = Ubt + ∆Ub . (6.24)

In the system equations (6.23), each of the terms has been assembledby looping over all the corresponding terms of the beam elements,including the condensed ones, of the bridge. Thus, [Mb] should beinterpreted as the mass matrix and [Kb] the stiffness matrix of thebridge, considering the contributions of both the VBI elements andthe parent elements are not directly acted upon by the wheel loads.The external loads Pbt+∆t and resistant forces Fbt on the right-hand side of Eq. (6.23) are constructed as follows:

Pbt+∆t =n.o.e∑

elm=1

(pbt+∆t + pwt+∆t) , (6.25a)

Fbt =n.o.e∑

elm=1

(fst + [kb]ubt) . (6.25b)

One exception is the damping matrix [Cb], which is constructed us-ing the procedure described in Appendix F for bridges of which thedamping property is assumed to be of the Rayleigh type.



6.6. Incremental Dynamic Analysis with Iterations

By Newmark’s β method, the system equations (6.23) can be ren-dered into a set of equivalent stiffness equations, from which thedisplacement increments ∆Ub of the bridge for a typical time stepcan be solved. By repeating such a procedure, the total bridge andvehicle responses can be computed as well. The accuracy of theincremental approach can be enhanced by inclusion of an iterativeprocedure to be presented below.

6.6.1. Equivalent Stiffness Equations for VBI System

First of all, the accelerations and velocities of the bridge at time t+∆t

can be related to the displacement increments ∆Ub and those ofthe bridge at time t using finite difference formulas of the Newmarktype similar to those given in Eq. (6.11) for the vehicles as

Ubt+∆t = a0∆Ub − a2Ubt − a3Ubt ,

Ubt+∆t = Ubt + a6Ubt + a7Ubt+∆t .(6.26)

By substitution of the expressions in Eq. (6.26), the system equations(6.23) can be manipulated to yield the following equivalent stiffnessequations:

[Kb]t+∆t∆Ub = Pbt+∆t − Fbt , (6.27)

where the effective stiffness matrix [Kb]t+∆t is

[Kb]t+∆t = a0[Mb] + a1[Cb] + [Kb] (6.28)

and the effective resistant force vector Fbt is

Fbt = Fbt + [Mb](a2Ubt + a3Ubt)

+ [Cb](a4Ubt + a5Ubt) . (6.29)



Both the effective stiffness matrix [Kb]t+∆t and the load vectorPbt+∆t can be regarded as constant within each time step, if suffi-ciently small time steps are used in the incremental analysis.

As was mentioned previously, the order of accuracy of the equiv-alent stiffness equations (6.27), including the condensation proce-dure involved in the derivation, is the same as that implied by theNewmark-type formulas given in Eq. (6.10). For most of the vehicle–bridge interaction problems encountered in practice, the accuracy ofthe present approach is satisfactory and no iterations are required, onthe condition that reasonably small time increments ∆t have beenselected. For this case, the displacement increments ∆Ub of thebridge can be solved directly from Eq. (6.27), which can then besubstituted into Eq. (6.24) to yield the total displacements Ubt+∆t

and into Eq. (6.26) to yield the accelerations Ubt+∆t and velocitiesUbt+∆t of the bridge at time t + ∆t. Accordingly, the displace-ments, velocities and accelerations of each element of the bridge,i.e., ubt+∆t, ubt+∆t and ubt+∆t, can be computed. By back-ward substitution, the displacement increments ∆z of the sprungmass unit can be computed from Eq. (6.15), with which the total re-sponse of the sprung mass unit at time t+∆t can be computed fromEqs. (6.5) and (6.11). This completes the cycle of analysis for thecurrent time step from t to t + ∆t. As a side note, the vehicle accel-eration z serves as a measure of the riding comfort for passengerscarried by the train.

As was stated previously, the condensation of the vehicle DOFsthat leads to the VBI element is performed on the element level. Itfollows that the conventional procedure for assembling the structuralmatrices from the element ones can be directly applied. For this rea-son, the amount of effort required in programming and computationis minimal, compared with approaches that perform condensation onthe structure level or with no condensation at all. Such an advantagebecomes clearer in the study of bridges subjected to a sequence ofmoving vehicular loads. In such cases, what one needs is a properbook-keeping scheme to identify the acting position of each wheelload of the railroad cars composing the train at each time step. Itshould be noted that the VBI element derived herein is applicable



not only for modeling the railroad cars with regular intervals andconstant sizes, but also for vehicles of varying sizes that constitute arandom traffic flow through the highway bridges.

6.6.2. Procedure of Iterations

During the passage of vehicles over a bridge, on the one hand, thevehicles excite the bridge through the interaction or contact forces,which change their action positions from time to time; on the otherhand, the bridge affects the behavior of the vehicles by its deflection,as well as by the contact forces. Such a phenomenon is a typicalnonlinear interaction problem, which can only be solved by proce-dures of incremental nature. For bridges whose displacements can beregarded as small under the moving vehicular loads, there is no needto perform iterations at each incremental step. However, for bridgeswhose displacements may not be regarded as small under the movingvehicular loads, iterations for removing the unbalanced forces at thestructural nodes for each incremental step should be conducted. Thefollowing is a summary of the procedure for iterations.

For the purpose of performing iterations, the equivalent stiffnessequations of the VBI system in Eq. (6.27), which has been presentedin incremental form, should be modified to include the feature ofiteration, that is,

[Kb]t+∆t∆Ubi = Pbt+∆t − Fbi−1t+∆t , (6.30)

in which the right superscript i on each symbol indicates the cur-rent number of iterations. The right hand side of Eq. (6.30) shouldnow be interpreted as the external load increments for the first iter-ation (i = 1) and as the system unbalanced forces for the followingiterations (i ≥ 2) (Yang and Kuo, 1994). The philosophy for modi-fying an equation originally presented in incremental form into oneincorporating the iterative measure is that all the terms originallyassociated with time t be interpreted as those for the last, i.e., the(i − 1)th, iterative step of the current incremental step, and all theterms originally associated with time t+∆t as those for the current,i.e., the ith, iterative step. For instance, the resistant force vector in



Eq. (6.29) should be modified for the iterative steps with i ≥ 2 asfollows:

Fbi−1t+∆t = Fbi−1

t+∆t + [Mb](a2Ubi−1t+∆t + a3Ubi−1

t+∆t)

+ [Cb](a4Ubi−1t+∆t + a5Ubi−1

t+∆t) . (6.31)

Equation (6.30) represents a typical nonlinear equation encounteredin the study of a great number of nonlinear problems. To solveproblems of this sort, the modified Newton–Raphson method thatperforms iterations at constant loads can be employed (Yang andKuo, 1994).

The initial conditions (i.e., for i = 1) to the system equations(6.30) are

Fb0t+∆t = Fbl

t ,

Ub0t+∆t = Ubl

t ,(6.32)

where Fblt has been given in Eq. (6.29) and the right superscript

l denotes the last iteration of the previous incremental or time step.Accordingly, the system equations (6.30) reduce to the following forthe first iteration (i.e., with i = 1) of the current incremental step:

[K]t+∆t∆Ub1 = Pbt+∆t − Fblt . (6.33)

Here, it should be noted that the effective stiffness matrix [K]t+∆t

remains constant within each incremental step, which need not be up-dated for the iterations involved using the modified Newton–Raphsonmethod.

For each iterative step, the displacement increments ∆Ubi ofthe bridge can be solved from Eq. (6.30). Accordingly, the totaldisplacements of the bridge for the current iterative step can be ac-cumulated as

Ubit+∆t = Ubi−1

t+∆t + ∆Ubi . (6.34)



The acceleration and velocity of the bridge can be computed fromEq. (6.26) with due account taken of the feature of iteration, that is,

Ubit+∆t = a0∆Ubi − a2Ubi−1

t+∆t − a3Ubi−1t+∆t ,

Ubit+∆t = Ubi−1

t+∆t + a6Ubi−1t+∆t + a7Ubi

t+∆t .(6.35)

Once the displacements Ubit+∆t of the bridge in global coordinates

are made available, the displacements ubit+∆t for each element of

the bridge can be computed accordingly. It follows that the vehicledisplacement increments ∆zi can be computed from Eq. (6.15), bytreating the terms with subscript t+∆t as those associated with theith iterative step, and the terms with subscript t as those associatedwith the (i − 1)th iterative step. The total responses of the sprungmass unit for the ith iterative step can be determined from Eqs. (6.5)and (6.11) as follows:

zit+∆t = zi−1

t+∆t + ∆zi , (6.36a)

zit+∆t = a0∆zi − a2zi−1

t+∆t − a3zi−1t+∆t , (6.36b)

zit+∆t = zi−1

t+∆t + a6zi−1t+∆t + a7zi

t+∆t , (6.36c)

with the following initial condition: z0t+∆t = zl

t.The following is a summary of the procedure for performing

the incremental-iterative analysis based on the modified Newton–Raphson algorithm:

(1) Read in all the structural and vehicle data.(2) Start with time t = 0 and adopt the following initial condi-

tions: Fbl0 = 0, Ubl

0 = Ubl0 = Ubl

0 = 0, andzbl

0 = zbl0 = zbl

0 = 0. Calculate the mass matrices[mb] for all elements of the bridge and assemble the mass matrix[Mb] for the bridge that is free of any vehicles. Select a propertime increment ∆t for the Newmark integration scheme.

(3) For each incremental or time step, check if the total duration t

for which the solution has been made available is greater thanthe duration desired. If yes, stop the incremental procedure.



Otherwise, let t = t + ∆t and use i = 1 for the first iteration.Determine the acting position xc of each wheel load and the railirregularity rc for each contact point.

(4) For the elements carrying no wheel loads, calculate the elementmatrix [kb]; and for the elements in direct contact with the wheelloads, calculate the modified element matrix [kb] using Eq. (6.19),in which the shape function Nc is known to be function of thecontact position xc.

(5) Assemble the bridge matrix [Kb] and load vector Pbt+∆t, usingEq. (6.25a) for the latter. The damping matrix [Cb] of the bridgeis assumed to be of the Rayleigh type, which can be computed asa linear combination of the mass matrix [Mb] and stiffness matrix[Kb] of the bridge free of the vehicles (see Appendix F). Computethe equivalent stiffness matrix [Kb]t+∆t using Eq. (6.28). For thepresent problem, the system matrices [Kb]t+∆t and Pbt+∆t aretreated as constant for each incremental or time step.

(6) Determine the resistant force vector Fbi−1t+∆t using Eq. (6.31).

For i ≥ 2, check if the unbalanced forces (Pbt+∆t − Fbi−1t+∆t)

are less than a given tolerance. If yes and if the contact conditionfc ≥ 0 as given in Eq. (6.2) is satisfied, go to step 3 for the nextincrement or time step. Otherwise, perform the following.

(7) Solve the displacement increments ∆Ubi from the system equa-tions (6.30). Determine the vehicle displacement increments∆zi from Eq. (6.15), where the terms with subscript t + ∆t

should be interpreted as those for the ith iterative step, and theterms with subscript t as those for the (i − 1)th step.

(8) Update the total displacements Ubit+∆t of the bridge using

Eq. (6.34) and zit+∆t of the vehicle using Eq. (6.36a). Com-

pute the displacement derivatives Ubit+∆t and Ubi

t+∆t for thebridge using Eq. (6.35) and zi

t+∆t and zit+∆t for each sprung

mass unit using Eqs. (6.36b) and (6.36c).(9) Let i = i + 1 and go to step 6 for the next iterative step.

As was stated in Sec. 6.5, a more efficient way for updating thestiffness matrix [Kb] for the VBI system at each time step is asfollows. First, one assembles the stiffness [Kb] for the bridge that



is free of any vehicles, using the element matrices [kb] for all thebridge elements. Then, at each incremental step, one checks theacting position xc of each sprung mass unit and adds only the inter-action (i.e., the second) part of the [kb] matrix of the VBI element inEq. (6.19) to the entries in [Kb] corresponding to the bridge elementon which the sprung mass unit is acting. By watching the movementof each sprung mass unit, one can either update the interaction partof the VBI element to account for the change in the acting positionof the sprung mass unit, or remove the interaction part from the [Kb]matrix, if the sprung mass unit is no longer acting on the element.

With the above procedure of iteration, the coupling effect betweenthe bridge and moving vehicles is considered through the condensedVBI elements. Because the same order of accuracy is maintainedboth in establishing the master–slave relations leading to the VBIelement and in discretizing the structural equations of motion, thenumber of iterations required for each time step is generally small,compared with approaches that do not rely on the condensation tech-nique. In addition to the bridge response, the present approach al-lows us to compute the vehicle response as a by-product, which servesas an indicator for the riding comfort of passengers carried by thetrain.

6.7. Numerical Verification

Three examples are prepared to verify the VBI element and the pro-cedure of solution presented in this chapter. First, the dynamic re-sponses solved by the present method for a simple beam subjectedto a moving sprung mass will be compared with those obtained byconsidering only the contribution of the first mode of vibration of thebeam. Second, by modeling a train as a sequence of moving lumpedloads or lumped masses, the impact responses of bridges excited bythe train will be investigated and compared with the existing solu-tions. Finally, the dynamic responses of a cantilever subjected tomoving vehicles that are simulated by different models will be stud-ied. In each case, the beam is represented by ten elements.



6.7.1. Simple Beam Subjected to Moving Sprung Mass

As shown in Fig. 6.3, a simple beam of span length L = 25 m issubjected to a moving sprung mass. The following data are adopted:Young’s modules E = 2.87 GPa, Poisson’s ratio v = 0.2, moment ofinertia I = 2.90 m4, mass per unit length m = 2303 kg/m, suspendedmass Mv = 5750 kg, suspension stiffness kv = 1595 kN/m, and speedv = 100 km/h. For this problem, the mass ratio of the vehicle tothe bridge is Mv/mL = 0.1. The frequency of vibration computedof the bridge is ω1 = 30.02 rad/s, and the one for the sprung massis ωv = 16.66 rad/s. By representing the vertical displacement ofthe beam as ub = qb(t) sin(πx/L), the displacement of the sprungmass as qv(t), and neglecting the effect of damping of the beam, theequations of motion for the beam and the sprung mass moving atspeed v can be given as (Biggs, 1964):

qb

qv

+

2ω2

v

Mv

mLsin2 πvt

L+ ω2

1 −2ω2v

Mv

mLsin

πvt

L

−ω2v

Mv

mLsin

πvt

Lω2

v

qb

qv

=

−2

Mvg

mLsin

πvt

L

0

. (6.37)

The dynamic responses of the midpoint vertical displacement of thebeam subjected to a moving load or sprung mass have been plottedin Fig. 6.4. As can be seen, the response obtained by the presentprocedure using the VBI element for the special case of sprung massagrees very well with the single mode solution to Eq. (6.37).

From the response of vertical acceleration for the midpoint of thebeam shown in Fig. 6.5, one observes that inclusion of the highermodes can result in drastic oscillation of the acceleration response,which was neglected in the solution to Eq. (6.37). The deflectionand vertical acceleration computed of the sprung mass have beenplotted in Figs. 6.6 and 6.7, respectively. In these two figures, thedifference between the present solution and that of Eq. (6.37) can beattributed mainly to the omission of higher vibration modes in the



latter. A comparison of Figs. 6.4 and 6.6 indicates that the sprungmass is more sensitive than the beam to the omission of higher modes.As was noted previously, the vertical acceleration of the sprung massserves as a measure of the riding comfort for passengers carried bythe vehicle.

Fig. 6.3. Beam with moving sprung mass.

Fig. 6.4. Midpoint vertical deflection of beam.



Fig. 6.5. Midpoint vertical acceleration of beam.

Fig. 6.6. Deflection of sprung mass.



Fig. 6.7. Vertical acceleration of sprung mass.

6.7.2. Simple Beam Subjected to Moving Train

Consider a simply-supported beam with the following properties:L = 20 m, I = 3.81 m4, E = 29.43 GPa, m = 34088 kg/m, anddamping ratio = 2.5%. The train traveling over the bridge contains10 wheel assemblies, which can be modeled as a sequence of movinglumped loads with regular nonuniform intervals arranged as follows:↓← Lc →↓← Ld →↓ . . . ↓← Ld →↓← Lc →↓ , where Lc = 18 m,Ld = 6 m, and “↓” indicates a lumped load. For this example, thelumped load is assumed as p = 215.6 kN, the car body mass asMv = 22000 kg and the wheel mass as mw = 0 kg. The impactfactor I as defined in Eq. (1.1) is adopted.

Two cases are considered herein. In the first case, the moving loadmodel is conceived by setting the suspension stiffness kv, dampingcv, and ballast stiffness kB all equal to zero. In the second case, themoving mass model is approximated by setting the vehicle stiffnesskv and ballast stiffness kB equal to a very large number, say, withkv = kB = 9.0 × 106 kN/m. The impact factors I calculated for themidpoint displacement of the beam subjected to the moving loads



Fig. 6.8. Impact response for bridge sustaining a moving train.

using the two models have been plotted in Fig. 6.8, with respectto the (first) speed parameter S, which, according to Eq. (1.4), isdefined as the ratio of the exciting frequency of the moving vehicle,πv/L, to the fundamental frequency ω of the bridge. Also shownin the figure are the results based on the analytical work of Yanget al. (1997b) or those presented in Sec. 2.6. Clearly, the presentsolutions correlate very well with those of the analytical study. Themoving mass model tends to reduce the frequency of vibration ofthe vehicle–bridge system, in the sense that the critical speed for thepeak response to occur shifts to a smaller value.

6.7.3. Free-Fixed Beam with Various Models for

Moving Vehicles

Figure 6.9 shows a cantilever subjected to a moving lumped mass.The following data are assumed: length L = 300 in (7.62 m); ve-locity v = 2000 in/s (50.8 m/s); flexural rigidity EI = 3.3 × 109

lbf-sq in (9.474 × 106 N-m2); mass of the beam per unit length



M=15 lbm

v=2000 in/s

L=300 in

Fig. 6.9. Cantilever with mass moving at constant speed.

Fig. 6.10. Comparison of results for the cantilever.

m = 0.006667 lbm/in (0.1192 kg/m); lumped mass Mv = 15 lbm(6.81 kg); and moving load = 5793 lbf (66.74 N). The results ob-tained for the free-end deflection of the cantilever using both themoving mass and moving load models have been plotted in Fig. 6.10,along with those of Akin and Mofid (1989). As can be seen, goodagreement has been achieved between the present solutions and thoseof Akin and Mofid (1989). It can be seen that throughout most of



the acting period of the vehicle on the beam, i.e., for t < 0.14 s, themoving load model tends to produce larger response, compared withthe moving mass model.

6.8. Parametric Studies

In the design of bridges for highways and railways, it is required thatthe stresses caused by the static live (vehicular) loads on the bridgebe increased by a dynamic allowance factor or impact factor to en-sure the capability of the bridge in resisting the impact loads. Someadditional requirements are placed on the design of high-speed rail-way bridges concerning the running safety of trains and the ridingcomfort of passengers. For instance, a limit of 0.05 g (= 0.49 m/s2)was previously used for the vertical acceleration of train cars travel-ing over bridges by the Bureau of Taiwan High Speed Railways. Dueto the relatively stringent requirements, the design of high-speed rail-way bridges is governed generally by the conditions of serviceability,rather than by the strength or yielding of materials, as encounteredin most highway bridges.

In this section, parametric studies will be carried out for both asimply-supported beam and a three-span continuous beam traveledby trains moving at high speeds, using the VBI element and proce-dure developed in the early part of this chapter, based primarily onthe paper by Yau et al. (1999). By high speeds, we mean that thetrain is allowed to travel in a range of speeds from 250 to 400 km/h,to reflect the current advances in power and control technologies onoperation of high-speed trains.

The numerical results to be discussed in the following subsectionsindicate that the moving load model is generally accurate if onlythe bridge response is desired. However, the use of the sprung massmodel is necessary whenever the riding comfort of passengers carriedby the train is of concern. Noteworthy is the fact that if the char-acteristic length, rather than the span length, is used in computingthe first speed parameter S for the continuous beam (see Eq. (1.4)for definition), then both the simple and continuous beams will at-tain their peaks at the same critical speed S, when subjected to the



same sequence of axle loads. Factors such as the rail irregularity,ballast stiffness, suspension stiffness and suspension damping can af-fect drastically the riding comfort of the train cars traveling oversimple beams. Their influence appears to be comparatively small forcontinuous beams.

6.8.1. Models for Bridge, Train and Rail

Irregularities

A simply-supported beam (SB) and a three-span continuous beam(CB) are considered in this section, which are made of prestressedconcrete with elastic constant E = 29.43 GPa and Poisson’s ratiov = 0.2. The damping of the bridge is assumed to be of the Rayleightype, with a damping ratio of 2.5% assumed for the first two modes.The properties of the beams have been given in Table 6.1, where l =the span length, L = characteristic length, A = cross sectional area, I

= moment of inertia, m = mass per unit length, and ω = frequencyof vibration. Using the characteristic length L for the continuousbeam and the fundamental frequency ω of vibration for the simpleand continuous beams given in Table 6.1, the vehicle velocity v canbe related to the first speed parameter S using Eq. (1.4) as v =1007.3 S km/h for the simple beam, and as v = 1017.6 S km/h forthe continuous beam.

In this section, a train is modeled as a sequence of sprung massesof regular intervals, as shown in Fig. 6.1(b). Two models of train sim-ilar to those commercially available are considered herein, of whichthe dynamic properties are given in Table 6.2, in which Mv denotesthe mass lumped from one half of the car body, mw the wheel mass,kv and cv respectively denote the stiffness and damping of the sus-pension devices, as defined in Fig. 6.2. The S25 model consists of 10axles with regular nonuniform intervals and the T18 model consistsof eight equidistantly-spaced axles.

Track irregularities may be caused by factors such as small im-perfections in materials, imperfections in manufacturing of rails andrail joints, terrain irregularities, and errors in surveying during de-sign and construction. The irregularity function proposed by Nielsen



Table 6.1. Properties of beams.

Beams l (m) L (m) A (m2) I (m4) m (kg/m) ω (rad/s)

Simple 30 30 7.94 8.72 36 056 29.3 117.0 263.3Continuous 25-40-25 30 8.66 14.05 37 784 29.6 60.4 70.7

Table 6.2. Dynamic properties of trains.

Model Axle arrangement Mv mw kv cv

(m) (kg) (kg) (kN/m) (kN-s/m)

S25 ↓18 ↓7 ↓18 ↓7 ↓18 ↓7 ↓18 ↓7 ↓18 ↓ 24 000 5000 1500 85T18 ↓18 ↓18 ↓18 ↓18 ↓18 ↓18 ↓18 ↓ 30 000 5000 1700 90

and Abrahamsson (1992) for the vertical profile of the rails will beadopted in this section:

r(x) = −r0

[1 − exp

(− x

x0

)3]

sin2πx

γ0, (6.38)

where x is the along-track distance (in m), x0 = 1.0 m, r0 (= 0.5 mm)is the amplitude of irregularities, and γ0 (= 1.0 m) the wavelength ofthe corrugation. As for the ballast stiffness kB , the value of 20 MN/mper rail has been used by Nielsen and Abrahamsson (1992). In thissection, the ballast stiffness kB is taken as 40 MN/m for two rails.

6.8.2. Moving Load versus Sprung Mass Model

To investigate the effect of different vehicle modelings on the bridgeresponse, the train is modeled either as a series of moving loadsor sprung masses, with no consideration made of rail irregularities.The impact factors I solved for the midpoint displacement of thesimple beam and three-span continuous beams have been plottedagainst the first speed parameter S in Figs. 6.11 and 6.12, for the twovehicle models T18 and S25, respectively. Evidently, the moving loadmodel can be reliably used to predict the bridge responses, as littledifference exists between the solutions obtained for the two types of



Fig. 6.11. Impact factor for midpoint deflection of beam (T18)-different bridgemodels.

Fig. 6.12. Impact factor for midpoint deflection of beam (S25)-different bridgemodels.



vehicle modeling. Moreover, the inclusion of the inertial effect ofthe moving vehicles, as represented by the sprung mass model, hasresulted in slight reduction of the peak response of the bridge, dueto the fact that the sprung mass behaves in some sense like a tunedmass to the bridge.

Figures 6.11 and 6.12 indicate that the simple beam reaches itspeak responses at the speeds S = 0.300 and 0.417 (or equivalentlyat v = 302 and 420 km/h for the case considered) under the movingaction of the two trains T18 and S25, respectively. These are exactlythe critical speeds (nondimensionalized) for the bridge to resonateunder the passage of the two trains, which correspond very well tothe values predicted from the formula given by Yang et al. (1997b)for the resonance of simple beams, i.e., the one given in Eq. (2.59)or S = d/(2L), where d is the train car length, which equals 25 mfor the S25 model and 18 m for the T18 model.

From Fig. 6.11, one observes that for the T18 train, whose axleloads are equidistantly-spaced, the continuous beam attains its peakresponse at exactly the same critical speed S as that for the simplebeam, although the peak amplitude has been drastically reduced dueto the restraint effect of the multi supports. No similar behavior isobserved from Fig. 6.12 for the S25 train, partly due to the factthat the axles are not uniformly-spaced for this case. Moreover, theimpact response appears to be much smaller for the continuous beamthan for the simple beam. Such an observation is consistent with theobservation made by Yang et al. (1995) for continuous beams underthe passage of a single HS20-44 truck.

6.8.3. Effect of Rail Irregularities

In general, the rail irregularities have little influence on the impactresponse of the bridges (see Figs. 6.11 and 6.12). However, the sameis not true with the moving vehicles or sprung masses. For the presentpurposes, let us adopt the rail irregularities given in Eq. (6.38). Forboth the simple beam (SB) and continuous beam (CB), the verticalacceleration of the sprung masses have been plotted in Figs. 6.13and 6.14 for the two train models T18 and S25, respectively. From



Fig. 6.13. Maximum acceleration of sprung mass (T18)-effect of irregularities.

Fig. 6.14. Maximum acceleration of sprung mass (S25)-effect of irregularities.



both figures, a critical speed of S = 0.05 can be identified, which isequivalent to v = 50.4 km/h for the simple beam and to 50.9 km/hfor the continuous beam. Let ωB denote the frequency of vibrationof the ballast, i.e.,

ωB =

√kB

mw= 89.44 rad/s . (6.39)

The speed at which the peak vehicle response occurs can be ex-plained as a phenomenon of resonance caused by the coincidenceof the ballast frequency ωB with the frequency of rail irregularitiesas represented by 2πv/γ0, where γ0 is the wavelength of corruga-tion (= 1.0 m). For instance, by letting 2πv/γ0 = ωB, one obtainsv = ωBγ0/2π. Hence,

S =πv

ωL=

ωBγ0

2ωL. (6.40)

By substitution of the values of ω, ωB, γ0, and L for the simple andcontinuous beams, the speeds at which the peak response occurs canbe computed, which turn out to be S = 0.05, coincident with thatobserved from Figs. 6.13 and 6.14. From Eq. (6.40), it is obvious thatthe larger the ballast stiffness, the greater is the speed for resonanceto occur. It should be added that the peak responses plotted inFigs. 6.13 and 6.14 for the vehicles exceed significantly the tolerancelimit of 0.49 m/s2 (= 0.05 g) previously used by the Bureau of TaiwanHigh Speed Railways.

6.8.4. Effect of Ballast Stiffness

To investigate the influence of ballast stiffness on the bridge response,three different values of ballast stiffness, 0.25kB , kB and 2.5kB , areused. For both the simple and continuous beams, the correspond-ing impact factors solved for the midpoint displacement have beenplotted in Figs. 6.15 and 6.16, for the two train models T18 andS25, respectively. As can be seen, softer ballast tends to reduce theimpact response of the bridges, although the degree of reduction ismarginal. One possible explanation for this is that the ballast layerserves to some extent as an energy dissipating mechanism for thebridge under the action of moving trains.



Fig. 6.15. Impact factor for midpoint deflection of beam (T18)-effect of ballaststiffness.

Fig. 6.16. Impact factor for midpoint deflection of beam (S25)-effect of ballaststiffness.



Fig. 6.17. Maximum acceleration of sprung mass (T18)-effect of ballast stiffness.

Unlike the case with the bridge response, the effect of ballast onthe vehicle response can be quite significant. For instance, the max-imum vertical accelerations of the sprung masses have been plottedin Figs. 6.17 and 6.18 for trains moving over the simple and contin-uous beams with different ballast stiffnesses, but no rail irregular-ities, for the two train models T18 and S25. The following obser-vations can be drawn from the two figures. First, for both simpleand continuous beams, resonant peaks exceeding the tolerance limitof 0.49 m/s2 (= 0.05 g) will be induced for train cars moving overharder ballast (i.e., with stiffness 2.50kB) in the low speed rangewith S < 0.15 (or v < 151.1 km/h). Second, softer ballast increasessignificantly the acceleration level of the sprung masses moving oversimple beams in the high-speed range, say, with S = 0.34 ∼ 0.50 (orv = 342.5 ∼ 503.7 km/h), especially for the S25 model. Finally, thedynamic response of the train cars moving over continuous beams isgenerally small for S > 0.20 (or v > 203.5 km/h) regardless of thestiffness of ballast. For the continuous beam, the use of softer ballastappears to be very effective for reducing the sprung mass response.



Fig. 6.18. Maximum acceleration of sprung mass (S25)-effect of ballast stiffness.

6.8.5. Effect of Vehicle Suspension Stiffness

Three values of suspension stiffness 0.25kv , kv and 2.5kv are assumedfor the two train models. As can be seen from Figs. 6.19 and 6.20, theinfluence of suspension stiffness of the train on the bridge responseis generally small for the simple beam and even invisible for thecontinuous beam, although a stiffer suspension system can slightlyreduce the bridge response. The conclusion here is that the effect ofsuspension stiffness of the moving train on the bridge response canbe generally ignored in a practical design.

However, the same is not true with the vertical acceleration of thetrain cars running over the bridge. As can be seen from Fig. 6.21for the T18 model and Fig. 6.22 for the S25 model, in addition tothe peak response caused by resonance with the ballast at S = 0.05(or v = 50.4 km/h), the use of stiffer suspension systems increasessignificantly the vertical acceleration of the train cars in running.This is certainly harmful concerning the riding comfort of passengers.



Fig. 6.19. Impact factor for midpoint deflection of beam (T18)-effect of suspen-sion stiffness.

Fig. 6.20. Impact factor for midpoint deflection of beam (S25)-effect of suspen-sion stiffness.



Fig. 6.21. Maximum acceleration of sprung mass (T18)-effect of suspensionstiffness.

Fig. 6.22. Maximum acceleration of sprung mass (S25)-effect of suspensionstiffness.



The reason behind this is that by making the suspension systemsstiffer, the frequency of the sprung mass, as represented by

√kv/Mv ,

tends to increase and approach that of the bridge. As the two fre-quencies become closer, the sprung mass will attain its maximumresponse, in a manner similar to that of the tuned mass for thebridge. For the continuous beam, although the same tendency canbe observed for the suspension stiffness from both figures, the level ofinfluence is comparatively small. Note that there also exists a peakaround S = 0.05 (or v = 50.9 km/h) for the continuous beam.

6.8.6. Effect of Vehicle Suspension Damping

Three values of suspension damping are considered, that is, 0.25cv ,cv, and 2.5cv . As can be seen from Figs. 6.23 and 6.24, for the twotrain models, by increasing the damping of the suspension systems,the response of the bridge will be reduced, although the degree ofinfluence is only marginal. However, as far as the dynamic responseof the train cars (or sprung masses) is concerned, it can be seen from

Fig. 6.23. Impact factor for midpoint deflection of beam (T18)-effect of suspen-sion damping.



Fig. 6.24. Impact factor for midpoint deflection of beam (S25)-effect of suspen-sion damping.

Fig. 6.25. Maximum acceleration of sprung mass (T18)-effect of suspensiondamping.



Fig. 6.26. Maximum acceleration of sprung mass (S25)-effect of suspensiondamping.

Figs. 6.25 and 6.26 for the T18 and S25 models, respectively, thatlarger values of suspension damping tend to drastically increase themaximum vertical acceleration of the train cars traveling over thesimple beam for a wide range of speeds that is of interest. Such afact must be carefully taken into account in the design of train cars.In contrast, the effect of suspension damping on the vehicle responseis comparatively small for the case of continuous beams, as can beseen from both figures.


In this chapter, the equations of motion for the vehicle is first dis-cretized using Newmark’s finite difference formulas and then con-densed to the bridge equations, considering the condition of contactbetween the bridge and moving vehicles. The vehicle–bridge interac-tion (VBI) element derived has the advantage that it possesses thesame number of DOFs as the parent element, while preserving the



properties of symmetry and bandedness in element matrices. Becauseof this, conventional assembly process can be applied to forming theequations of motion for the entire VBI system. The present approachallows us to compute not only the bridge response, but also the ve-hicle response. The applicability of the derived element has beendemonstrated in the numerical examples.

The following conclusions can be extracted from the parametricstudies for the simple and three-span continuous beams using theVBI element derived: (1) The moving load model can be reliablyused to predict the bridge response. However, the use of the sprungmass model enables us to compute the vehicle response, in addi-tion to the bridge response. (2) Resonant response may be inducedon the bridge by the train within the range of speeds of operation.The critical speed for resonance solved by the present approach co-incides very well with that predicted by the analytical formula, asgiven in Eq. (2.59). (3) If the characteristic length, rather than thespan length, is used for a continuous beam, then both the simpleand continuous beams will attain their peaks at the same criticalspeed S, when subjected to moving loads of constant intervals. (4)The effect of rail irregularities is generally small on the bridge re-sponse. However, it can affect drastically the vertical acceleration ofthe sprung masses, which serves as a measure of the riding comfortfor passengers carried by the train. (5) Softer ballast tends to reducethe impact response of the bridges, although the degree of reductionis marginal. For simple beams, the use of ballast that is too hard ortoo soft is not good for running vehicles concerning the riding com-fort. (6) The influence of suspension stiffness on the bridge responseis generally small, which can be neglected in practice. However, theuse of stiffer suspension devices can dramatically increase the verticalacceleration of the train cars. (7) Increasing the suspension damp-ing can only result in marginal reduction of the bridge response, butcan adversely magnify the dynamic response of the train cars. (8)In general, much smaller response can be expected of a continuousbeam, compared with that of the simple beam.



Chapter 7

Vehicle–Bridge Interaction ElementConsidering Pitching Effect

Vehicle–bridge interaction (VBI) elements that were derived by treat-ing a vehicle as discrete lumped masses, such as the one shown inChapter 6, has the apparent drawback that the pitching effect ofthe car body caused by the differential deflections of the suspensionsystems was not duly taken into account. To overcome this draw-back, a vehicle with two suspension systems, each for the front andrear wheel assemblies, is modeled as a rigid beam supported by twospring-dashpot units in this chapter. The equations of motion writ-ten for the vehicle and the bridge (beam) elements are coupled dueto the existence of two interacting forces acting through the contactpoints. Following the procedure presented in the preceding chapter,the vehicle equations are first reduced to a set of equivalent stiffnessequations using Newmark’s finite difference scheme. The vehicle de-grees of freedom (DOFs) are then condensed to those of the beamelements in contact. The rigid vehicle–bridge interaction elementsderived is good for computing not only the bridge response, but alsothe vehicle response, with the vehicle’s pitching effect duly taken intoaccount.

7.1. Introduction

Previous research on the interaction between a bridge and the vehi-cles traveling over it has been abundant, which continues to growin recent years due to the booming demands of high-speed rail-ways in several European and Asian countries. Some of the related

199



research works have been reviewed in Chapter 1. When studyingthe dynamic response of a vehicle–bridge interaction system, twosets of second-order differential equations can be written, one for themoving vehicles and the other for the bridge. It is the interactionforces existing at the contact points of the two subsystems that makethe two sets of equations coupled. As the contact points are time-dependent, whose acting positions move from time to time, so arethe system matrices, which therefore must be updated and factorizedat each time step. To solve these two sets of second-order differen-tial equations, procedures of an iterative nature are often adopted.For instance, by first assuming a trial solution for the displacementsof the contact points, the contact forces can be solved from the ve-hicle equations. Then, by substituting these forces into the bridgeequations, an improved solution for the displacements of the contactpoints can be solved. One drawback with iterative approaches of thistype is that the convergence rate of iteration is likely to be low, whendealing with the more realistic case of a bridge traveled by a seriesof vehicles that make up a train, for there exists a great number ofcontact points.

The condensation method has been demonstrated to be an ef-ficient method for solving the vehicle–bridge interaction problems.Garg and Dukkipati (1984) used the Guyan reduction scheme tocondense the degrees of freedom (DOFs) of the vehicles to the DOFsof the bridge. In the paper by Yang and Lin (1995), the dynamiccondensation method was used instead. However, because of theapproximation made in relating the vehicle (slave) to the bridge(master) DOFs, this approach is good only for computing the bridgeresponse, but not for the vehicle response; the latter serves as theindicator of riding comfort useful to the design of high-speed railroadbridges. To overcome this drawback, the vehicle equations were firstreduced to a set of equivalent stiffness equations using Newmark’sfinite difference scheme, by which the vehicle DOFs are condensed tothose of the bridge element in contact. The vehicle–bridge interaction(VBI) element derived is featured by the fact that the symmetry andbandedness of the element matrices are retained, while the vehicle


Vehicle–Bridge Interaction Element Considering Pitching Effect 201

response can be computed simply by back substitution (Yang andYau, 1997). Also, it possesses the same order of accuracy as thatimplied by the Newmark scheme used in the solution of the entirevehicle–bridge system.

The aforementioned VBI element is not perfect, however, sinceno account has been made of the pitching effect of the car bodywith respect to the front and rear wheels, which appears to be ofparamount importance in computation of the vehicle response andmay be aggravated in the presence of track irregularities. Previously,a series expansion approach was conducted by Wen (1960) to analyzethe dynamic response of beams traversed by two-axle loads. Thischapter can be regarded as an extension of the theory presentedin Chapter 6 toward the development of accurate and efficient VBIelements. It surpasses most existing VBI elements, in that the vehicleis modeled as a rigid beam supported by two suspension units, ratherthan as one or two discrete sprung masses. Further extension of thecondensation technique presented herein is not impossible, say, toinclude the rolling motion or other three-dimensional effects of themoving vehicle.

Theoretically, more sophisticated models that contain dozens ofDOFs can be developed for the vehicle, similar to those presentedin Chu et al. (1986), and Huang and Novak (1991). However, dueto their complexities and the large amount of computation required,these models are most suitable for the simulation of a single vehicletraveling over bridges or for the case where the vehicle response isof major concern. Since we are interested in the simulation of themore realistic case of a bridge traveled by a series of vehicles thatconstitute a train, the use of too complicated a vehicle model willmake the derivation of the VBI element a formidable task. It iswith these considerations that the vehicle model presented in thischapter, along with the special condensation technique, is consideredappropriate. It should be noted that the materials presented in thischapter are based generally on the paper by Yang et al. (1999),except those in Secs. 7.5.4–7.5.6, which are based on the thesis byChang (1997).



7.2. Equations of Motion for the Vehicle and Bridge

Figure 7.1 shows a train moving with speed v over a bridge, in whicha bridge is modeled as a number of beam elements and each car of thetrain as a uniform rigid beam supported by two suspension (spring-dashpot) units, as an improvement over the sprung mass model usedin the preceding chapter. A schematic of the VBI system consideredis given in Fig. 7.2, in which r(x) denotes the track irregularity andmw the (unsprung) mass of the bogie. The motion of the rigid beamis described by the generalized coordinates yT = yvθv, with yv

denoting the vertical displacement and θv the rotation about thecenter point. In this chapter, the subscripts i and j respectively are

... ...

Rail withIrregularities

Car BodyCar Body

Suspension System

TruckWheelset

Fig. 7.1. Train–bridge system.

y

z

x

Rail Irregularity

element ielement j

mwmw

kv cvcv

Iv

Mv

r x

v

yv

θv

kv

Lc

Fig. 7.2. Typical VBI system.



y

z

x

element i

element j

fc2fc3 fc1fc4 WW

xcixcj

ui

u j

rirj

xci

xcj

WW

2W

yv

θv

Fig. 7.3. Free body diagrams for components of the VBI system.

used to denote quantities associated with the front and rear wheels(or wheel assemblies) of the vehicle under consideration. As can beseen from Fig. 7.2, both elements i and j of the bridge, on which thevehicle is acting will be affected by the pitching motion of the carbody via the two suspension units. The reverse is also true.

Figure 7.3 shows the free body diagrams for the components of theVBI system that are of interest, in which 2W denotes the weight ofthe car body (i.e., W = 0.5 Mvg, with Mv indicating the total massof the car body, and g the acceleration of gravity), xc the contactposition of each set of wheels (or wheel assemblies) on the beamelement, and u the vertical displacement of the beam element. Ascan be seen, the (rigid) car body is acted upon by the contact forcesfc1, fc2, fc3, fc4, and the front and rear wheels by the contact forces



(fc1 + fc2 + W ) and (fc3 + fc4 + W ), respectively. For the presentcase, all the component forces can be given as follows:

fc1 = cv(ui − yv − 0.5Lcθv) ,

fc2 = kv(ui + ri − yv − 0.5Lcθv) ,

fc3 = cv(uj − yv + 0.5Lcθv) ,

fc4 = kv(uj + rj − yv + 0.5Lcθv) ,

(7.1)

where an overdot denotes differentiation with respect to time t; kv,cv = the spring constant and damping coefficient of the suspensionunit; Lc = the axle distance of the vehicle; ri, rj = the track irreg-ularities of elements i and j at the contact points; and ui, uj = thedisplacements of elements i and j evaluated at the contact points.The equations of equilibrium for the rigid car body are

fc1 + fc2 + fc3 + fc4 = Mvyv ,

0.5d(fc1 + fc2 − fc3 − fc4) = Iv θv ,(7.2)

where Iv = the rotatory inertia of the car body. Substituting thecomponent forces in Eq. (7.1) into the preceding two equations inEq. (7.2) yields the equations of motion for the vertical and rotationaldisplacements of the car body as:[

Mv 00 Iv

]yv

θv

+[

2cv 00 0.5cvL

2c

]yv

θv

+[

2kv 00 0.5kvL

2c

]yv

θv

=

fver

frot

, (7.3)

where the term on the right-hand side denotes the acting forces re-sulting from the two suspension (spring-dashpot) units,

fver

frot

=

cv(ui + uj) + kv(ui + uj) + kv(ri + rj)

0.5Lc[cv(ui − uj) + kv(ui − uj) + kv(ri − rj)]

.

(7.4)



As can be seen, the vertical force fver and rotational moment frot ex-perienced by the car body are caused by the velocity u, displacementu, and track irregularity r of the bridge and transmitted through thesuspension units at the contact points.

In this chapter, the most commonly used 12-DOF beam elementwill be used to simulate the bridge structure, of which the axialdisplacement is interpolated by linear functions, and the transversedisplacements by cubic interpolation (Hermitian) functions. For in-stance, the y-direction displacement u of the element at section x

(in local coordinate) can be related to the nodal DOFs of the beamelement as u = 〈N〉uy, where uy denotes the y-direction relatednodal DOFs, uyT = 〈uyA lθzA uyB lθzB〉, and the associated inter-polation function N is

N =⟨

1 − 3(x

l

)2+ 2(x

l

)3, x

(1 − x

l

)2, 3

(x

l

)2

− 2(x

l

)3,(x

l− 1) x2

l

⟩T

, (7.5)

where l denotes the length of the element. See Appendix F for no-tation of the nodal DOFs of the beam element, and Yang and Kuo(1994), Chapter 2, for more details on the interpolation functionsused by the beam element. Assuming that Rayleigh damping is validfor the bridge, the equations of motion can be written for elements i

and j of the bridge as

[mi]ui + [ci]ui + [ki]ui = −piNci ,

[mj ]uj + [cj ]uj + [kj ]uj = −pjNcj ,(7.6)

where u = the displacement vector; [m], [c], and [k] = the mass,damping, and stiffness matrices of the element [see for instance Ap-pendix F for details of these matrices]; Nc = the Hermitian func-tions evaluated for the DOFs associated with the y-direction displace-ment at the contact point (i.e., at x = xc), which can be obtainedby substituting the contact point position xc into Eq. (7.5), but withthose associated with the x- and z-direction displacements set to zero;



and p = the contact force for each of the two elements considered,

pi = fc1 + fc2 + 0.5Mvg + mw(g + ui) ,

pj = fc3 + fc4 + 0.5Mvg + mw(g + uj) ,(7.7)

with mw denoting the mass of the wheel assemblies.In Eq. (7.6), the terms on the right-hand side should be recognized

as the consistent nodal loads acting on the two elements due to thecontact forces. Besides, a comparison of the vehicle equations, i.e.,Eq. (7.2), with the bridge equations, i.e., Eq. (7.6), together withthe contact force expressions in Eq. (7.7), indicates that the twosubsystems are coupled via the spring and damping forces, i.e., fc1,fc2, fc3, fc4, of the suspension units.

By substitution of the spring and damping forces in Eq. (7.1), thecontact forces in Eq. (7.7) can be rewritten,

pi = cv(ui − yv − 0.5Lcθv) + kv(ui + ri − yv − 0.5Lcθv)

+ 0.5Mvg + mw(g + ui) ,

pj = cv(uj − yv + 0.5Lcθv) + kv(uj + rj − yv + 0.5Lcθv)

+ 0.5Mvg + mw(g + uj) .

(7.8)

As can be seen, the contact forces acting on the two elements i andj are composed of four components: (1) the static weights associ-ated with the car body and wheel assemblies, as represented by Mvg

and mwg; (2) the damping forces resulting from the relative velocityof the car body to the bridge elements, as indicated by the termscontaining cv; (3) the elastic forces resulting from the relative dis-placement of the car body to the bridge elements, as indicated bythe terms involving kv; and (4) the inertial forces due to the verticalacceleration of the bridge elements, as indicated by the term mwu.It is these coupling forces existing between the two subsystems thatmake the vehicle–bridge interaction a problem difficult to solve. Inthe preceding chapter, it was demonstrated that such a problem canbe effectively resolved by first reducing the vehicle equations intoa set of equivalent stiffness equations using Newmark’s finite dif-ference scheme, and then by condensing the vehicle DOFs to those



of the bridge element in contact, which will result in the so-calledVBI element. Such an approach will be generally followed in thischapter.

7.3. Rigid Vehicle–Bridge Interaction Element

The system equations as given in Eqs. (7.3) and (7.6) for the car bodyof the moving vehicle and the associated bridge elements, along withthe acting forces in Eq. (7.4) and the contact forces in Eq. (7.8), arenonlinear in nature, as the contact points move from time to time,which can only be solved by incremental methods. Consider a typicalincremental step from time t to t + ∆t. The system equations, i.e.,Eqs. (7.3) and (7.6), together with the contact forces in Eqs. (7.4)and (7.8), should now be interpreted as those established for thestructure in the current configuration at time t + ∆t.

Accordingly, the equations of motion for the car body, i.e.,Eq. (7.3), can be rewritten for the current time step, with t + ∆t

clearly inserted as the subscript as:

[Mv 00 Iv

]yv

θv

t+∆t

+[

2cv 00 0.5cvL

2c

]yv

θv

t+∆t

+[

2kv 00 0.5kvL

2c

]yv

θv

t+∆t

=

fver

frot

t+∆t

, (7.9)

where the acting forces are given as

fver

frot

t+∆t

=

cv(ui + uj) + kv(ui + uj) + kv(ri + rj)

0.5Lc[cv(ui − uj) + kv(ui − uj) + kv(ri − rj)]

t+∆t

.

(7.10)



Similarly, the equations of motion for the bridge elements i and j, asgiven in Eq. (7.6), can be rewritten for the current time step t+∆t as

[mi]uit+∆t + [ci]uit+∆t + [ki]uit+∆t = −pi,t+∆tNci ,

[mj ]ujt+∆t + [cj ]ujt+∆t + [kj ]ujt+∆t = −pj,t+∆tNcj ,(7.11)

where the associated contact forces are

pi,t+∆t = [cv(ui − yv − 0.5Lcθv) + kv(ui + ri − yv − 0.5Lcθv)

+ 0.5Mvg + mw(g + ui)]t+∆t ,

pj,t+∆t = [cv(uj − yv + 0.5Lcθv) + kv(uj + rj − yv + 0.5Lcθv)

+ 0.5Mvg + mw(g + uj)]t+∆t .

(7.12)

Following the procedures of the preceding chapter, the vehicle equa-tions, Eq. (7.9), which are of second order, will first be reduced to aset of equivalent stiffness equations using Newmark’s finite differencescheme. Then, the vehicle DOFs will be condensed to those of thebridge elements in contact.

Let yt denote the car body displacements at time t. The fol-lowing are the finite difference equations proposed by Newmark:

yt+∆t = yt + [(1 − γ)yt + γyt+∆t]∆t ,

yt+∆t = yt + yt∆t + [(0.5 − β)yt + βyt+∆t]∆t2 ,(7.13)

where β = 0.25 and γ = 0.5 will be adopted, for they imply anintegration scheme of constant average acceleration within the timestep, which has the advantage of being unconditionally stable. FromEq. (7.13), the following can be solved,

yt+∆t = a0(yt+∆t − yt) − a2yt − a3yt ,

yt+∆t = a1(yt+∆t − yt) − a4yt − a5yt ,(7.14)



where the coefficients and those to appear are defined as

a0 =1

β∆t2, a1 =

γ

β∆t, a2 =

1β∆t

,

a3 =12β

− 1 , a4 =γ

β− 1 , a5 =

∆t

2

(γ

β− 2)

,

a6 = ∆t(1 − γ) , a7 = γ∆t .

(7.15)

By the relations in Eq. (7.14), the car body equations, i.e., Eq. (7.9),can be manipulated to yield a set of equivalent stiffness equations asfollows:[

D11 00 D22

]yv

θv

t+∆t

=

fver

frot

t+∆t

+

fver

frot

t

, (7.16)

where

D11 = a0Mv + 2a1cv + 2kv ,

D22 = a0Iv + 0.5a1cvL2c + 0.5kvL

2c ,

(7.17)

and the last term on the right-hand side denotes the equivalent nodalloads resulting from the vehicle responses at time t,

fver

frot

t

=

Mv(a0yv + a2yv + a3yv) + 2cv(a1yv + a4yv + a5yv)

Iv(a0θv + a2θv + a3θv) + 0.5cvL2c(a1θv + a4θv + a5θv)

t

.

(7.18)

Solving Eq. (7.16) yields the car body displacements yv θvT at timet + ∆t as

yv

θv

t+∆t

=1D

D22fver

D11frot

t+∆t

+1D

D22fver

D11frot

t

, (7.19)



where D is the determinant of the system matrix in Eq. (7.16), D =D11D22. Once the vehicle displacements yv θvT are made availableat time t + ∆t, the car body velocities at the same moment can becomputed using Eq. (7.14) as

yv

θv

t+∆t

=a1

D

D22fver

D11frot

t+∆t

+

f1

f2

t

, (7.20)

where

f1

f2

t

=1D

D22(−2a1kvyv + a0Mvyv − 2a4kvyv

+a0a6Mvyv − 2a5kvyv)

D11(−0.5a1kvL2cθv + a0Iv θv − 0.5a4kvL

2c θv

+a0a6Iv θv − 0.5a5kvL2c θv)

t

.

(7.21)

In the meantime, the car body accelerations can be computed as

yv

θv

t+∆t

=a0

D

D22fver

D11frot

t+∆t

−

f3

f4

t

, (7.22)

where

f3

f4

t

=1D

2D22(a0kvyv + a0cvyv + a2kv yv

+a0a6cvyv + a3kv yv)

0.5L2cD11(a0kvθv + a0cv θv + a2kv θv

+a0a6cv θv + a3kv θv)

t

. (7.23)

As can be seen from Eqs. (7.19), (7.20) and (7.22), one feature withNewmark’s finite difference equations is that all the quantities to besolved at time t + ∆t can be related exclusively to those existing atthe beginning and ending points, i.e., at time t and t + ∆t, of thecurrent time step.



By substitution of Eqs. (7.19) and (7.20), the car body displace-ments yv and θv can be eliminated from the associated bridge elementequations, Eq. (7.11), resulting in

[[mi] + [mii] [0]

[0] [mj ] + [mjj]

] uiuj

t+∆t

+[

[ci] + [cii] [cij ]

[cji] [cj ] + [cjj]

] uiuj

t+∆t

+[

[ki] + [kii] [kij ]

[kji] [kj ] + [kjj]

] uiuj

t+∆t

= −

piNcipjNcj

. (7.24)

These are exactly the equations of motion for the two elements i andj on which the front and rear wheels (or wheel assemblies) of thevehicle are directly acting. The elements as presented in Eq. (7.24)are referred to as the rigid vehicle–bridge interaction elements, whichare characterized by the fact that the car body dynamic propertieshave been included through the condensation process, and that theelement has the same order of accuracy as that implied by the New-mark finite difference scheme. In Eq. (7.24), [0] is a 12 × 12 matrixcontaining only zero entries, the vehicle-induced matrices appear-ing on the first row of Eq. (7.24) associated with element i can begiven as

[mii] = mwNci〈Nci〉 ,

[cii] = cvξNci〈Nci〉 ,

[cij ] = cvηNci〈Ncj〉 ,

[kii] = kvξNci〈Nci〉 ,

[kij ] = kvηNci〈Ncj〉 ,

(7.25)



and those on the second row associated with element j as

[mjj] = mwNcj〈Ncj〉 ,

[cjj ] = cvξNcj〈Ncj〉 ,

[cji] = cvηNcj〈Nci〉 ,

[kjj ] = kvξNcj〈Ncj〉 ,

[kji] = kvηNcj〈Nci〉 .

(7.26)

It is easy to see that the terms on the right-hand side of the equal signof Eq. (7.24) represent the consistent nodal loads acting on elementsi and j, resulting from the following contact force pi for element i:

pi = −[kv(ςri + ηrj) + 0.5Mvg + mwg]t+∆t

+[cv(f1 + 0.5Lcf2) +

kv

D(D22fver + 0.5D11Lcfrot)

]t

(7.27)

and the following for element j:

pj = −[kv(ςrj + ηri) + 0.5Mvg + mwg]t+∆t

+[cv(f1 − 0.5Lcf2) +

kv

D(D22fver − 0.5D11Lcfrot)

]t

, (7.28)

where

ς =a0

D[(a1cv + kv)(0.25MvL

2c + Iv) + a0MvIv] ,

η =a0

D(a1cv + kv)(0.25MvL2

c − Iv) .(7.29)

As can be seen from Eq. (7.24), the mass, damping and stiffnessmatrices derived for the rigid vehicle–bridge interaction elements dif-fer from those of the parent (beam) elements in the addition of thematrices given in Eq. (7.25) for element i, those in Eq. (7.26) forelement j, to account for the effect of interaction from the movingvehicle. In particular, the matrices [cij ] and [kij ] represent the pitch-ing action of the car body via the front wheels on element i, and thematrices [cji] and [kji] via the rear wheels on element j. It shouldbe noted that because of the existence of the transposition relations:



[cij ] = [cji]T and [kij ] = [kji]T , the system matrices [M ], [C], [K] as-sembled from all the element matrices, including those that are freeof moving vehicles, retain the desired property of symmetry.

7.4. Equations of Motion for the VBI System

When a train is traveling over a bridge, the elements of the bridgethat are directly acted upon by the wheels (or wheel assemblies) ofthe vehicles constituting the train at the instant considered should bemodeled by the VBI elements derived, and the remaining by the con-ventional (parent) beam elements. Here, two things need be noted.First, the vehicle loads can only transmit through the nodal pointsof the VBI elements, but not of the conventional beam elements.Second, the vehicle-induced matrices that are special for the VBI el-ements, when summed up all over the entire VBI system, turn out tobe symmetric on the global level. Based on these considerations, thefirst step in analyzing a VBI system is to construct the mass matrix,damping matrix, and stiffness matrix for the primary bridge struc-ture with no vehicles acting on it. The next step is to determine theacting positions (xc) of the front and rear wheels (or wheel assem-blies) of each vehicle, given its speed and acceleration. The vehicle-induced matrices and the contact forces p of the VBI elements, whichare considered “extra” to those of the primary structure, can then bedetermined using Eqs. (7.25)–(7.26) and Eqs. (7.27)–(7.28), respec-tively, depending on whether they are acted upon by the front orrear wheels. In particular, the contact forces p can be transformedinto the (consistent) nodal loads of the VBI elements as given onthe right-hand side of Eq. (7.24). All these “extra” terms, which areto account for the VBI effect, including the pitching actions, shouldbe added to the entries in the system matrices corresponding to thenodal DOFs of each of the VBI elements considered.

For illustration, let us consider a simple beam traveled by a two-axle vehicle. By dividing the beam into three elements, the equationsof motion, in terms of [M ], [C] and [K], are first established for theprimary structure that is free of any vehicles, with the load vector



element i

(a)

+

+

=

U

U

U

U

U

U

U

U

U

U

U

U

b

b

b

b

b

b

b

b

b

b

b

b

1

2

3

4

1

2

3

4

1

2

3

4

p Nbi cikiiciimii

(b)

Fig. 7.4. Vehicle with front wheels only acting on the bridge.

P initialized as 0. For the case when only the front wheels ofthe vehicle are acting on the bridge, say, with coordinate xci knownfor the acting position on element i (which becomes now a VBI el-ement), the vehicle nodal loads piNci, where Nci is evaluatedusing Eq. (7.5) by setting x = xci, and the vehicle-induced matrices[mii], [cii], [kii] should be added to the rows and columns correspond-ing to the nodal DOFs of element i in the system equations, as wasdepicted in Fig. 7.4. Several things should be noted here. First, forthis case, the pitching actions vanish, i.e., [cij ] = [kij ] = [0]. Sec-ond, the vehicle load piNci and the vehicle-induced matrices [mii],[cii], [kii] should be adjusted according to the acting position xci ofthe front wheels at each time step. Finally, all these “extra” termsshould be removed from the system matrices associated with elementi once the front wheels move to the neighboring element.

For the case when both the front and rear wheels of the vehicle areacting on the bridge, say, with the front wheels acting at coordinatexci on element i and the rear wheels at coordinate xcj on elementj (here both elements i and j are the VBI elements), as shown in



element j element i

(a)

+

+

=

U

U

U

U

U

U

U

U

U

U

U

U

b

b

b

b

b

b

b

b

b

b

b

b

1

2

3

4

1

2

3

4

1

2

3

4

p Nbj cj

p Nbi ci

c jj c ji

cijcii

kjikjj

kiikijmii

mjj

(b)

Fig. 7.5. Vehicle with both front and rear wheels acting on the bridge.

Fig. 7.5, the vehicle nodal loads piNci and pjNcj, which areevaluated by setting x = xci and xcj for elements i and j, respectively,and the vehicle-induced matrices [mii], [mjj], [cii], [cij ], [cji], [cjj],[kii], [kij ], [kji] and [kjj ] should be added to the rows and columnscorresponding to the nodal DOFs of elements i and j in the systemequations. As was stated, all the load vectors and vehicle-inducedmatrices should be adjusted at each time step according to the actingpositions of the vehicle axles, and removed from the columns androws in the system matrices corresponding to each element if thatelement is no longer acted upon by the front or rear wheels.

The third case occurs when only the rear wheels of the vehicleare acting on the bridge, as shown in Fig. 7.6. For this case, thereis no pitching actions, i.e., [cij ] = [kij ] = [0]. By setting the act-ing position to be x = xcj, the vehicle nodal loads pjNcj andthe vehicle-induced matrices [mjj], [cjj], [kjj] can be evaluated andsubstituted into the rows and columns corresponding to the nodalDOFs of element j in the system equations. The other treatmentsare similar to those stated above.



element j

(a)

+

+

=

U

U

U

U

U

U

U

U

U

U

U

U

b

b

b

b

b

b

b

b

b

b

b

b

1

2

3

4

1

2

3

4

1

2

3

4

p Nbj cjk jjc jjmjj

(b)

Fig. 7.6. Vehicle with rear wheels only acting on the bridge.

Based on the above considerations, the system equations can beestablished for the VBI system at time t + ∆t as follows:

[M ]Ut+∆t + [C]Ut+∆t + [K]Ut+∆t = Pt+∆t , (7.30)

in which U = the displacement vector, P = the external loadvector, and [M ], [C] and [K] = the mass, damping and stiffnessmatrices of the VBI system. By substitution of the Newmark-typeequations similar to those given in Eq. (7.14) for the velocity and ac-celeration vectors, Ut+∆t and Ut+∆t, the preceding second-orderequations can be reduced to a set of equivalent stiffness equations as

[K]Ut+∆t = P , (7.31)

where the effective stiffness matrix [K] is

[K] = a0[M ] + a1[C] + [K] (7.32)

and the effective load vector [P ] is

P = Pt+∆t + [M ](a0Ut + a2Ut + a3Ut)

+ [C](a1Ut + a4Ut + a5Ut) . (7.33)



From Eq. (7.31), the displacements Ut+∆t for the bridge can besolved. Consequently, the velocities and accelerations of the bridge,i.e., Ut+∆t and Ut+∆t, can be computed using equations similarto Eq. (7.14). After the bridge responses are made available, the carbody responses can be computed simply by back substitution. Inthis regard, one may first compute the car body forces fver and frot

using Eq. (7.10), with which the car body displacements, velocitiesand accelerations can be computed using Eqs. (7.19), (7.20), and(7.22).

7.5. Numerical Studies

In this section, the VBI element derived in this chapter and theassociated computer program developed will be verified through thestudy of some examples. Thus, all the examples presented in thischapter are characterized by the fact that the pitching motion of themoving vehicles is duly taken into account. The time step size usedthroughout all time histories of the examples is ∆t = 0.001 s.

7.5.1. Simple Beam Traveled by a Two-Axle System

Consider a simple beam traveled by a two-axle system in Fig. 7.7.The frequencies of the vertical and rotational vibration of the carbody are as follows (Yau and Yang, 1998):

ωv =√

2kv

Mv,

ωθ =

√kvd

2

2Iv,

(7.34)

where Mv = the mass, Iv = rotatory inertia, kv = spring constant ofthe car body; and Lc = the axle distance. The following data are as-sumed: Mv = 180 t, Iv = 4600 t-m2, kv = 13783 kN/m, Lc = 17.5 m,and v = 100 km/h (= 27.78 m/s). Correspondingly, the frequenciesfor the two-axle system are: ωv = 12.38 rad/s; ωθ = 21.42 rad/s.The following data are assumed for the beam: E (Young’s modulus)



d 17 5. m

L 30m

kvkv

mw mw

Mv Iv

Fig. 7.7. Simple beam traveled by a two-axle system.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Time (s)

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

Mid

poin

t D

efle

ctio

n (

mm

)

Moving Load

Moving Two-Axle System

Rigid Car Body

Fig. 7.8. Midpoint deflection for the two-axle system.

= 29.43 GPa; v (Poisson’s ratio) = 0.2; L = 30 m; A (cross-sectionalarea) = 2.14 m2; Iy (moment of inertia) = 2.88 m4; m (mass per unitlength) = 12 t/m; and zero damping. The first frequency computedof the bridge is ω1 = 29.15 rad/s. For the present case, the vehicle-to-bridge mass ratio is Mv/mL = 0.5. In the finite element analysis,10 elements were used for the beam.

The midpoint displacement of the beam calculated using thepresent VBI element has been plotted in Fig. 7.8, along with the



analytical solution given by Yau and Yang (1998) for the two-axlesystem and the solution of the moving load model, obtained by set-ting the suspension stiffness kv of the sprung mass model equal toa very large value. It is observed that the present solution agreesvery well with the analytical one, from which the accuracy of thepresent VBI element in simulating the vehicle response is confirmed.However, slight deviation exists between the present solution andthe one based on the moving load model. This can be attributedto the fact that the moving load model loses its accuracy for largevehicle-to-bridge mass ratios; for the present case, the mass ratiois 0.5.

7.5.2. Simple Beam Traveled by a Train Consisting of

Five Identical Cars

Figure 7.9 shows a train consisting of five identical cars, which trav-els over a simple beam represented by 10 elements. The follow-ing data are assumed for each car: Mv = 48 t, Iv = 2500 t-m2,kv = 1500 kN/m, cv = 85 kN-s/m, mw = 5 t, Lc (axle distance) =18 m, and d (car length) = 25 m. The frequencies of vibration cal-culated for the car body are: ωv = 7.9 rad/s; ωθ = 9.86 rad/s.The data assumed below for the simple beam are close to thoseused in the high-speed railway bridges under construction in Taiwan:E = 29.43 GPa; v = 0.2; L = 30 m; A = 7.94 m2; Iz = 8.72 m4;m = 36 t/m; and damping coefficient = 2.5%. The first frequencycomputed of the bridge is ω1 = 29.30 rad/s.

For illustration, the time-history response of the midpoint dis-placement of the bridge traveled by the train at speeds v = 140 and

25m 25m 25m

25m

18m

Fig. 7.9. A train consisting of five identical cars.



0.0 1.0 2.0 3.0 4.0 5.0 6.0

Time (s)

-1.5

-1.0

-0.5

0.0

0.5M

idp

oin

t D

efle

ctio

n (

mm

)

v = 140 km/hr

v = 105 km/hr

Exit

Exit

Fig. 7.10. Time-history responses of the bridge excited by moving trains.

105 km/h have been plotted in Fig. 7.10. Assuming that the trainsenter the bridge at t = 0 s, they will leave the bridge at t = 3.81 sand 5.07 s for the speeds 140 and 105 km/h, respectively, which werealso indicated in the figure. It is interesting to note that the maxi-mum response of the bridge excited by the train moving at the lowerspeed, i.e., v = 105 km/h, is higher than that at the higher speed, i.e.,v = 140 km/h. The reason can be given as follows. From Eq. (2.59)of Chapter 2, it is known that the dimensionless resonant speed isSr = d/2nL, where d is the car length, L the bridge span length, andn a positive integer. With d = 25 m, L = 30 m, ω1 = 29.30 rad/s, onecan obtain the resonant speed from Eq. (1.4) as vr = 116.6/n (m/s)= 419.7/n (km/h). For n = 3, 4, the resonant speed vr turns out tobe 140, 105 km/h, implying that the two speeds analyzed equal theresonant speeds. On the other hand, from Eq. (2.61) of Chapter 2,it is also known that the dimensionless speed for the waves gener-ated by the continuously moving axles loads to cancel each other isSc = 1/(2i − 1), where i is a positive integer. Equivalently, the can-cellation speed is vc = 279.8/(2i − 1) (m/s) = 1007/(2i − 1) (km/h).Here, it can be seen that the traveling speed 140 km/h of the train



0.0 0.1 0.2 0.3 0.4 0.5

Speed parameter

0.0

0.5

1.0

1.5Im

pac

t F

acto

r Rigid Car Bodies

Sprung Lumped Masses

I

S

Fig. 7.11. Impact factor for midpoint displacement of simple beam.

is close to the cancellation speed of 144 km/h obtained by lettingi = 4, implying that the bridge response excited by the train mov-ing at 140 km/h will be significantly suppressed, as the cancellationcondition is approximately met. However, the response of the bridgetraversed by the train at 105 km/h will not be suppressed, since it isnot close or equal to any cancellation speed. This explains why thebridge response is greater for the train traveling at speed 105 km/h,rather than at 140 km/h.

The deflection response computed for the midpoint of the beamhas been plotted with respect to the speed parameter S in Fig. 7.11,in which the solution obtained alternatively by modeling each vehi-cle as two discrete sprung masses was also shown. One observationherein is that for the case with no track irregularities, the influenceof the vehicle’s pitching effect on the bridge response can basicallybe neglected. From Fig. 7.11, the critical speed parameter for theresonant response to occur is found as Sr = 0.417, which is consis-tent with that predicted analytically using Eq. (2.59) of Chapter 2:S = d/2nL with n = 1. Correspondingly, the resonant velocity ofthe train can be computed from Eq. (1.4) as vr = 420 km/h.



0.0 0.1 0.2 0.3 0.4 0.5

Speed parameter

0.0

0.1

0.2

0.3

0.4M

ax.

Ver

tica

l A

ccel

erat

ion

(m

/s2)

Rigid Car Bodies

Sprung Lumped Masses

S

Fig. 7.12. Maximum vertical acceleration of train with no track irregularities.

The vertical acceleration of the moving vehicles has been regardedas an indicator of the riding comfort or runnability of high-speedtrains. Due to the stringent requirements placed for the bridge re-sponse and riding comfort, the design of high-speed railroad bridgesis generally governed by the conditions of serviceability, rather thanby strength. For the present example, the maximum vertical acceler-ation computed for the train versus the speed parameter S has beenplotted in Fig. 7.12, along with the solution based on the sprung massmodel. As can be seen, the omission of the effect of vehicle pitch-ing or the interaction between the front and rear wheels, as impliedby the sprung mass model, may result in significant underestimateof the vehicle response even in the absence of track irregularities,which is not conservative from the design point of view. The effectof pitching of the car body appears to be especially important in therange from S = 0.2 to S = 0.4, which may be encountered by mosthigh-speed trains.

In this subsection and those to follow, the maximum vertical ac-celeration of the train denotes the maximum of the vertical acceler-ations computed for all parts of each vehicle constituting the train.



Therefore, it should be interpreted as the absolute vertical accelera-tion of the train.

7.5.3. Riding Comfort in the Presence of Track

Irregularities

For the present purposes, the irregularity function proposed byNielsen and Abrahamsson (1992) for the vertical profile of the rail-road is adopted:

r(x) = −r0

[1 − exp

(− x

x0

)3]

sin2πx

γ0, (7.35)

where x is the along-track distance (in m), x0 = 1.0 m, r0 (= 0.5 mm)= amplitude of irregularities, and γ0 (= 1.0 m) = wavelength of thecorrugation. The train model and bridge model are identical to thoseof the preceding example. The maximum vertical accelerations of thetrain computed for the cases with and with no track irregularitieshave been compared in Fig. 7.13. As can be seen, even for the casewith very small track irregularities, i.e., with r0 = 0.5 mm, signifi-cant amplification on the vehicle acceleration can be observed, whichis harmful to the riding comfort and, in some cases, to the runningsafety of the high-speed trains. Obviously, the importance of main-taining a smooth track surface in high-speed railroad engineeringcannot be overstressed.

7.5.4. Effect of Elasticity of the Suspension System

Let us consider again the problem of a train consisting of five identi-cal cars traveling over a simply-supported bridge. The data for thetrain and bridge are identical to those used in Sec. 7.5.2. To inves-tigate the effect of elasticity of the suspension system, three valuesof suspension stiffness, i.e., kv, 2kv, 3kv, are assumed. The resultscomputed for the impact factor of the midpoint displacement of thebridge, the maximum vertical acceleration and maximum rotationalacceleration of the train have been plotted against the speed parame-ter S in Figs. 7.14–7.16. From Fig. 7.14, one observes that the impact



0.0 0.1 0.2 0.3 0.4 0.5

Speed parameter

0.0

0.1

0.2

0.3

0.4

Max

. V

erti

cal

Acc

eler

atio

n (

m/s

2)

Smooth Surface

Irregular Surface

S

Fig. 7.13. Maximum vertical acceleration of train with track irregularities.

0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.0

0.5

1.0

1.5

2.0

Imp

act

Fac

tor

Rigid Car Bodies with kv

Rigid Car Bodies with 2kv


I

S

Fig. 7.14. Impact factor for the midpoint displacement of the bridge.



0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.00

0.25

0.50

0.75

1.00

Max

. V

erti

cal

Acc

eler

atio

n (

m/s

2)




Allowable Max. Acceleration = 0.05g (= 0.49 m/s2)

S

Fig. 7.15. Maximum vertical acceleration of the train.

0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.00

0.02

0.04

0.06

0.08

Max

. R

ota

tio

nal

Acc

eler

atio

n (

rad

./s2

)




S

Fig. 7.16. Maximum rotational acceleration of the train.



factor for the midpoint displacement of the bridge increases gener-ally following the increase in the speed parameter S of the train, andreaches the peak when S = 0.417 for all the three values of suspen-sion stiffness considered. In general, the influence of the suspensionstiffness on the bridge response is quite little, although larger valuesof suspension stiffness can induce marginally smaller bridge response.

Concerning the train vibrations, it can be seen from Figs. 7.15and 7.16 that the suspension stiffness can drastically affect the trainresponse. The larger the suspension stiffness, the greater is the trainresponse, which is consistent with our common understanding that itis uncomfortable to ride on a vehicle with flat tires. The speed param-eter for the peak response to occur is around S = 0.417 and becomessmaller for larger suspension stiffness. Such a trend is valid for boththe vertical acceleration and rotation acceleration of the train. Infact, the allowable maximum acceleration of 0.05 g (= 0.49 m/s2)previously used by the Taiwan High-Speed Railway was exceeded inthe case with 3kv .

7.5.5. Effect of Damping of the Suspension System

Same data as those used in Sec. 7.5.2 for the train and bridge areadopted herein. Let us consider three values of suspension damping,i.e., cv , 2cv , 3cv . The results computed for the impact response of themidpoint displacement of the bridge, the maximum vertical acceler-ation and maximum rotational acceleration of the train were shownin Figs. 7.17–7.19. As can be seen from Fig. 7.17, the general trendfor the bridge midpoint displacement impact is similar to the oneshown in Fig. 7.14, except that larger suspension damping will resultin slightly smaller impact response for the bridge. From Figs. 7.18and 7.19, one observes that the suspension damping can affect sig-nificantly the train response. The larger the damping coefficient ofthe suspension system, the greater is the train response, whether itis vertical or rotational acceleration. The magnification effect of sus-pension damping on the train response seems to be coupled by theresonance effect, as indicated by the drastic increase in train responsefor the speed parameter range with S greater than 0.32.



0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.0

0.5

1.0

1.5

2.0

Imp

act

Fac

tor

Rigid Car Bodies with cv

Rigid Car Bodies with 2cv


I

S

Fig. 7.17. Impact factor of the midpoint displacement of the bridge.

0.0 0.1 0.2 0.3 0.4 0

Speed Parameter

.5

0.00

0.25

0.50

0.75

1.00

Max

. V

erti

cal

Acc

eler

atio

n (

m/s

2)





S




0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.00

0.02

0.04

0.06

0.08

Max

. R

ota

tional

Acc

eler

atio

n (

rad./

s2)




S


0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.0

0.5

1.0

1.5

2.0

Imp

act

Fac

tor

with r0

with 2r0

with 3r0

I

S

Fig. 7.20. Impact factor for the midpoint displacement of the bridge.



7.5.6. Effect of Track Irregularity

Again, the same data as those used in Sec. 7.5.2 for the train andbridge are adopted herein. To investigate the effect of track irreg-ularity, we shall consider three values of irregularity amplitude r0,2r0, 3r0. In Figs. 7.20–7.22, the impact factor of the midpoint dis-placement of the bridge, the maximum vertical acceleration and themaximum rotational acceleration of the train have been plotted withrespect to the speed parameter S. As can be seen from Fig. 7.20, theinfluence of track irregularity on the bridge response is so small thatit can virtually be neglected. On the other hand, from Figs. 7.21 and7.22, one observes that the increase in the magnitude of track irreg-ularity will result in significant magnification of the train response,no matter it is vertical or rotational acceleration.


In this chapter, a vehicle is modeled as a rigid beam supported bytwo suspension units, and a bridge by a number of beam elements.The equations of motion written for the rigid beam (actually the carbody) are first reduced to a set of equivalent stiffness equations usingNewmark’s finite difference equations, by which the car body DOFsare condensed to those of the bridge elements in contact, followingbasically the procedures in Chapter 6. The VBI elements derived arecharacterized by the fact that the pitching or linking action of thecar body is duly taken into account, while the property of symmetryis retained in the system matrices. The applicability of the derivedelement has been demonstrated in the numerical studies concerningboth the bridge and the vehicle responses.

Some conclusions can be made from the numerical studies con-ducted in this chapter. (1) For the case with no track irregularities,the pitching effect of vehicles on the bridge response can be generallyneglected. (2) Wherever the vehicle response is of major concern, itis necessary to consider the effect of pitching of the car body. Theuse of sprung mass model for the vehicles is inadequate and not con-servative from the point of view of design. (3) The resonant speed



0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.00

0.25

0.50

0.75

1.00

Max

. V

erti

cal

Acc

eler

atio

n (

m/s

2)

with r0

with 2r0

with 3r0


S


0.0 0.1 0.2 0.3 0.4 0.5

Speed Parameter

0.00

0.02

0.04

0.06

0.08

Max

. R

ota

tional

Acc

eler

atio

n (

rad./

s2)

with r0

with 2r0

with 3r0

S




parameter predicted using the present finite element model agreesvery well with that predicted analytically. (4) The vehicle responsewill be drastically amplified in the presence of track irregularities andthrough consideration of the pitching effect (i.e., using the presentVBI elements), which is harmful to both the riding comfort and run-ning safety of high-speed trains. (5) The effects of both elasticityand damping of the suspension system on the bridge response areso small that they can be virtually neglected. However, increasingthe stiffness or damping coefficient of the suspension system tendsto increase largely the maximum train response. (6) The trend forthe track irregularity is similar to that for the suspension stiffnessor damping concerning the bridge and train responses. Naturally,greater magnitude of track irregularity will induce larger trainresponse.



Chapter 8

Modeling of Vehicle–BridgeInteractions by the Concept of

Contact Forces

Unlike Chapter 6 that deals with the vehicle–bridge interactions(VBIs) by condensing the vehicle degrees of freedom (DOFs) to thebridge DOFs, in this chapter we present a procedure that utilizes thecontact forces as the interface for treating the same problem. Cen-tral to the present formulation is the adoption of Newmark’s finitedifference scheme for discretizing the vehicle equations of motion, bywhich the contact forces are solved and expressed in terms of thewheel displacements. Through the use of no-jump condition for ve-hicles, the contact forces can then be related to the displacementsof the contact points of the bridge. As such, a VBI element thatconsiders all the interaction effects can be derived from the bridgeequations, with which the vehicle response, contact forces and bridgeresponse can be computed with no iterations required. The presentprocedure is versatile in that there is virtually no limit on the levelof complexity of the vehicles simulated, which may range from themoving load, moving mass, sprung mass, to suspended rigid bar, andso on. The capability of the present procedure is demonstrated inthe study of a number of VBI problems, including those caused byvehicles in braking.

8.1. Introduction

The vibration of bridges caused by the moving vehicles or trains hasbeen a subject of continuous research since the nineteenth century,as was revealed by the review presented in Chapter 1. In the past

233



three decades, due to the construction of high-speed railways andthe upgrading of existing railways worldwide, the problem of train–bridge interactions has received more attention from engineers andresearchers than ever (Richardson and Wormley, 1974; Matsuura,1976; Cherchas, 1979; Chu et al., 1979; Aida et al., 1990; Wakuiet al., 1995; Yang et al., 1997b). Previously, vehicles have oftenbeen approximated as moving loads, which in many cases offer afeasible means for obtaining solutions in closed form. To considerthe inertial effect of the moving vehicles, the moving mass model hasbeen adopted instead (Stanisic and Hardin, 1969; Ting et al., 1974;Sadiku and Leipholz, 1987; Akin and Mofid, 1989). However, forthe case where the riding comfort or vehicle response is of concern,it is necessary to consider the effect of suspension systems of thevehicles as well. The simplest model that can be conceived in thisregard is a lumped mass supported by a spring-dashpot unit, referredto as the sprung mass model (Tan and Shore, 1968b; Genin et al.,1975; Blejwas et al., 1979). Although more sophisticated models canstill be devised for simulating the vehicle structures, the efficiency ofsolution of the vehicle–bridge interaction (VBI) system becomes anissue of great concern, especially when there exists a large number ofvehicles, e.g., for the case of a train consisting of a number of vehiclesin connection. By taking into account the VBI, it is realized that allthe contact forces vary not only in magnitudes, but also in actingpositions.

In analyzing the VBI systems, two sets of second-order differen-tial equations can be written each for the vehicles and for the bridge.It is the interaction forces or contact forces existing at the contactpoints that make the two subsystems coupled. As the contact pointsmove from time to time, the system matrices are time-dependentand must be updated and factorized at each time step in an incre-mental time-history analysis. To solve these two sets of equations,procedures of iterative nature are often adopted (Hwang and Nowak,1991; Yang and Fonder, 1996; Yang and Yau, 1998). For instance,by first assuming the displacements for the contact points, one cansolve the vehicle equations to obtain the interaction (contact) forcesand then proceed to solve the bridge equations for improved values


Modeling of Vehicle–Bridge Interactions 235

of displacements for the contact points. This completes the first cy-cle of iteration. One drawback with methods of iterative nature isthat the convergence rate is likely to be low when dealing with themore realistic case of a bridge sustaining a large number of vehiclesin motion. In the literature, Lagrange’s equation with multipliersand constraint equations has also been used (Blejwas et al., 1979).However, the use of Lagrange multipliers increases the number of un-knowns and thus the effort of computation, especially for problemsinvolving a large number of moving vehicles.

Still, another category of methods exists for solving the VBIproblems, e.g., those based on the condensation method. Garg andDukkipati (1984) used the Guyan reduction technique to condensethe vehicle degrees of freedom (DOFs) to the associated bridge DOFs.In the paper by Yang and Lin (1995), the dynamic condensationmethod was used to eliminate all the vehicle DOFs on the elementlevel. These methods have been demonstrated to be efficient for com-puting the bridge response. However, because of the approximationsmade in relating the vehicle (slave) DOFs to the bridge (master)DOFs, they are not adequate for computing the vehicle response,which serves as an indicator of the riding comfort generally requiredin the design of high-speed railway bridges. By using the Newmarkfinite difference scheme to discretize the vehicle equations, ratheraccurate master–slave relations have been established and used ineliminating the vehicle DOFs from the bridge equations. Such rela-tions have been employed in deriving the VBI elements in Chapters 6and 7, which are good for computing both the vehicle and bridge re-sponses.

The objective of this chapter is to develop a more general ap-proach for solving both the vehicle and bridge responses that are ofparticular interest in high-speed railways, but not to study solely thebridge impact response, as was the case for highway bridges (Nassifand Nowak, 1995; Kim and Nowak, 1997). First of all, the second-order differential equations for the moving vehicles will be renderedinto a set of equivalent stiffness equations using Newmark’s finitedifference scheme, as presented in Chapter 6. However, insteadof establishing the master–slave relations, we shall concentrate on



computation of the contact forces from the equivalent vehicle equa-tions, which are first expressed in terms of the wheel displacements,and then, through enforcement of the constraint equations, related tothe bridge displacements at contact points. With the contact forcesmade available, the VBI element can be derived from the bridge equa-tions by treating the contact forces as consistent nodal loads. Theadvantage of the present procedure is its versatility in dealing withvehicle structures with practically no limit on the level of complex-ity, which may range from the moving loads, moving masses, sprungmasses, to rigid car bodies supported by spring-dashpot units, andso on. The capability and reliability of the present procedure will bedemonstrated in the study of a number of examples close to those en-countered in practice. The materials presented in this chapter followbasically those of Yang and Wu (2001).

8.2. Vehicle Equations and Contact Forces

In this study, matrices, column and row vectors will be representedby quantities enclosed by [ ], and 〈〉, respectively. Consider asimply-supported beam that is under the action of a moving vehiclein Fig. 8.1. The vehicle is assumed to be composed of two parts.

Fig. 8.1. Schematic of vehicle–bridge interaction.



The upper or noncontact part consists of the car body, suspensionsystems, and bogies, which has a total of k DOFs, as indicated bythe vector du. The wheel or contact part consists of n wheelsets.Assuming that each wheelset is represented by one vertical DOF, thewheel part can be denoted as dw = 〈vw1vw2 . . . vwi . . . vwn〉T , wherevwi denotes the displacement of the ith wheel. Correspondingly, thereexist n contact points on the bridge, of which the displacement maybe denoted as dc = 〈vc1vc2 . . . vci . . . vcn〉T , where vci denotes thedisplacement at the ith contact point.

Let [mv], [cv], and [kv ] respectively denote the mass, damping andstiffness matrices of the vehicle, and dv the displacement vector ofthe vehicle, i.e., dv = 〈dudw〉T . The following is the equationof motion for the vehicle:

[mv]dv + [cv ]dv + [kv ]dv = fv , (8.1)

where fv is the force vector, which can be decomposed into twoparts,

fv = fe + [l]fc . (8.2)

Here, fe denotes the external force components excluding thecontact forces, fc the contact forces acting through the wheels,fc = 〈V1V2 . . . Vi . . . Vn〉T , where Vi is the force acting at the ithcontact point of the bridge, and [l] is a transformation matrix. Thewheel displacements dw can be related to the contact displacementsdc of the bridge by the constraint conditions,

dw = [Γ]dc + r , (8.3)

where [Γ] should be interpreted as a unit matrix for the case where nojumps occur between the vehicle’s wheels and the bridge, as is con-sidered herein, and r is a vector representing the rail irregularity(or pavement roughness for highway bridges) at the contact points.The irregularity or roughness vector r depends on the condition ofcontact between the wheels and bridge and the level of complexityof the problem considered, that is, whether the problem is 2D or 3Din nature, as will be studied in depth in Chapters 9 and 10 to follow.



In this section and the section to follow, only contact forces thatact along the vertical (gravity) direction will be considered. The pro-cedure presented in these two sections will be generalized in Sec. 8.5to include the horizontal contact forces, which may be generated asthe frictional forces between the wheels and rails.

The VBI system will be analyzed in an incremental manner intime domain. Assuming that all the information of the system attime t is known and ∆t is a small time increment, we are interestedin the behavior of the system at time t + ∆t. The vehicle equationsgiven in Eqs. (8.1) can be rewritten for time t + ∆t with partitionsin matrices for the upper and wheel parts as

[[muu] [muw]

[mwu] [mww]

] dudw

t+∆t

+[

[cuu] [cuw]

[cwu] [cww]

] dudw

t+∆t

+[

[kuu] [kuw]

[kwu] [kww]

] dudw

t+∆t

= fuefwe

t+∆t

+[

[lu]

[lw]

]fct+∆t , (8.4)

where fue and fwe respectively denote the external forces actingon the upper and wheel parts of the vehicle. The first row in Eq. (8.4)relates to the behavior of the upper or non-contact part of the vehicle,and the second row of the wheels or contact part. Since only thewheels will be acted upon by the contact forces, the submatrix [lu]should be set to [0]. Expanding the first row of Eq. (8.4) yields

[muu]dut+∆t + [cuu]dut+∆t + [kuu]dut+∆t

= fuet+∆t − quct+∆t , (8.5)

where

quct+∆t = [muw]dwt+∆t+[cuw]dwt+∆t+[kuw]dwt+∆t . (8.6)



Let ∆du denote the increment in the upper-part vehicle displace-ment du occurring during the time step from t to t+∆t. By New-mark’s finite difference scheme (see Appendix B), the vector duand its derivatives at the instant t + ∆t can be related to those attime t as

dut+∆t = b0∆du − b1dut − b2dut ,

dut+∆t = dut + b3dut + b4dut+∆t ,

dut+∆t = dut + ∆du ,

(8.7)

where the variables with subscript t denote the quantities occurringat time t, which are assumed to be known. Using Newmark’s pa-rameters β and γ, the coefficients and those to be used later can begiven as

b0 =1

β∆t2, b1 =

1β∆t

, b2 =12β

− 1 ,

b3 = (1 − γ)∆t , b4 = γ∆t , b5 =γ

β∆t,

b6 =γ

β− 1 , b7 =

∆t

2

(γ

β− 2)

,

(8.8)

which are equivalent to the coefficients a0 ∼ a7 presented inAppendix B. Substituting Eqs. (8.7) into the differential equationsfor the upper part of the vehicle in Eq. (8.5), one obtains after somemanipulations the following,

[Ψuu]∆du = fuet+∆t − quct+∆t + qut , (8.9)

where

[Ψuu] = b0[muu] + b5[cuu] + [kuu] , (8.10a)

qut = [muu](b1dut + b2dut)

+ [cuu](b6dut + b7dut) − [kuu]dut . (8.10b)



From Eq. (8.9), the displacement increments ∆du for the upperpart of the vehicle can be solved as

∆du = [Ψuu]−1(fuet+∆t − quct+∆t + qut) . (8.11)

With the use of Eq. (8.11), the displacement vector dut+∆t and itsderivatives for the upper part of the vehicle can be obtained as

dut+∆t = b0[Ψuu]−1(fuet+∆t − quct+∆t + qut)

− b1dut − b2dut ,

dut+∆t = b5[Ψuu]−1(fuet+∆t − quct+∆t + qut)

− b6dut − b7dut ,

dut+∆t = [Ψuu]−1(fuet+∆t − quct+∆t + qut) + dut ,

(8.12)

of which the order of accuracy is the same as that implied by theNewmark finite difference equations presented in Eq. (8.7).

8.3. Solution of Contact Forces from VehicleEquations

One key step in solution of the VBI systems is to solve for the contactforces existing between the two subsystems, i.e., the moving vehiclesand the bridge. By substituting Eq. (8.12) into the second row ofthe vehicle equations in Eq. (8.4), one obtains the contact forcesfct+∆t as

fct+∆t = [mc]dwt+∆t + [cc]dwt+∆t + [kc]dwt+∆t

+ pct+∆t + qct , (8.13)

where the contact matrices [mc], [cc] and [kc] are

[mc] = [lw]−1([mww] − [Ψwu][Ψuu]−1[muw]) ,

[cc] = [lw]−1([cww] − [Ψwu][Ψuu]−1[cuw]) ,

[kc] = [lw]−1([kww] − [Ψwu][Ψuu]−1[kuw]) ,

(8.14)



the load vectors pct+∆t and qct are

pct+∆t = [lw]−1([Ψwu][Ψuu]−1fuet+∆t − fwet+∆t) ,

qct = [lw]−1([Ψwu][Ψuu]−1qut − qwt) ,(8.15)

and

[Ψwu] = b0[mwu] + b5[cwu] + [kwu] , (8.16a)

qwt = [mwu](b1dut + b2dut)

+ [cwu](b6dut + b7dut) − [kwu]dut . (8.16b)

From Eq. (8.13), it can be seen that the contact forces fct+∆t

depend not only on the wheel response and the forces acting on thevehicle at time t+∆t, but also on those at time t. By the constraintcondition of no jumps for the vehicles in Eq. (8.3), i.e., dw ≡ dc,the contact forces fct+∆t can be expressed in terms of the contactdisplacements dc of the bridge as follows:

fct+∆t = [mc]dct+∆t + [cc]dct+∆t + [kc]dct+∆t

+ pct+∆t + qct , (8.17)

from which each of the contact forces Vi,t+∆t, with i = 1, . . . , n, canbe given as

Vi,t+∆t = pci,t+∆t + qci,t

+n∑

j=1

(mcij dcj,t+∆t + ccij dcj,t+∆t + kcijdcj,t+∆t) , (8.18)

where mcij, ccij and kcij represent the entry located at the ith rowand jth column of the matrices [mc], [cc], and [kc], respectively, andpci,t+∆t, qci,t are the ith entry of the vectors pct+∆t and qct,respectively.



8.4. VBI Element Considering Vertical ContactForces Only

Consider that at time t+∆t, there are n wheels acting simultaneouslyon the e1, e2, . . . , enth elements of the bridge, which will be referredto as the VBI elements as they are directly under the action of thewheel loads. For the time being, we shall consider only the verticalcomponents of the contact forces and assume that the eith elementis acted upon by the ith contact force Vi,t+∆t. Consequently, theequation of motion for the eith element of the bridge at time t + ∆t

can be written as follows:

[mbi]dbit+∆t + [cbi]dbit+∆t + [kbi]dbit+∆t

= fbit+∆t − fbcit+∆t , (8.19)

where [mbi], [cbi] and [kbi] denote the mass, damping and stiffnessmatrices of the eith element of the bridge, dbi the nodal displace-ment vector, fbi the vector of external forces directly acting on thenodal points, and fbci the vector of consistent nodal forces resultingfrom action of the ith vertical contact force Vi,t+∆t,

fbcit+∆t = NvciVi,t+∆t , (8.20)

where Nvci denotes the interpolation vector of the eith bridge

element in which all entries are set to zero except for thoseassociated with the vertical displacements, which are represented bycubic interpolation (Hermitian) functions. The subscript c indicatesthat the interpolation vector Nv

ci is evaluated at the contact point,i.e.,

Nvci = Nv(xi) , (8.21)

where xi is the local coordinate of the ith contact point on the eithelement. Note that in Eq. (5.19) the centrifugal and Coriolis forceeffects induced on the beam by the moving vehicle loadings have beenneglected, as they are usually small in realistic bridge structures. By



using Eqs. (8.18) and (8.20), one can rewrite the bridge equations,Eq. (8.19), as follows:

[mbi]dbit+∆t + [cbi]dbit+∆t + [kbi]dbit+∆t

= fbit+∆t −n∑

j=1

([m∗cij]dbj + [c∗cij]dbj + [k∗

cij]dbj)

−p∗cit+∆t − q∗cit , (8.22)

where the matrices with an asterisk represent the interaction effectof the VBI elements due to the interlocking action of the movingvehicle via the front and rear wheelsets,

[m∗cij] = Nv

cimcij〈Nvcj〉 ,

[c∗cij ] = Nvciccij〈Nv

cj〉 ,

[k∗cij ] = Nv

cikcij〈Nvcj〉 ,

(8.23)

and the equivalent nodal loads are

p∗cit+∆t = Nvcipci,t+∆t ,

q∗cit = Nvciqci,t .

(8.24)

Evidently, the effect of interaction with the moving vehicles has beenconsidered in Eq. (8.22) for the eith element of the bridge through theasterisked matrices and vectors. The equations of motion as givenin Eq. (8.22) will be referred to as the condensed equation of motionfor the VBI element, as all the vehicle DOFs in contact with thebridge element have been eliminated. It should be noted that all theasterisked matrices and vectors in Eqs. (8.23) and (8.24) are time-dependent, since they are all functions of the contact positions, asimplied by the shape function Nv

ci. Because of this, all these matri-ces and vectors should be updated at each time step in an incrementalanalysis. Besides, the accuracy of the VBI element derived, as givenin Eq. (8.22), is of the same order as that of the Newmark equationsgiven in Eq. (8.7). Because of this, the VBI element derived herein



is quite accurate and can be used to solve the time-history responseof most VBI systems with no iterations required.

8.5. VBI Element Considering General ContactForces

The equations of motion as derived in Eq. (8.22) for the VBI elementconsider only the action of the vertical contact forces Vi,t+∆t. For thecase where acceleration or deceleration are involved in the movementof the vehicles over the bridge, it is necessary to consider the actionof the horizontal contact forces Hi,t+∆t along the bridge axis, asshown in Fig. 8.1. To this end, the contact force vector fbcit+∆t inEq. (8.20) should be generalized as

fbcit+∆t = NhciHi,t+∆t + Nv

ciVi,t+∆t , (8.25)

where Nhci denotes the interpolation vector of the eith (VBI) ele-

ment in which all entries are set to zero except those associated withthe axial displacements, which are represented by linear functionsand evaluated at the contact point xi. The horizontal componentsHi,t+∆t of the contact forces may be generated by the rolling, ac-celerating or braking action of the wheels. All these actions can bereferred to as the variational forms of the frictional force, which varyaccording to the cohesion existing between the wheels and rails. Forthe present purposes, one may relate the ith horizontal contact forceHi to the corresponding vertical contact force Vi simply as

Hi = µiVi , (8.26)

where µi is the frictional coefficient for the ith wheel, which equalseither the coefficient of braking or the coefficient of acceleration; thefrictional coefficient in rolling is so small that it can be neglected inpractice.

Considering the more general expression for the contact forces inEq. (8.25), along with Eq. (8.26), one can proceed to derive a setof condensed equations of motion for the eith element of the bridgeidentical in the form of Eq. (8.22), but with the asterisked matrices



modified as

[m∗cij] = Nh

ciµimcij〈Nvcj〉 + Nv

cimcij〈Nvcj〉 ,

[c∗cij] = Nhciµiccij〈Nv

cj〉 + Nvciccij〈Nv

cj〉 ,

[k∗cij] = Nh

ciµikcij〈Nvcj〉 + Nv

cikcij〈Nvcj〉 ,

(8.27)

and the asterisked vectors as

p∗cit+∆t = Nhciµipci,t+∆t + Nv

cipci,t+∆t ,

q∗cit = Nhciµiqci,t + Nv

ciqci,t .(8.28)

Again, it should be noted that all the matrices and vectors with anasterisk in Eqs. (8.27) and (8.28) are time-dependent. In this chap-ter, the general equations as given in Eqs. (8.27) and (8.28), whichconsider the effect of both the vertical and horizontal contact forces,will be used in studying the vehicle–bridge interactions whenever theacceleration or braking of the moving vehicles are involved. The twoequations given in Eqs. (8.23) and (8.24), which consider only thevertical contact forces, will be used for the case when the vehiclesare allowed to travel at a constant speed.

8.6. System Equations and Structural Damping

As was stated previously, a bridge element that is directly under theaction of wheel loads is referred to as a VBI element. Now, let usconsider a bridge that is traveled by a connected line of vehicles, asthe case with railway bridges. At some instant, say, at time t + ∆t,only parts of the bridge will be directly acted upon by the wheelloads, which should be modeled by the VBI elements. However, forthe remaining parts of the bridge that are not in direct touch withthe wheel loads, they should be modeled by the original (parent)bridge element. Following the conventional finite element procedure,all the VBI elements and bridge elements can be assembled to yield



the following equations:

[M ]Dt+∆t + [C]Dt+∆t + [K]Dt+∆t

= Fbt+∆t − P ∗c t+∆t − Q∗

ct , (8.29)

where Dt+∆t denotes the displacements of the entire VBI system,[M ], [C] and [K] the mass, damping and stiffness matrices, Fbt+∆t

the external loads, and P ∗c t+∆t and Q∗

ct the equivalent contactforces in global coordinates. A convenient way to construct the sys-tem matrices [M ], [C] and [K] is first to assemble the matrices [Mb],[Cb] and [Kb] for the bridge that is free of any wheel actions, andthen to add to them the interaction effects of vehicles contributed bythe VBI elements, as represented by the terms with an asterisk inEqs. (8.23) or (8.27), i.e.,

[M ] = [Mb] + [M∗c ] =

∑[mbi] +

∑∑[m∗

cij] ,

[C] = [Cb] + [C∗c ] =

∑[cbi] +

∑∑[c∗cij ] ,

[K] = [Kb] + [K∗c ] =

∑[kbi] +

∑∑[k∗

cij ] .

(8.30)

Similarly, the equivalent contact forces P ∗c t+∆t and Q∗

ct are

P ∗c t+∆t =

∑p∗cit+∆t ,

Q∗ct =

∑q∗cit .(8.31)

In Eqs. (8.30) and (8.31), all the terms or components with an as-terisk should be interpreted as those generated by the interactioneffect of the moving vehicles. Therefore, they should be looped overthe VBI elements only. As the wheel loads move from time to time,it is necessary to check at each time step whether a bridge elementbecomes a VBI element and vice versa, and to update (i.e., to add ordelete) the entries of the system matrices and vectors, concerning thecontribution of the terms or components identified by an asterisk inEqs. (8.30) and (8.31), for the DOFs that are directly affected by thevehicle actions, according to the variation of the contact positions.

One feature with the present formulation is that the total numberof DOFs of the VBI system remains identical to that of the originalbridge, regardless of the consideration of vehicles interaction effects.



The symmetry property of the system matrices is also preserved.Furthermore, the present procedure can be adopted to deal withthe vehicle models of various complexities, which may vary from thesimplest case of moving load to models with dozens of DOFs. Aswill be demonstrated in the numerical examples, with the presentapproach there is virtually no limit on the number of DOFs used todescribe the vehicle structure.

In this chapter, Rayleigh damping is assumed for the bridge,namely, the damping matrix [Cb] of the bridge that is free of anyvehicle actions can be computed as a linear combination of the massmatrix [Mb] and stiffness matrix [Kb],

[Cb] = α0[Mb] + α1[Kb] , (8.32)

where, given the damping ratio ξ, the two coefficients α0 and α1 are

α0 =2ξω1ω2

ω1 + ω2,

α1 =2ξ

ω1 + ω2.

(8.33)

Here, ω1 and ω2 are the first and second frequencies of vibration ofthe bridge.

Also, the system equations as given in Eq. (8.29) will be solvedin an incremental sense using Newmark’s β method. The parametricvalues of β = 0.25 and γ = 0.5 are used throughout, implying thatthe marching scheme is unconditionally stable. By this method, thesystem accelerations D, velocities D and displacements D attime t + ∆t can first be discretized and related to those at time t

in exactly the same manner as in Eq. (8.7), with which the systemequation (8.29) can be reduced to an equivalent stiffness equationand be solved (see Appendix B for more details).

8.7. Procedure of Time-History Analysis for VBISystems

The following is a summary of the procedure for incremental analysisof the VBI system based on Newmark’s β method:



(1) Input the fundamental data of the bridge and vehicles.(2) Construct the mass matrix [Mb] and stiffness matrix [Kb] for

the bridge that is free of any vehicles. Perform an eigenvalueanalysis to obtain the first two frequencies ω1 and ω2. Giventhe damping ratio ξ, determine the damping matrix [Cb] of thebridge using Eqs. (8.32) and (8.33).

(3) Specify the initial conditions for the bridge and the vehicles, andthe position xi0 of each wheel at time t = 0, i.e., D0 = D0 =D0 = 0 for the bridge and Dv0 = Dv0 = Dv0 = 0for all the vehicles. Specify the following data: (a) the initialvelocity v0 and acceleration a of the vehicles; (b) the positionxbs0 for the first wheel to start acceleration or braking; (c) theposition xbsf for the first wheel to stop acceleration or braking,xbsf > xbs0; (d) the frictional coefficient for each wheel µi = µ

(µ = µb for braking and µ = µs for acceleration); (e) the endingtime tend for analysis or ending position xend of the first wheel;and (g) the time increment ∆t.

(4) Construct the matrices and vectors for the vehicles equations,i.e., Eq. (8.4). Compute [Ψuu] and [Ψwu], using Eqs. (8.10a)and (8.16a), and the contact matrices [mc], [cc] and [kc] usingEq. (8.14).

(5) For a new time with t′ = t + ∆t, check if t′ > tend or x1 > xend

for the first wheel. If yes, stop analysis; otherwise, proceed toStep 6.

(6) Calculate the global position xi of each wheel, identify the ele-ment ei in contact, and compute the values of the interpolationfunctions Nv

ci and Nhci at the contact point.

(7) Compute qut and qwt using Eqs. (8.10b) and (8.16b), andpct+∆t and qct using Eq. (8.15).

(8) If x1 < xbs0 or x1 > xbsf , which means that the vehiclesare moving with constant speed, use Eqs. (8.23) and (8.24)to compute the vehicle-related matrices [m∗

cij], [c∗cij], [k∗cij ] and

vectors p∗cit+∆t, q∗cit for each element ei in contact. Ifxbs0 ≤ x1 ≤ xbsf , which means that the vehicles are in acceler-ation or in braking, then use Eqs. (8.27) and (8.28) to computethe matrices [m∗

cij], [c∗cij], [k∗cij] and vectors p∗cit+∆t, q∗cit for



each element ei in contact. In each case, update the global ma-trices [M∗

c ], [C∗c ], [K∗

c ] and vectors P ∗c t+∆t, Q∗

ct accordingly.Note that for vehicles with acceleration a < 0, use µ = µb < 0,and for acceleration a > 0, use µ = µs > 0.

(9) Summing up [Mb], [Cb] and [Kb] respectively with [M∗c ], [C∗

c ],[K∗

c ], one can obtain the matrices [M ], [C] and [K] for the entireVBI system following Eq. (8.30).

(10) Solve Eq. (8.29) by Newmark’s β method for the displace-ment increments ∆D. Using equations identical in form toEq. (8.7), compute the displacements D, velocities D andaccelerations D of the bridge.

(11) Compute the response for all the contact points of the bridge,i.e., dc, dc and dc. Compute the contact force Vi,t+∆t foreach wheel using Eq. (8.18). Based on the no-jump conditionfor vehicles, dw ≡ dc, one can compute quct+∆t fromEq. (8.6) and ∆du from Eq. (8.9). Consequently, the responseof the upper or noncontact part of the vehicle, i.e., du, duand du, can be computed from Eq. (8.7).

(12) Go to Step 5 to perform the next time increment of analysis.

8.8. Numerical Examples and Verification

In order to demonstrate the capability and reliability of the presentprocedure in dealing with various vehicle models, six typical exampleswill be studied in this section. In each case, the beam is modeled as10 elements.

8.8.1. Cantilever Beam Subjected to a Moving Load

As shown in Fig. 8.2(a), a cantilever beam of length L = 300 in.is subjected to a moving load p. The following data are adopted:flexural rigidity EI = 3.3 × 109 lb-in.2, mass per unit length m =0.00667 lbm/in., damping ratio ξ = 0, moving load p = 5793 lb andvelocity v = 2000 in./s. For this case, the moving load p shouldbe interpreted as a vehicle with only one wheel (n = 1) of zero masswhich has a single DOF denoted as vw and is subjected to an external



L

(a)

L

M

(b)

vM

wM

vkvc

L

wv

uv

(c)

Fig. 8.2. VBI models: (a) moving load, (b) moving mass, (c) sprung mass, and(d) suspended rigid beam.



vM

wM

vkvc

L

wM

vk vc

vI

cL

1wv2wv

uv

uϕ

sd

(d)

Fig. 8.2. (Continued).

load of magnitude p. The partitioned matrices corresponding to theupper part of the vehicle have to be eliminated from the relevantequations, and the remaining matrices corresponding to the wheelcan be given as follows:

[mww] ≡ 0 ,

[cww] ≡ 0 ,

[kww] ≡ 0 ,

fwet+∆t ≡ −p ,

[lw] ≡ 1 ,

[Γ] ≡ 1 .

(8.34)

By using Eq. (8.34), together with Eqs. (8.14) and (8.15), we canobtain from Eq. (8.18) the contact force V1,t+∆t = p, as it shouldbe for the moving load case. The free end displacements of thecantilever caused by load p moving from the left to the right and inthe reversed direction were plotted in Fig. 8.3. As can be seen, theresults obtained by the present procedure agree well with those ofAkin and Mofid (1989).



-2

0

2

4

6

8

10

12

14

16

18

20

0 0.025 0.05 0.075 0.1 0.125 0.15

Time (s)

Dis

pla

cem

ent at F

ree E

nd (in)

Fixed-Free Beam (Present)

Fixed-Free Beam (Akin & Mofid)

Free-Fixed Beam (Present)

Free-Fixed Beam (Akin & Mofid)

Fig. 8.3. Free-end responses of cantilever (moving load model).

8.8.2. Cantilever Beam Subjected to a Moving Mass

Consider a cantilever beam subjected to a lumped mass of M =15 lbm moving at v = 2000 in./s, as shown in Fig. 8.2(b). Theproperties of the beam are identical to those used in the precedingexample. The moving mass should now be treated as a vehicle withnothing but a single wheel of mass M , which has a single DOF de-noted as vw, and one contact point. The corresponding partitionedmatrices for the wheel are

[mww] ≡ M ,

[cww] ≡ 0 ,

[kww] ≡ 0 ,

(8.35)

fwet+∆t ≡ −Mg ,

[lw] ≡ 1 ,

[Γ] ≡ 1 .

(8.36)



-2

0

2

4

6

8

10

12

0 0.025 0.05 0.075 0.1 0.125 0.15

Time (s)

Dis

pla

cem

ent at F

ree E

nd (in)

Fixed-Free Beam (Present)

Fixed-Free Beam (Akin & Mofid)

Free-Fixed Beam (Present)

Free-Fixed Beam (Akin & Mofid)

Fig. 8.4. Free-end responses of cantilever (moving mass model).

Similarly, by using Eqs. (8.35), (8.36) and relevant equations, onecan obtain the contact force as V1,t+∆t = Mvw,t+∆t + Mg, of whichthe first term represents the inertial effect and the second term themoving load effect. The dynamic responses of the free end of thecantilever caused by the mass M moving from the fixed to the freeend and in the reverse direction have been plotted in Fig. 8.4. Again,the responses obtained by the present procedure agree well with thoseof Akin and Mofid (1989). From the contact force response plottedin Fig. 8.5, one observes that the contact force fluctuates and doesnot remain equal to the static weight of the mass, implying thatthe effect of inertia is significant and cannot be neglected for thepresent case. Moreover, for the case with the mass moving from thefree to the fixed end, the contact force encounters a serious drop atthe beginning, then escalates and eventually drops again to negativevalues near the fixed end. The occurrence of negative contact forcesshould be recognized as part of the effect of no-jump assumption,which enforces the mass to keep in contact with the beam at alltimes.



-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.025 0.05 0.075 0.1 0.125 0.15

Time (s)

Conta

ct F

orc

e / S

tatic W

eig

ht

Fixed-Free Beam

Free-Fixed Beam

Fig. 8.5. Contact force response (moving mass model).

8.8.3. Simple Beam Subjected to a Moving Sprung

Mass

Figure 8.2(c) shows a simply-supported beam subjected to a movingsprung mass system, of which the vehicle (or sprung) mass Mv issupported by a dashpot-spring unit of spring constant kv and damp-ing cv, which is further supported by a wheel mass of Mw. The dataidentical to those of Sec. 6.7.1 are adopted herein: Young’s modulesE = 2.87 GPa kN/m2, Poisson’s ratio v = 0.2, moment of inertiaI = 2.90 m4, per-unit-length mass m = 2.303 t/m, length of beamL = 25 m; sprung mass Mv = 5.75 t, and suspension spring constantkv = 1595 kN/m. For illustration, the effect of the dashpot dampingand the wheel mass will be neglected, i.e., by letting cv = 0 kN-s/m,Mw = 0 t. By using vu and vw to denote the DOF associated withthe car body and wheel masses, respectively, the partitioned matricescorresponding to the two DOFs can be given as follows:

[muu] ≡ Mv ,

[muw] = [mwu]T ≡ 0 ,

[mww] ≡ Mw ,

(8.37)



[cuu] ≡ cv ,

[cuw] = [cwu]T ≡ −cv ,

[cww] ≡ cv ,

(8.38)

[kuu] ≡ kv ,

[kuw] = [kwu]T ≡ −kv ,

[kww] = kv ,

(8.39)

fuet+∆t ≡ 0 ,

fwet+∆t ≡ −Mvg − Mwg ,

[lw] ≡ 1 ,

[Γ] ≡ 1 .

(8.40)

For this particular problem, an approximate analytical solution canbe established by considering only the first mode of vibration, i.e., byrepresenting the deflection of the beam as vb(x, t) = qb(t) sin(πx/L).The general procedure of solution was outlined in Biggs (1964).

For the sprung mass with speed v = 27.78 m/s (= 100 km/h),the midpoint displacement of the beam obtained based either on thesprung mass or moving load assumption has been plotted in Fig. 8.6.As can be seen, the response obtained by the present VBI elementbased on the sprung mass assumption agrees well with the analyticalsolution. Moreover, from the vertical acceleration of the midpointof the beam shown in Fig. 8.7, one observes that inclusion of thehigher vibration modes can result in drastic, but local oscillationaround the acceleration obtained by the analytical approach thatconsiders only the first mode. On the other hand, the response of thevertical acceleration of the sprung mass has been plotted in Fig. 8.8,from which the effect of higher modes on the general trend can beappreciated. This figure also indicates that the acceleration of thevehicle or sprung mass is quite sensitive to the omission of higher-order terms, which appears to be reasonable as the vehicle can insome sense be regarded as a tuned mass to the bridge. It is interesting



-0.003

-0.0025

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Mid

poin

t D

ispla

cem

ent (

m)

Analytical (Sprung Mass, 1 Mode)

Present (Sprung Mass)

Present (Moving Load)

Fig. 8.6. Midpoint displacement of simple beam (sprung mass model).

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Mid

poin

t A

ccele

ration (m

/s^2

)



Present (Moving Load)

Fig. 8.7. Midpoint acceleration of simple beam (sprung mass model).



-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Spru

ng-M

ass A

ccele

ration (m

/s^2)



Fig. 8.8. Vertical acceleration of sprung mass (sprung mass model).

to note that the acceleration of the sprung mass has been taken asa measure of the passengers’ riding comfort in the design of high-speed railways. Clearly, as far as the riding comfort is concerned, itis necessary to take into account the higher mode vibrations of thebeam.

8.8.4. Simple Beam Subjected to a Moving Rigid Bar

Supported by Spring-Dashpot Units

Let us consider a simple beam with the following properties: L =30 m, E = 2.943 × 107 kN/m2, Poisson’s ratio v = 0.2, I = 8.65 m4,m = 36 t/m, ξ = 0, and cross-sectional area A = 5.16 m2. Thevehicle moving over the beam is modeled as a rigid bar supportedby two identical spring-dashpot units, as shown in Fig. 8.2(d). Withthis model, the interlocking effect of the car body on the front andrear wheels can be duly taken into account. The following data areadopted for the vehicle: rigid bar mass Mv = 540 t, mass moment ofinertia Iv = 13800 t-m2, spring stiffness kv = 41350 kN/m, dashpotcoefficient cv = 0 kN-s/m, wheel mass Mw = 0 t, wheel-to-wheeldistance Lc = 17.5 m, and vehicle speed v = 27.78 m/s. For this



example, the DOFs of the rigid bar are denoted as vu and ϕu andthe DOFs of the front and rear wheels as vw1 and vw2, respectively.Using the present notation, the dynamic matrices and relevant termsfor the vehicle model considered can be written as follows:

[muu] =[

Mv 00 Mv

],

[muw] = [mwu]T = [0] ,

[mww] =[

Mw 00 Mw

],

(8.41)

[cuu] =[

2cv 00 0.5d2cv

],

[cuw] = [cwu]T =[ −cv −cv

−0.5dcv 0.5dcv

],

[cww] =[

cv 00 cv

],

(8.42)

[kuu] =[

2kv 00 0.5d2kv

],

[kuw] = [kwu]T =[ −kv −kv

−0.5dkv 0.5dkv

],

[kww] =[

kv 00 kv

],

(8.43)

fuet+∆t = 0 ,

fwet+∆t =−0.5Mvg − Mwg

−0.5Mvg − Mwg

,

[lw] =[

1 00 1

],

[Γ] =[

1 00 1

].

(8.44)



The deflection and acceleration of the midpoint of the beam and thevertical acceleration of the center of gravity of the rigid bar havebeen plotted in Figs. 8.9–8.11, along with analytical solutions givenby Yau and Yang (1998) based on the first mode approximation. Ascan be seen, good agreement has been achieved between the presentsolutions and the analytical ones. Like the sprung mass example,the differences in these figures between the present solutions and theanalytical ones can be attributed mainly to the omission of highermodes in the latter.

For comparison, the rigid bar model has also been approximatedby two identical moving loads each of 2648.7 kN (= 0.5× 540× 9.81)or as two suspended masses each of 270 t (= 0.5×540) with identicalinterval Lc. The results obtained for the midpoint displacement ofthe simple beam by the moving load, sprung mass and suspendedrigid bar models have been shown in Fig. 8.12. Here, one observesthat the maximum response obtained by the moving load modelis similar to that of the suspended rigid bar model and both aregreater than that of the sprung mass model. In Fig. 8.13, the results

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Nondimensional Time (vt /L )

Mid

po

int

Dis

pla

ce

me

nt

(m

)

Analytical (1 Mode)

Present

Fig. 8.9. Midpoint displacement of simple beam (rigid beam model).



-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Nondimensional Time (vt/L )

Mid

po

int

Acce

lera

tio

n

(m/s

^2

)

Analytical (1 Mode)

Present

Fig. 8.10. Midpoint acceleration of simple beam (rigid beam model).

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Rig

id-B

ar

Ce

ntr

al V

ert

ica

l A

cce

lera

tio

n

(m/s

^2

)

Analytical (1 Mode)

Present

Fig. 8.11. Vertical acceleration of rigid beam.



-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Mid

po

int

Dis

pla

ce

me

nt

(m

)

Moving Load

Sprung Mass

Suspended Rigid Bar

Fig. 8.12. Comparison of midpoint responses of simple beam.

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Ve

hic

le A

cce

lera

tio

n

(m/s

^2

)

First Sprung Mass

Second Sprung Mass

Front (Suspended Rigid Bar)

Rear (Suspended Rigid Bar)

Fig. 8.13. Responses of vehicle based on rigid beam model.



obtained for the rigid bar at points corresponding to the front andrear wheels using the suspended rigid bar model and the discretesprung mass model have been plotted. As can be seen, the max-imum acceleration of the vehicle computed using the suspendedrigid bar model appears to be much larger than those using thesprung mass models, indicating that the interlocking effect of therigid bar tends to increase the vehicle response, and that using thesprung mass model to compute the vehicle response is generallynonconservative.

8.8.5. Bridge Subjected to a Vehicle in Deceleration

Let us now proceed to investigate the behavior of a bridge caused bya vehicle in deceleration. The bridge is modeled as a simple beamand the vehicle as a suspended rigid bar. The data adopted for thebeam and the suspended rigid bar are the same as those used in thepreceding example, except for inclusion of the damping ratio ξ =0.025. Here, we shall use V to denote the initial velocity of the rigidbar, a the acceleration, xbs0 the initial position of deceleration, xbsf

the final position of deceleration, and µ the frictional coefficient. Thefollowing three cases are considered: (1) V = 50 m/s, a = −10 m/s2,(2) V = 100 m/s, a = −10 m/s2, (3) V = 100 m/s, a = −20 m/s2,where in each case the following are adopted: xbs0 = 0 m, xbsf = 60 mand µ = −0.26.

The midpoint deflection and acceleration respectively of the beamfor the three cases have been plotted in Figs. 8.14 and 8.15, and thevertical acceleration of the vehicle was plotted in Fig. 8.16. As canbe seen, the responses for Cases (2) and (3) are close to each other,which are larger than those for Case (1). This has the indication thatthe response of the bridge is generally not influenced by the actionof braking, but by the initial velocity before braking is applied. Thereason is that the change in speed is quite limited during the passageof the vehicle over the beam, as the acting time is so short. The effectof braking will be evident, however, in the case of a beam traveled bya train of which the speed can change significantly during a longeracting time.



-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Mid

po

int

Dis

pla

ce

me

nt

(m

)

V=50 m/s , a= -10 m/s^2

V=100 m/s , a= -10 m/s^2

V=100 m/s , a= -20 m/s^2

Fig. 8.14. Midpoint displacement of simple beam for vehicles in braking.

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Mid

po

int

Acce

lera

tio

n

(m/s

^2

)

V=50 m/s , a= -10 m/s^2

V=100 m/s , a= -10 m/s^2

V=100 m/s , a= -20 m/s^2

Fig. 8.15. Midpoint acceleration of simple beam for vehicles in braking.



-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Rig

id-B

ar

Ce

ntr

al V

ert

ica

l A

cce

lera

tio

n

(m/s

^2

)

V=50 m/s , a= -10 m/s^2

V=100 m/s , a= -10 m/s^2

V=100 m/s , a= -20 m/s^2

Fig. 8.16. Vertical acceleration of vehicles in braking.

-2000

-1500

-1000

-500

0

500

1000

1500

2000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Hin

ge

d-S

up

po

rt H

ori

zo

nta

l R

ea

ctio

n (

kN

)

V=50 m/s , a= -10 m/s^2

V=100 m/s , a= -10 m/s^2

V=100 m/s , a= -20 m/s^2

Fig. 8.17. Horizontal reaction at hinged support of simple beam.



1700

1900

2100

2300

2500

2700

2900

3100

3300

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2


Bra

kin

g F

orc

e

(kN

)

Front Wheel (V=50 m/s , a= -10 m/s^2)

Rear Wheel (V=50 m/s , a= -10 m/s^2)





Fig. 8.18. Braking forces of vehicles.

Figure 8.17 depicts the horizontal reaction force occurring at thehinged support for the above three cases of deceleration. As canbe seen, the reaction force for each case increases significantly atthe instant when the front wheel or rear wheel enters the beam.It decreases rapidly at the instant when the front wheel leaves thebeam and rebounds immediately to the maximum value. Thereafterit fluctuates around the value of force caused only by the rear wheel.When the rear wheel leaves the beam, it oscillates around the valueof zero. Besides, it can be observed that the reaction forces for thethree cases are nearly the same before the front wheel leaves thebeam.

The braking forces existing between each of the two wheels andthe beam have been plotted in Fig. 8.18. It indicates that the brakingforces for the cases with higher initial speed, i.e., Cases (2) and (3),are much larger than those for the case with lower initial speed, i.e.,Case (1). The maximum braking force of the rear wheel is also largerthan the front wheel for each case and will occur at some time afterthe front wheel leaves the beam.



8.8.6. Bridges Subjected to a Train Consisting of

10 Identical Cars

The train considered herein consists of 10 identical cars, each of whichis modeled as a suspended rigid bar as shown in Fig. 8.2(d). Thebridge is modeled as a simple beam with the following properties:E = 2.87 × 107 kN/m2, v = 0.2, I = 6.635 m4, m = 32.4 t/m,L = 30 m, and ξ = 0.025. Ten elements are used for the beam.The data adopted for the cars are extracted from Iwnick (1998) forthe Manchester vehicle model, i.e., Mv = 32 t, Iv = 1970 t-m2,kv = 430 kN/m, cv = 20 kN-s/m, Mw = 6.241 t, Lc = 19 m, andds = 3 m (the distance from the front or rear wheels of a car to itsnearest end, see Fig. 8.2(d)). Thus, the total length d of each caris 19 + 2 × 3 = 25 m. Two different speeds are considered for thetrain, i.e., v = 60 and 100 km/h. In each case, the train is consideredentering the bridge when the train head reaches x = 0 m, i.e., leftend of the bridge, and departing from the bridge when it reachesx = 300 m.

The response of the midspan displacement and acceleration of thebridge for the two speeds were plotted in Figs. 8.19 and 8.20. As can

-0.0016

-0.0014

-0.0012

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0 1 2 3 4 5 6 7 8 9

Nondimesional Time (vt/L )

Bri

dg

e M

idsp

an

Dis

pla

ce

me

nt

(m

)

10

v= 60 km/h

v=100 km/h

Fig. 8.19. Midpoint displacement of simple beam due to a moving train.



-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 1 2 3 4 5 6 7 8 9


Bri

dg

e M

idsp

an

Acce

lera

tio

n

(m/s

^2

)

10

v= 60 km/h

v=100 km/h

Fig. 8.20. Midpoint acceleration of simple beam due to a moving train.

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0 1 2 3 4 5 6 7 8 9


Ve

hic

le V

ert

ica

l A

cce

lera

tio

n

(m/s

^2

)

10

1st car (v=60 km/h)

10th car (v=60 km/h)

1st car (v=100 km/h)

10th car (v=100 km/h)

Fig. 8.21. Vertical acceleration of train cars.



be seen, the bridge vibrates in a periodic manner as a result of thepassage of each car of the train. Moreover, the response of the bridgefor the train with higher speeds is larger than that with lower speeds.This is especially true for the acceleration response. The results forthe vertical acceleration of the first and tenth (last) cars of the trainwere shown in Fig. 8.21. It can be observed that the train vibratesmore severely at higher speeds, and that the response of the last caris larger than that of the first car at the same speed.


In this chapter, an effective procedure hinging on computation ofthe contact forces is proposed for solving the interaction equationsof general vehicle–bridge systems. One key step herein is the dis-cretization of the second-order equations of motion for the vehiclesusing Newmark’s finite difference scheme, which enables us to re-late the contact forces to the wheel displacements, and then to thecontact displacements of the bridge through enforcement of the no-jump condition for the vehicles. As the contact forces have beenmade available, the VBI element can be directly derived from thebridge equations by the concept of consistent nodal forces. The VBIelement so derived possesses the same number of DOFs as its parentelement, while possessing the property of symmetry. Since the VBIeffect has been duly taken into account, the derived element can beused to compute not only the bridge response, the contact force, butalso the vehicle response, the latter serves as a measure of passengers’riding comfort.

The versatility and applicability of the proposed procedure havebeen demonstrated in the numerical examples. It is concluded thatthe high modes of vibration of the bridge can affect more signifi-cantly the vehicles than the bridge response. The vehicle responseis underestimated when using the sprung mass model, in compari-son with those using the suspended rigid bar model. Using a singlemoving vehicle, it was demonstrated that the action of braking hasno substantial influence on the response of the bridge due to therelatively small change in vehicle speed and relatively short acting



time. The vertical response of the bridge is determined mainly bythe vehicle speed before braking is applied. However, the horizontalreaction at the hinged support of the bridge is affected mainly by theentrance and departure of each of the two wheels from the beam. Fora bridge traveled by a train with multi cars, periodic response canbe observed following the passage of each car. Higher accelerationscan be expected for the cars of a train moving with a higher speed,and the last car of a train tends to vibrate more seriously than thefirst car.



Chapter 9

Vehicle–Rails–Bridge Interaction —Two-Dimensional Modeling

This chapter deals with the steady-state response and riding comfortof a train moving over a series of simply-supported railway bridges,together with the impact response of the rails and bridges, using thetwo-dimensional model. The dynamic response of the vehicle–rails–bridge interaction system is solved by the condensation techniquepresented in Chapter 8. For the moving train to achieve the steady-state response, a bridge segment consisting of a minimal numberof bridge units should be considered. Track irregularity with ran-dom nature is considered through the use of a power spectral density(PSD) function. The steady-state response of the train, rails andbridges, together with the fast Fourier transform (FFT) of the re-sponse, are computed and discussed. The impact responses of therails and bridges under different train speeds are investigated usingthe impact factor. The maximum response of the train caused by thetrain–rails–bridge resonance is identified. Finally, the riding comfortof the trains moving over tracks of different classes of irregularitiesis assessed using Sperling’s ride index.

9.1. Introduction

In regions where the ground traffic is congested, elevated bridges con-sisting of a series of simply-supported beams are often adopted as thesupporting structure for railways, particularly for the high-speed rail-ways, to offer unobstructed right of way, while minimizing the totalarea of lands used in construction. For this kind of problems, the

271



bridge units are considered separate, but the track structure sup-ported by the bridge units that carries the trains are treated as con-tinuous. Thus, a train traveling over such a bridge system can excitecertain interactions between the various components of the bridge.The dynamic behavior of a train traveling over a multi-unit bridgesystem as mentioned here is different from that over a short bridge.For the latter case, the responses of the train and bridge are gen-erally of the transient nature, as the duration of the train travelingover the bridge is too short for the response to accumulate. Most ex-isting researches in the literature are in this category (Fryba, 1972;Matsuura, 1976; Chu et al., 1979; Bhatti et al., 1985; Wakui et al.,1995; Yau et al., 1999). For the former case, however, due to therepetitive nature of the span units and the continuity nature of thetrack, the response of the train moving over the bridge can reach asteady state if the travel time of the train is long enough. To theauthors’ knowledge, such a problem was not thoroughly studied inthe literature.

Smith et al. (1975) analyzed the response of the moving vehiclesinteracting with single, multiple and continuous span elevated struc-tures, where the operating conditions for the occurrence of multi-span resonance were identified, along with the resonant amplitudescomputed. Chen and Li (2000) studied the dynamic response of sin-gle and three-span uniform railway bridges subjected to three typesof high-speed trains. In their work, only the response of the bridgewas obtained, as the train was simplified as a series of moving loads.Based on the Lagrangian approach, Cheung et al. (1999) analyzedthe vibration of multi-span nonuniform bridges due to the movingvehicles and trains, and presented the dynamic magnification fac-tor for the bridge under different velocity ratios. Dukkipati andDong (1999) studied the idealized steady state interaction betweenthe railway vehicle and track, with the track deflection and contactforce computed for the steady state.

In the aforementioned works, none has dealt with the steady-statevibration of the train moving over multi-unit railway bridges usingrealistic vehicle and bridge models that take into account the dy-namic effect of the track system, which is assumed to be infinitely


Vehicle–Rails–Bridge Interaction — Two-Dimensional Modeling 273

continuous. For the purpose of assessing the riding comfort of passen-gers, while enhancing the maneuverability of the train, it is necessaryto investigate the dynamic response of the train, track and bridge as awhole, considering the interaction among the various components ofthe train–railway bridge system, including in particular the steady-state response.

The interaction between the moving train and the sustainingbridge makes the equations of motion for the two subsystems coupledand nonstationary. To solve this problem, a dynamic condensationtechnique has been developed by Yang and Wu (2001) for decouplingthe two sets of equations of motion, which is basically the techniquepresented in Chapter 8. This chapter has been rewritten primarilyfrom the work by Wu and Yang (2003), which can be regarded as anextension of the theoretical framework established by Yang and Wu(2001) in dealing with the two-dimensional steady-state response ofa train moving over a bridge system that consists of a number ofseparate bridge units, which are connected to the track structure.This is a problem commonly encountered in railway engineering, butnot very well studied previously. The riding comfort of the train willbe evaluated using a comfort index based on the vertical accelerationresponse of the train.

9.2. Train and Bridge Models and Minimal BridgeSegment

Figure 9.1 shows the two-dimensional view of a train traveling overa multi-unit railway bridge with constant speed v. The train is sim-plified as a sequence of identical vehicles, each of which comprisesone car body, two bogies and four wheels, as shown in Fig. 9.2.Each of the suspension systems of the vehicle is represented by aspring-dashpot unit. The bridge consists of many identical units,each of which is made of a concrete box-girder of constant sec-tion and is simply-supported at both ends. The track system lyingon the bridges, particularly the rails, is simplified as an infinitebeam resting on uniformly-distributed spring-dashpot units. Be-cause the track structure is continuous, it serves as a medium for



L

train

bridge unit

track

shift

L

train

bridge unit

track

CFR element

bridge element

LSR element RSR element

central track segmentleft track segment right track segment

bridge segment

bridge segment

Fig. 9.1. A train moving on a multi-unit railway bridge.

G

cl

tl

sl

ev

1wv2wv3wv4wv

vehicle body

ballast

l

1V2V3V

4V

4cv 3cv 2cv1cv

pk

sk sc

pcfront bogierear bogie

tfvtfϕtrvtrϕ

eϕ

bridge

rail

bvcbvk

sleeperwheel

bhk

bhc

Fig. 9.2. Vehicle, track and bridge models.



transmitting the kinetic energy brought by the moving trains to thevarious components of the multi-unit bridge. All the wheels are as-sumed to be in close contact with the rails as they roll over. Thephysical parameters of the vehicle, track and bridge interaction sys-tem were indicated in Fig. 9.2 and summarized in Tables 9.1 and9.2.

The car body of the vehicle is represented by a rigid bar, while therails and bridges by beam elements of the Bernoulli–Euler type. It isrealized that the equations of motion of the vehicle–rails–bridge inter-action system are coupled and time-dependent, which will be solvedusing the dynamic condensation technique presented in the preced-ing chapter. By this technique all the degrees of freedom (DOFs) ofthe vehicle (i.e., the moving subsystem) are condensed into those ofthe rail elements in contact to form a condensed set of equations ofmotion, which are expressed in terms of only the DOFs of the railsand bridge units (i.e., the supporting subsystem).

For a moving train to reach the steady-state response, the minimalnumber Nmin of the bridge units that should be considered for the

Table 9.1. The properties of vehicle model.

Item Notation Value*

Mass of vehicle body Mc 41.75 tMass moment of inertia of vehicle body Ic 2080 t-m2

Mass of bogie Mt 3.04 tMass moment of inertia of bogie It 3.93 t-m2

Mass of wheel Mw 1.78 tStiffness of primary suspension system kp 1180 kN/mDamping of primary suspension system cp 39.2 kN-s/mStiffness of secondary suspension system ks 530 kN/mDamping of secondary suspension system cs 90.2 kN-s/mHalf of longitudinal distance between centers of gravity

of front and rear bogies lc 8.75 mHalf of wheelbase lt 1.25 mLongitudinal distance between center of gravity of

bogie and nearest side of vehicle body ls 3.75 m

∗For SKS series 300 vehicle model, extracted from Wakui et al. (1995) with somemodifications.



Table 9.2. The properties of track and bridge unit.


TrackYoung’s modulus of rail Er 210 GPaPoisson’s ratio of rail vr 0.3Per-unit-length mass of rail

(including mass of sleeper) mr 0.2995 tSectional area of rail Ar 7.686 × 10−3 m2

Flexural moment of inertia of rail Ir 3.055 × 10−5 m4

Per-unit-length vertical stiffness of ballast layer kbv 1.04 × 105 kN/mPer-unit-length vertical damping of ballast layer cbv 50 kN-s/mPer-unit-length horizontal stiffness of ballast layer kbh 1.04 × 104 kN/mPer-unit-length horizontal damping of ballast layer cbh 50 kN-s/m

Bridge unitYoung’s modulus of concrete Eb 28.2 GPaPoisson’s ratio of concrete vb 0.2Per-unit-length mass

(including mass of ballast layer) mb 31.4 tSectional area Ab 7.94 m2

Flexural moment of inertia Ib 8.72 m4

Length of bridge unit L 30 m

∗The data for the bridge unit were modified from those used by the Taiwan-HSR.

bridge segment included in analysis is:

Nmin =∥∥∥∥nv × d

L

∥∥∥∥+ 2 , (9.1)

where nv denotes the number of vehicles comprising the train, d isthe length of the vehicle, i.e., d = 2(lc + ls) as indicated in Fig. 9.2, L

is the length of each bridge unit, and ‖a‖ represents the integer thatis less than but nearest to a(a 0). As can be seen from Fig. 9.1, thenumber Nmin given in Eq. (9.1) is the maximum number of bridgeunits that can be occupied by the train at any instant. On the otherhand, since all the vehicles constituting the train can affect each otherthrough the bridge units, which are connected by the track structure,it is necessary to consider a minimum of Nmin bridge units so thatthe response of all the vehicles can be properly captured. The partof the bridge formed integrally by these Nmin bridge units is referred



to as the minimal bridge segment that should be included in analysisin order for the moving train to reach the steady-state response. Inreality, the minimal bridge segment moves following the motion ofthe train but its size remains unchanged.

9.3. Vehicle’s Equations of Motion and Contact Forces

Let [mv], [cv ], and [kv] respectively denote the mass, damping andstiffness matrices of the vehicle, and dv the displacement vector ofthe vehicle. The equation of motion for the vehicle can be written as

[mv]dv + [cv ]dv + [kv ]dv = fv , (9.2)

where fv is the external force vector, which can be decomposedinto two parts,

fv = fe + [l]fc . (9.3)

Here, fe denotes the external force components excluding the con-tact forces, [l] is a transformation matrix, and fc represents thecontact forces acting through the wheels. As shown in Fig. 9.2,for a vehicle with four wheels, the contact force vector is fc =〈V1 V2 V3 V4〉T , where Vi is the ith contact force.

For the present purposes, we shall decompose the vehicle struc-ture in Fig. 9.2 into two parts, i.e., the upper (or noncontact) andwheel (or contact) parts. The upper part consists of the car body,the primary and secondary suspension systems and the front andrear bogies, which has a total of six DOFs as denoted by du.The wheel part consists of four wheels, which can be representedas dw = 〈vw1 vw2 vw3 vw4〉T with vwi denoting the vertical DOF ofthe ith wheel. Correspondingly, the vertical displacements of the fourcontact points on the rails are denoted as dc = 〈vc1 vc2 vc3 vc4〉T .The total displacement DOFs of the vehicle can therefore be pre-sented as dv = 〈dudw〉T .

The vehicle–bridge interaction (VBI) system will be analyzed inan incremental manner in time domain. Assuming that all the in-formation of the system at time t is known and ∆t is a small timeincrement, we are interested in the behavior of the system at time



t + ∆t. The equations of motion as given in Eq. (9.2) for the vehiclecan be rewritten for time t+∆t with proper partitions in the systemmatrices to account for the upper and wheel parts as

[[muu] [muw]

[mwu] [mww]

] dudw

t+∆t

+[

[cuu] [cuw]

[cwu] [cww]

] dudw

t+∆t

+[

[kuu] [kuw]

[kwu] [kww]

] dudw

t+∆t

= fuefwe

t+∆t

+[

[lu]

[lw]

]fct+∆t , (9.4)

where [muu], [muw], [mwu] and [mww] denote the partitioned massmatrices of the vehicle, [cuu], [cuw], [cwu] and [cww] the dampingmatrices, and [kuu], [kuw], [kwu] and [kww] the stiffness matrices;fue and fwe the external forces acting on the upper and wheelparts of the vehicle, respectively; and [lu] and [lw] the transforma-tion matrices for the two parts. Since only the wheels are actedupon by the contact forces, the matrix [lu] is set to [0]. All the par-titioned matrices and vectors were listed in Appendix G, in whichW = (Mw +0.5Mt +0.25Mc)g represents the static axle load and g isthe acceleration of gravity. Here, we use Mw, Mt, and Mc to denotethe mass of the wheel, bogie, and car body, respectively.

The wheel displacements dw can be related to the contact dis-placements dc of the rails by the constraint equation,

dw = [Γ]dc + r , (9.5)

where [Γ] is the constraint matrix and r is the vector of trackirregularity, r = 〈r1 r2 r3 r4〉T , where ri = r(xi) is the trackprofile evaluated at the ith contact point. Assuming the wheels tobe in full contact with the rails, [Γ] reduces to a unit matrix [I]. Bythe procedure presented in Chapter 8, the ith contact force Vi of the



vehicle can be solved from Eqs. (9.4) and (9.5) as:

Vi,t+∆t = pci,t+∆t + qci,t

+4∑

j=1

(mcij vcj,t+∆t + ccij vcj,t+∆t + kcijvcj,t+∆t) , (9.6)

where mcij, ccij and kcij represent the entries located at the ith rowand jth column of the contact matrices [mc], [cc], and [kc], respec-tively, as given in Eq. (8.14), and pci,t+∆t and qci,t the ith entries ofthe load vectors pct+∆t and qct resulting from the contact forces.Both the load vectors pct+∆t and qct are available in Eq. (8.15).

9.4. Rails and Bridge Element Equations

With reference to Fig. 9.1, the track is divided into three typicalparts, i.e., the central, left and right track segments. The centraltrack segment refers to the part of the track within the bridge range,in which the rails interact with the bridge through the ballast layer.The left and right track segments correspond to the semi-infiniteparts of the track outside the bridge range. The track structurein the central track segment is modeled as a set of rail elementssupported by spring-dashpot units and in turn by bridge elements.Each of the rail elements is of identical length and referred to as thecentral finite rail (CFR) element. The track structures in the twoside segments are modeled as semi-infinite rail elements supportedby distributed spring-dashpot units, which have been referred to asthe left semi-infinite rail (LSR) element and right semi-infinite rail(RSR) element.

9.4.1. Central Finite Rail (CFR) Element and Bridge

Element

As shown in Fig. 9.3(a), a CFR element and underlying bridge ele-ment are connected by a uniformly-distributed spring-dashpot-unitlayer, both of which are of length l and are modeled as conventionalbeam elements. The nodal DOFs of the rail element can be denoted



as dr = 〈ur1 vr1 θr1 ur2 vr2 θr2〉T , and those of the bridge elementas db = 〈ub1 vb1 θb1 ub2 vb2 θb2〉T . By the principle of virtual work,the equation of equilibrium for the rail element can be written as

∫ l

0Er(2Ar)u′

rδu′rdx +

∫ l

0Er(2Ir)v′′r δv′′r dx

= −∫ l

0(2mr)urδurdx −

∫ l

0(2cr)urδurdx

+∫ l

0kbh(ub − ur)δurdx +

∫ l

0cbh(ub − ur)δurdx

−∫ l

0(2mr)vrδvrdx −

∫ l

0(2cr)vrδvrdx

+∫ l

0kbv(vb − vr)δvrdx +

∫ l

0cbv(vb − vr)δvrdx

+ 〈δdr〉fr , (9.7)

where x is the local coordinate, 0 x l, δdr denotes thevariation of the rail element displacement vector dr, fr thecorresponding nodal loads acting on the rail element, i.e., fr =〈Hr1 Vr1 Mr1 Hr2 Vr2 Mr2〉T , (ur, vr) the axial and vertical displace-ments of the rail element, and (ub, vb) the corresponding displace-ments of the bridge element. The displacement fields of the rail andbridge elements can be expressed in terms of the nodal DOFs as

ur(x) = 〈Nu〉dr ,

vr(x) = 〈Nv〉dr ,

ub(x) = 〈Nu〉db ,

vb(x) = 〈Nv〉db ,

(9.8)

where 〈Nu〉 = 〈N1 0 0 N2 0 0〉 denotes the interpolation vector forthe axial displacement, 〈Nv〉 = 〈0 N3 N4 0 N5 N6〉 the interpolationvector for the vertical displacement, N1 and N2 are linear, and N3,



1ru

1rv

1rθ 2ru

2rv

2rθ

1bu

1bv

1bθ 2bu

2bvbvk bhk1 2

1 2bridge

rail

ballast

2bθ

bhcbvc

l

(a)

2ru

2rv

2rθ

2rail

ballastbvkbhkbhc

bvc

(b)

1ru

1rv

1rθ1

rail

ballastbvk bhkbhc

bvc

(c)

Fig. 9.3. Rail element: (a) central, (b) left, and (c) right.

N4, N5 and N6 are cubic Hermitian functions. Substituting the pre-ceding displacement fields into Eq. (9.7), one can obtain the equationof motion for the CFR element as follows:

[mr]dr + [cr]dr + [kr]dr = fr + [cd]db + [ks]db , (9.9)



where dr and db denote the nodal DOFs of the CFR elementand bridge element, respectively, fr the external nodal forces. Themass, damping and stiffness matrices, [mr], [cr] and [kr], are

[mr] = 2mr[ψ0u] + 2mr[ψ0

v ] ,

[cr] = 2cr[ψ0u] + 2cr[ψ0

v ] + cbh[ψ0u] + cbv [ψ0

v ] = [cr0] + [crb] ,

[kr] = 2ErAr[ψ1u] + 2ErIr[ψ2

v ] + kbh[ψ0u] + kbv[ψ0

v ] .

(9.10)

The damping matrix [cr] is composed of the rail material dampingmatrix [cr0] and the ballast damping matrix [crb]. The other param-eters in Eq. (9.10) have been defined in Table 9.2. The damping andstiffness matrices, [cd] and [ks], due to interaction with the bridgeelement, are

[cd] = cbh[ψ0u] + cbv[ψ0

v ] ,

[ks] = kbh[ψ0u] + kbv[ψ0

v ] ,(9.11)

where the matrices [ψ0u], [ψ1

u], [ψ0v ] and [ψ2

v ] are defined as

[ψ0u] =

∫ l

0Nu〈Nu〉dx ,

[ψ1u] =

∫ l

0N ′

u〈N ′u〉dx ,

[ψ0v ] =

∫ l

0Nv〈Nv〉dx ,

[ψ2v ] =

∫ l

0N ′′

v 〈N ′′v 〉dx .

(9.12)

The results for all these matrices have been listed in Appendix H.Similarly, the equation of motion for the bridge element can be

derived as

[mb]db + [cb]db + [kb]db = fb + [cd]dr + [ks]dr . (9.13)



Here, fb denotes the external nodal forces. The mass, dampingand stiffness matrices, [mb], [cb] and [kb], are defined as follows:

[mb] = mb[ψ0u] + mb[ψ0

v ] ,

[cb] = cb[ψ0u] + cb[ψ0

v ] + cbh[ψ0u] + cbv [ψ0

v ] = [cb0] + [cbr] ,

[kb] = EbAb[ψ1u] + EbIb[ψ2

v ] + kbh[ψ0u] + kbv[ψ0

v ] ,

(9.14)

where the damping matrix [cb] consists of the material damping ma-trix [cb0] and the ballast damping matrix [cbr], with [cbr] = [crb]. Thetwo matrices [cd] and [ks] in Eqs. (9.9) and (9.13) are to account forthe interaction between the rails and the bridge, which were given inEq. (9.11).

9.4.2. Left Semi-Infinite Rail (LSR) Element

The LSR element is idealized as a semi-infinite beam with a singlenode, as shown in Fig. 9.3(b). The nodal displacements may bedenoted as drl = 〈ur2 vr2 θr2〉T and the nodal external forces asfrl = 〈Hr2 Vr2 Mr2〉T . By the virtual work principle, one can write∫ 0

−∞Er(2Ar)u′

rδu′rdx +

∫ 0

−∞Er(2Ir)v′′r δv′′r dx

= −∫ 0

−∞(2mr)urδurdx −

∫ 0

−∞(2cr)urδurdx

−∫ 0

−∞kbhurδurdx −

∫ 0

−∞cbhurδurdx

−∫ 0

−∞(2mr)vrδvrdx −

∫ 0

−∞(2cr)vrδvrdx

−∫ 0

−∞kbvvrδvrdx −

∫ 0

−∞cbv vrδvrdx + 〈δdrl〉frl , (9.15)

where 〈δdrl〉 = 〈δur2 δvr2 δθr2〉 denotes the nodal virtual displace-ments, and x is the local coordinate, −∞ < x 0. The horizontaland vertical displacement fields of the element can be described us-ing the relations such as: ur = 〈Nu〉drl, vr = 〈Nv〉drl, where



〈Nu〉 = 〈N2 0 0〉 and 〈Nv〉 = 〈0 N5 N6〉 denote the interpolationvectors. Because of the semi-infinite nature of the LSR element, theinterpolation functions should be obtained from the static solution tothe problem of a beam resting on the Winkler foundation subjectedto a unit load (Hetenyi, 1979), that is,

N2 = eλux ,

N5 = eλvx(cos λvx − sin λvx) ,

N6 =1λv

eλvx sin λvx ,

(9.16)

where λu and λv denote the horizontal and vertical characteristicnumbers of the beam-Winkler foundation system,

λu =√

kbh

Er(2Ar),

λv = 4

√kbv

4Er(2Ir).

(9.17)

By using Eq. (9.16) and the definition of the interpolation functions,Eq. (9.15) can be manipulated to yield the equation of motion forthe LSR element,

[mrl]drl + [crl]drl + [krl]drl = frl . (9.18)

Here, the mass, damping and stiffness matrices [mrl], [crl] and [krl]are

[mrl] = 2mr[ψ0u]l + 2mr[ψ0

v ]l ,

[crl] = 2cr[ψ0u]l + 2cr[ψ0

v ]l + cbh[ψ0u]l + cbv [ψ0

v ]l

= [crl0] + [crlb] ,

[krl] = 2ErAr[ψ1u]l + 2ErIr[ψ2

v ]l + kbh[ψ0u]l + kbv[ψ0

v ]l ,

(9.19)

where [ψ0u]l, [ψ1

u]l, [ψ0v ]l, and [ψ2

v ]l can be found in Appendix H.



9.4.3. Right Semi-Infinite Rail (RSR) Element

As shown in Fig. 9.3(c), the RSR element is also idealized as asemi-infinite beam with a single node. The nodal displacementsare drr = 〈ur1 vr1 θr1〉T and the nodal forces are frr =〈Hr1 Vr1 Mr1〉T . The horizontal and vertical displacements in theelement can be related to the nodal DOFs as ur = 〈Nu〉drr,vr = 〈Nv〉drr, where 〈Nu〉 = 〈N1 0 0〉 and 〈Nv〉 = 〈0 N3 N4〉.Here the interpolation functions can also be determined from thestatic solution as

N1 = e−λux ,

N3 = e−λvx(cos λvx + sinλvx) ,

N4 =1λv

e−λvx sin λvx ,

(9.20)

where λu and λv are the same parameters as those given in Eq. (9.17).Following the same procedure as that for the LSR element, the equa-tion of motion for the RSR element can be derived as

[mrr]drr + [crr]drr + [krr]drr = frr . (9.21)

The mass, damping and stiffness matrices [mrr], [crr] and [krr] are

[mrr] = 2mr[ψ0u]r + 2mr[ψ0

v ]r ,

[crr] = 2cr[ψ0u]r + 2cr[ψ0

v ]r + cbh[ψ0u]r + cbv [ψ0

v ]r

= [crr0] + [crrb] ,

[krr] = 2ErAr[ψ1u]r + 2ErIr[ψ2

v ]r + kbh[ψ0u]r + kbv[ψ0

v ]r ,

(9.22)

where the matrices [ψ0u]r, [ψ1

u]r, [ψ0v ]r, and [ψ2

v ]r have been listed inAppendix H. In a step-by-step time-history analysis, the equations ofmotion in Eqs. (9.9), (9.13), (9.18), and (9.21) should be interpretedas those established for the deformed position of the system at timet + ∆t.



9.5. VRI Element Considering Vertical ContactForces Only

Assume that at time t+∆t, the four wheelsets of the vehicle are actingsimultaneously at the e1, e2, e3 and e4th rail elements, which will begrossly referred to as the vehicle–rails interaction (VRI) elements, asthey are directly under the action of the wheel loads. Consider theeith element that is acted upon only by the vertical component ofthe ith contact force Vi,t+∆t. The equation of motion for the eith railelement at time t + ∆t can be written as follows:

[mri]drit+∆t + [cri]drit+∆t + [kri]drit+∆t

= frit+∆t + εi([cd]dbi + [ks]dbi)−frcit+∆t , (9.23)

where [mri] = [mr], [cri] = [cr], [kri] = [kr], dri = dr and εi =1 for the case with the contact force acting on the CFR element;[mri] = [mrl], [cri] = [crl], [kri] = [krl], dri = drl and εi = 0 forthe case with the LSR element; [mri] = [mrr], [cri] = [crr], [kri] =[krr], dri = drr and εi = 0 for the case with the RSR element;and frci denotes the vector of equivalent nodal forces resultingfrom the action of the ith vertical contact force Vi,t+∆t,

frcit+∆t = NviVi,t+∆t , (9.24)

where Nv denotes the interpolation vector for the vertical displace-ment of the eith element, which varies according to the type of ele-ments, i.e., CFR, RSR or LSR, to which the contact force is acting.The subscript i indicates that the vector Nvi is evaluated at theith contact point, i.e.,

Nvi = Nv(xi) , (9.25)

where xi is the local coordinate of the ith contact point on the eithelement. By using Eqs. (9.24) and (9.6), Eq. (9.23) can be rewritten



as follows:

[mri]drit+∆t + [cri]drit+∆t + [kri]drit+∆t

= frit+∆t + εi([cd]dbit+∆t + [ks]dbit+∆t)

−4∑

j=1

([m∗cij ]drjt+∆t + [c∗cij]drjt+∆t

+ [k∗cij ]drjt+∆t) − p∗cit+∆t − q∗cit , (9.26)

where the asterisked matrices represent the linking action transmit-ted through the car body by the ejth element (under the jth wheelload) on the eith element (under the ith wheel load),

[m∗cij ] = Nvimcij〈Nvj〉 ,

[c∗cij ] = Nviccij〈Nvj〉 ,

[k∗cij ] = Nvikcij〈Nvj〉 ,

(9.27)

and the equivalent nodal loads resulting from the contact forces are

p∗cit+∆t = Nvipci,t+∆t ,

q∗cit = Nviqci,t .(9.28)

The equation of motion as given in Eq. (9.26) will be referred to as thecondensed equation of motion for the VRI element, as all the relevantvehicle DOFs have been eliminated. Note that all the asteriskedmatrices and load vectors involved in Eqs. (9.27) and (9.28), whichare representative of the interaction effects, are time-dependent, sinceNvi varies as the contact point moves. They should be updated ateach time step in an incremental analysis.

9.6. VRI Element Considering General ContactForces

The equation of motion as derived in Eq. (9.26) for the VRI elementconsiders only the action of the vertical contact forces Vi,t+∆t. Forthe case where acceleration or deceleration are involved in the motion



of vehicles over the bridge, it is necessary to consider the horizon-tal contact forces Hi,t+∆t as well. In accordance, the contact forcefrcit+∆t in Eq. (9.23) should be modified as

frcit+∆t = NuiHi,t+∆t + NviVi,t+∆t , (9.29)

where Nu denotes the interpolation vector for the axial displace-ment of the eith element, which varies according to the type of railelements, i.e., CFR, RSR or LSR elements, and Nui = Nu(xi)is evaluated at the ith contact point. The horizontal contact forcesHi,t+∆t may be generated by the rolling, accelerating or braking ac-tion of the wheelsets. All these actions can be regarded as parts ofthe frictional force, depending on the cohesion between the wheelsand the rails. For convenience, the ith horizontal contact force Hi

may be related to the vertical force Vi as

Hi = µiVi , (9.30)

where µi is the frictional coefficient for the ith wheelset, which equalsthe coefficient of braking or acceleration. The frictional coefficient inrolling is neglected in this study, since it is quite small.

Considering the more general expression in Eq. (9.29) for the con-tact forces, along with Eq. (9.30), one can derive a condensed equa-tion of motion for the eith rail element that is identical in form toEq. (9.26), but with the asterisked matrices given as

[m∗cij] = Nuiµimcij〈Nuj〉 + Nvimcij〈Nvj〉 ,

[c∗cij ] = Nuiµiccij〈Nuj〉 + Nviccij〈Nvj〉 ,

[k∗cij ] = Nuiµikcij〈Nuj〉 + Nvikcij〈Nvj〉 ,

(9.31)

and the following for the nodal loads resulting from the contactforces,

p∗cit+∆t = Nuiµipci,t+∆t + Nvipci,t+∆t ,

q∗cit = Nuiµiqci,t + Nviqci,t .(9.32)



Again, all the asterisked matrices and vectors in Eqs. (9.31) and(9.32) are time-dependent and should be updated at each time in-crement.


Let us consider a train moving on a multi-unit railway bridge. Atinstant t+∆t, the parts of the rails directly acted upon by the wheelloads should be modeled by the VRI elements, while the remainingparts of the rails by the rail (beam) elements. All the VRI elements,rail elements and bridge elements can be assembled to yield the sys-tem equations:

[M ]Dt+∆t + [C]Dt+∆t + [K]Dt+∆t

= Ft+∆t − P ∗c t+∆t − Q∗

ct , (9.33)

where D = 〈Dr Db〉T denotes the nodal DOFs of the entire rail-way bridge, with Dr, Db denoting those of the rails and bridge,respectively; [M ], [C] and [K] the system mass, damping and stiff-ness matrices; F = 〈FrFb〉T the external nodal loads, withFr and Fb denoting those acting on the rails and on the bridge;P ∗

c t+∆t and Q∗ct+∆t the equivalent nodal (contact) forces in

global coordinates.A convenient way to construct the system matrices [M ], [C] and

[K] is first to assemble the matrices [M0], [C0] and [K0] for the railwaybridge that is free of any vehicle actions, i.e.,

[M0] =[

[Mr] 0

0 [Mb]

]

=

[mrl] + [mrr] +nb∑

m=1

[mr]m 0

0nb∑

m=1

[mb]m

, (9.34)



[C0] = [C00] + [C0b]

=[

[Cr0] 0

0 [Cb0]

]+[

[Crb] −[Cd]

−[Cd]T [Cbb]

]

=

[crl0] + [crr0] +nb∑

m=1

[cr0]m 0

0nb∑

m=1

[cb0]m

+

[crlb] + [crrb] +nb∑

m=1

[crb]m −nb∑

m=1

[cd]m

−nb∑

m=1

[cd]Tm

nb∑m=1

[cbb]m

, (9.35)

[K0] =[

[Kr] −[Ks]

−[Ks]T [Kb]

]

=

[krl] + [krr] +nb∑

m=1

[kr]m −nb∑

m=1

[ks]m

−nb∑

m=1

[ks]Tm

nb∑m=1

[kb]m

, (9.36)

where nb is the number of the CFR (or bridge) elements consideredwithin the minimal bridge segment. Then we can add to the preced-ing matrices the interaction effects of vehicles contributed throughthe VRI elements, as represented by the asterisked terms in Eq. (9.27)or Eq. (9.31), depending on the type of acting forces, namely,

[M ] =[

[Mr] + [M∗c ] 0

0 [Mb]

]

=

[Mr] +

nv∑k=1

4∑

i=1

4∑j=1

[m∗cij]

k

0

0 [Mb]

, (9.37)



[C] = [C00] +[

[Crb] + [C∗c ] −[Cd]

−[Cd]T [Cbb]

]

= [C00] +

[Crb] +

nv∑k=1

4∑

i=1

4∑j=1

[c∗cij ]

k

−[Cd]

−[Cd]T [Cbb]

, (9.38)

[K] =[

[Kr] + [K∗c ] −[Ks]

−[Ks]T [Kb]

]

=

[Kr] +

nv∑k=1

4∑

i=1

4∑j=1

[k∗cij ]

k

−[Ks]

[Ks]T [Kb]

, (9.39)

where nv is the number of vehicles. The equivalent nodal forcesP ∗

c t+∆t and Q∗ct+∆t resulting from the contact forces are

P ∗c t+∆t =

nv∑k=1

(4∑

i=1

p∗cit+∆t

)k

0

, (9.40)

Q∗ct =

nv∑k=1

(4∑

i=1

q∗cit

)k

0

. (9.41)

In Eqs. (9.37)–(9.39), all the asterisked terms or components repre-sent exactly the interaction effects caused by the linking action ofthe car bodies. In particular, the subscript k that represents the kthvehicle should be looped over from 1 to nv. As the wheel loads movefrom time to time, it is necessary to check at each time step whethera rail element changes into a VRI element and vice versa, and toupdate the entries of the system matrices and vectors, concerningthe contribution of the asterisked terms in Eqs. (9.37)–(9.39) and(9.40)–(9.41), according to the acting positions of the contact forces.One feature with the present procedure is that the total number ofDOFs of the system remains unchanged, regardless of the interaction



effects of moving vehicles. The symmetry property of the originalsystem is also preserved.

Conventionally, the structural damping has been computed onthe structure level. Based on the definition of Rayleigh damping, thedamping matrix [C00] of the railway bridge in Eq. (9.38) is computedas follows:

[C00] = α0[M0] + α1[K0] . (9.42)

Given the damping ratio ξ, the two coefficients α0 and α1 can bedetermined as

α0 =2ξω1ω2

ω1 + ω2, α1 =

2ξω1 + ω2

, (9.43)

where ω1 and ω2 are the first two frequencies of vibration of therailway bridge.

The system equations as given in Eq. (9.33) will be solved usingNewmark’s β method with β = 0.25 and γ = 0.5. By this method,the system responses D, D and D at time t + ∆t can firstbe discretized and related to those at time t, with which the systemequations in Eq. (9.33) can be reduced to an equivalent stiffnessequation and solved. Details of such a procedure are available inAppendix B.

9.8. Shift of Bridge Segment and Renumbering ofNodal Degrees of Freedom

Once the train moves out of the right boundary of the bridge segmentconsidered, a shift in the element mesh is performed, i.e., by addinga new bridge unit to the segment on the right-hand side, and bydeleting the leftmost unit from the segment (see Fig. 9.1). By doingso, the train can interact continuously with the rails and bridge, whilethe size of the bridge segment remains unchanged. On the otherhand, since the bridge segment has shifted one unit rightward, thenodal DOFs of the rails and bridge elements have to be renumberedsuch that consistence is maintained for the nodal response beforeand after shifting. Accordingly, the global coordinates of the element



nodes and the positions of the contact points between the wheels andrails should also be updated. Besides, the irregular track profile, ifthere is any, has to be introduced for the newly-added bridge unit.Note that the system matrices of the rails–bridge model remain thesame regardless of the element shifting, as the same properties areused for each bridge unit, which therefore need not be updated.

9.9. Verification of Proposed Procedure

For the purpose of verification, let us consider a simply-supportedbeam of length L = 25 m subjected to a moving sprung mass(Fig. 9.4), with the following properties: Young’s modus E =2.87 GPa, Poisson’s ratio v = 0.2, moment of inertia I = 2.90 m4,mass per unit length m = 2.303 t/m, suspended mass Mv = 5.75 t,suspension stiffness kv = 1595 kN/m, and speed v = 27.78 m/s.The first frequency computed of the beam is ω1 = 30.02 rad/s,and the frequency of the sprung mass is ωv = 16.66 rad/s. Thedamping of the beam is neglected. To make use of the present pro-cedure, the following data are assumed for the vehicle and bridge:Mc = Mv = 5.75 t, Ic = 1 t-m4, Mt = 10−5 t, It = 1 t-m4, Mw = 10−5 t, kp = 1010 kN/m, ks = 0.5kv = 797.5 kN/m,cp = cs = 0 kN-s/m, and lc = lt = ls = 0 m; Er = 210 GPa, vr = 0.3,Ir = 10−10 m4, mr = 10−10 t/m, kbv = 1010 kN/m/m, cbv = 0 kN-s/m/m; Eb = E = 2.87 GPa, vb = v = 0.3, Ib = I = 2.90 m4,mb = m = 3.303 t/m, ξ = 0. The bridge is modeled as 10 ele-ments. The frequency ω∗

1 of the vehicle obtained from the eigen-value analysis is the same as the frequency ωv of the sprung mass,

vM

vk

LEI

Fig. 9.4. A simple beam subjected to a moving sprung mass.



-3.0E-03

-2.5E-03

-2.0E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Mid

po

int

Dis

pla

ce

me

nt

of

be

am

(m

)

semi-analytical (first mode)

present (10 elements)

Fig. 9.5. The midpoint displacement of beam.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Time (s)

Acce

lera

tio

n o

f S

pru

ng

Ma

ss

(m/s

2)

semi-analytical (first mode)

present (10 elements)

Fig. 9.6. The acceleration response of sprung mass.



i.e., ω∗1 = ωv = 16.66 rad/s, indicating that the data assumed for the

vehicle are equivalent to those adopted for the sprung mass model.The dynamic response of the midpoint displacement of the beam

has been plotted in Fig. 9.5, along with the analytical solution con-sidering only the first mode (Biggs 1964). Evidently, good agreementhas been achieved between the present and analytical solutions. FromFig. 9.6, one observes that the acceleration of the mass computed bythe present approach agrees generally well with the analytical one,with slight deviation due to consideration of only the first mode ofvibration of the beam in the latter. This example serves to illustratethe reliability of the present procedure.

9.10. Numerical Studies

The procedure presented above is applied to studying: (1) the steady-state behavior and riding comfort of a train moving over a multi-unitrailway bridge, and (2) the impact response of the bridge duringthe passage of the train. The data adopted for the train cars areextracted from those for the SKS Series 300 rail cars (Wakui et al.,1995) (see Table 9.1). The rails are assumed to be of the UIC-60 type. The bridge units comprising the track supporting systemare assumed to be simply-supported and of constant cross sections.The properties of the rails, ballast layer and bridge units have beenlisted in Table 9.2. Ten vehicles are considered for the train, i.e.,nv = 10. Each bridge unit, assumed to be of length L = 30 m, ismodeled by 10 elements, i.e., with element length l = 3 m. Accordingto Eq. (9.1), ten bridge units are used for the bridge segment inanalysis, i.e., Nmin = 10. In addition, a time increment of ∆t =0.005 s and damping ratio of ξ = 0.025 are used. No braking oraccelerating actions are considered for the train (µ = 0). The firstand second natural frequencies of the bridge segment obtained fromthe eigenvalue analysis are: ω1 = 26.87 rad/s (4.28 Hz) and ω2 =107.42 rad/s (17.10 Hz). The fundamental frequency is close to themean value of those measured from several concrete railway bridgesin service (Fryba, 1996), indicating the adequacy of the data used inthe present analysis.



9.10.1. Steady-State Responses of the Train, Rails

and Bridge

Figure 9.7 shows the steady-state vertical accelerations of the carbody of the 5th vehicle of the train at speeds v = 300 and 375 km/h.Obviously, the train undergoes a periodic vibration at the steadystate. The dominant frequencies of the acceleration response forv = 300 km/h obtained by the fast Fourier transform (FFT) are 2.83and 5.56 Hz and those for v = 375 km/h are 3.52 and 7.03 Hz. Itis found that the vibration frequency fv of the vehicle, the lengthL (= 30 m) of the bridge unit and the train speed v satisfy therelation: fv = n × v/L, n = 1, 2, 3, . . . ,∞, where relatively smallcontributions are made by harmonic components with n ≥ 3. Inaddition, the response for v = 375 km/h appears to be much largerthan that for v = 300 km/h, the reason for which will be explainedlater.

The midspan responses of the bridge unit to the passage of thetrain have been plotted in Fig. 9.8. As can be seen from Fig. 9.8(a)for the displacement, the bridge unit vibrates periodically during

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4 5 6 7 8 9 10


Ve

rtic

al A

cce

lera

tio

n o

f th

e 5

th V

eh

icle

of

tra

in

(m/s

2)

11

v = 300 km/h (normal operating speed)

v = 375 km/h (resonant speed)

v =300 km/h

dominant freqs.= 2.83, 5.56 Hz

maximum acc.= -0.06 m/s2

v =375 km/h



Fig. 9.7. The vertical acceleration of the 5th vehicle of train.



-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0 1 2 3 4 5 6 7 8 9 10 1


Ve

rtic

al D

isp

lace

me

nt

at

the

Mid

sp

an

of

Bri

dg

e U

nit

(m)

1



v =300 km/h

dominant freq.= 3.32 Hz

maximum disp.= -1.4 mm

v =375 km/h

dominant freq.= 4.10 Hz


(a)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10


Ve

rtic

al A

cce

lera

tio

n a

t th

e M

idsp

an

of

Bri

dg

e U

int

(m

/s2)

11

v= 300 km/h (normal operating speed)

v= 375 km/h (resonant speed)

v =375 km/h


maximum acc.= 1.91 m/s2

v =300 km/h



(b)

Fig. 9.8. The vertical response at the midspan of bridge unit: (a) displacementand (b) acceleration.



the passage of the train, implying a dominant frequency of 3.32 Hzfor v = 300 km/h and 4.10 Hz for v = 375 km/h, both ofwhich satisfy the relation for the frequency fb of the bridge unit:fb = n × v/d, n = 1, 2, 3, . . . ,∞, where d denotes the vehicle length(= 25 m). The contribution from the higher harmonic components,i.e., with n > 2, to the displacement can virtually be ignored. Similartrend exists for the acceleration response in Fig. 9.8(b), except thatthe contribution of the second dominant frequency can be readilyobserved, which is equal to 33.4 Hz for v = 300 km/h and 41.6 Hzfor v = 375 km/h. The second frequency is about 10 times the valueof the first frequency, which satisfies the following relation for thefrequency f ′

b of the bridge unit: f ′b = n × v/dwb, n = 1, 2, 3, . . . ,∞,

where dwb denotes the wheelbase of each wheelset (= 2lt = 2.5 m),as indicated in Fig. 9.2.

Evidently, the response of the bridge unit for v = 375 km/h ismuch larger than that for v = 300 km/h and increases as there aremore vehicles passing the bridge unit. This is an indication of theoccurrence of resonance on the bridge unit for the train moving atspeed v = 375 km/h. The same is not true for v = 300 km/h. Itshould be added that due to its interaction with the bridge units thetrain also exhibits resonance at v = 375 km/h, as indicated by therather large response in Fig. 9.7. This type of resonance, referred toas the train–rails–bridge resonance, occurs if the train speed v (m/s),the vehicle length d (m) and the loaded fundamental frequency ofthe bridge unit ω∗

1 (rad/s) satisfy the following conditions (Li andSu, 1999):

ω∗1d

2v= nπ , n = 1, 2, 3, . . . , (9.44)

where ω∗1 = ω1 × [m/(m+M/L)]1/2, M = Mc +2Mt +4Mw (see Ta-

bles 9.1 and 9.2 for definition of symbols). According to Eq. (9.44),the primary resonant speed for the present case can be predicted as374 km/h (= 103.9 m/s) with ω∗

1 = 26.1 rad/s, d = 25 m and n = 1,which is very close to the value of 375 km/h indicated above. Besides,it is noted that the bridge unit exhibits certain positive displace-ments at the resonant speed of 375 km/h, implying the occurrence of



negative moments in the bridge unit. This is an issue that should betaken into account in the design of bridges. No similar phenomenoncan be observed for the train moving at speed v = 300 km/h.

Figure 9.9 shows the vertical response of the rails at the pointright above the midspan of the bridge unit. The displacement of therails is generally similar to that of the bridge unit due to the strongconstraint effect of the ballast layer, where pronounced positive dis-placements (implying negative moments) are also observed for theresonant speed. Unlike that for the response of the bridge unit, thecontribution of the second mode to the rails response is not negligi-ble, which equals three times the fundamental one, as revealed by aFFT analysis. Because of the filtering effect of the ballast layer, thesecond frequency does not contribute to the response of the bridgeunit. In contrast, the acceleration of the rails in Fig. 9.9(b) showseven much larger contribution from the higher modes, in compari-son with the bridge response. Note that the dominant frequency ofthe acceleration of the rails at v = 375 km/h is lower than that atv = 300 km/h, contrary to the case for the displacement.

As shown in Fig. 9.10, the contact force between the rails and thefirst wheel of the 5th vehicle is quite different for the two speeds. Dueto occurrence of the train–rails–bridge resonance, the maximum andmean values of the contact force for v = 375 km/h are greater thanthose for v = 300 km/h. The dominant frequency and maximumvalue of the responses of the 5th vehicle, rails, bridge unit and contactforce have been indicated in the figure.

9.10.2. Impact Response of Rails and Bridge Under

Various Train Speeds

The impact response of the rails or bridge is represented by the im-pact factor I as defined in Eq. (1.1), i.e., I = [Rd(x)−Rs(x)]/Rs(x),where Rd(x) and Rs(x) denote the maximum dynamic and staticresponses, respectively, of the rails or the bridge at section x.The impact factors computed for the bridge, including the dis-placements and internal forces, traveled by the train moving atspeeds 0 ∼ 400 km/h have been plotted in Fig. 9.11. As can be



-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0 1 2 3 4 5 6 7 8 9 10 1


Ve

rtic

al D

isp

lace

me

nt

of

Ra

ils a

bo

ve

th

e

Mid

sp

an

of

Bri

dg

e U

nit

(m)

1

v= 300 km/h (normal operating speed)

v= 375 km/h (resonant speed)

v =375 km/h

dominant freq.= 4.10, 12.5 Hz


v =300 km/h

dominant freq.= 3.32, 9.96 Hz


(a)

-25

-20

-15

-10

-5

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8 9 10


Ve

rtic

al A

cce

lera

tio

n o

f R

ails

ab

ove

th

e

Mid

sp

an

of

Bri

dg

e U

nit

(m/s

2)

11



v =375 km/h



v =300 km/h



(b)

Fig. 9.9. The vertical response of rails above the midspan of bridge unit: (a)displacement and (b) acceleration.



110

120

130

140

150

160

170

180

0 1 2 3 4 5 6 7 8 9 10


Co

nta

ct

Fo

rce

of

the

1st

Wh

ee

l o

f th

e 5

th

Ve

hic

le

(kN

)

11



static wheel load (135 kN)

v =300 km/h


maximum value = 148.3 kN

mean value = 140.3 kN

v =375 km/h


maximum value = 162.8 kN

mean value = 145.4 kN

Fig. 9.10. The contact force of the 1st wheel of the 5th vehicle.

-0.5

0

0.5

1

1.5

2

2.5

0 50 100 150 200 250 300 350 400

Train Speed (km/h)

Imp

act

Fa

cto

r fo

r th

e R

esp

on

se

of

Bri

dg

e

Unit

vertical disp. (midspan)

rotational angle (hinged support)

bending moment (midspan)

shearing force (hinged support)

Japan's provision

Eurocode

vertical disp. (midspan)(no track)

Maximum static response

(absolute)

vertical disp. = 1.12 mm

rotational angle = 0.00012 rad

bending moment = 3058 kN-m

shearing force = 399 kN

Fig. 9.11. The impact factor for the response of bridge unit under various trainspeeds.



seen, the impact factor increases as the train speed increases andexhibits a peak value at a speed around 375 km/h, indicating theoccurrence of the train–bridge resonance at that speed. Little dif-ference can be observed between the impact factors for the verticaldisplacement, rotational angle and bending moment. The impactfactor for the shearing force is much smaller than those for the bend-ing moment and displacement. A comparison of the present resultswith existing specifications, such as Japan’s provision and Eurocode,indicates that the impact factor exceeds the allowable limit set bythe specifications as the train speed exceeds 350 km/h or reaches theresonant speed, where the impact factor can be as high as 2.2.

Figure 9.12 shows the impact factor for the response of the railsat the points right above the midspan and hinged support of thebridge unit under different train speeds. In general, the impact fac-tor increases as the train speed increases. The impact factor forthe displacements is higher than that for the internal forces. Thedisplacement impact factor reaches a peak at the resonant speed of375 km/h, while such a peak does not exist for the internal forces.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 50 100 150 200 250 300 350 400

Train Speed (km/h)

Imp

act

Fa

cto

r fo

r R

esp

on

se

of

Ra

il

vertical disp. (above midspan)

rotational angle (above hinged support)

bending moment (above midspan)

shearing force (above hinged support)

Maximum static response

(absolute)

vertical disp. = 1.64 mm

rotational angle = 0.00064 rad

bending moment = 4.7 kN-m

shearing force = 3.7 kN

Fig. 9.12. The impact factor for the response of rails under various train speeds.



The implication is that the extent of amplification for the internalforces (including the moment and shear) is smaller than that for thedisplacement at resonant speeds.

9.10.3. Response of Train to Track Irregularity and

Riding Comfort of Train

Track irregularity is one of the factors that can amplify the vibrationresponse of the train. In this study, the track irregularity is assumedto be of random nature and characterized by the following powerspectral density (PSD) function S(Ω) (White et al., 1978):

S(Ω) =AvΩ2

c

(Ω2 + Ω2r)(Ω2 + Ω2

c), (9.45)

where Ω(= 2π/λr) denotes the spatial frequency (rad/m), λr thewavelength of the irregularity (m), and Av (m), Ωr (rad/m) andΩc (rad/m) are relevant parameters. Table 9.3 contains the values ofthe parameters in Eq. (9.45) for track classes 4, 5 and 6 designated bythe Federal Railroad Administration (FRA), with class 6 indicatingthe best quality (Fries and Coffey, 1990). By using PSD functionin Eq. (9.45) with 0.209 rad/m ≤ Ω ≤ 209.441 rad/m (0.03 m ≤λr ≤ 30 m) and the method proposed in Chen and Zhai (1999), theirregular profiles r(x) of the three classes of tracks are computed andplotted in Fig. 9.13 with the maximum deviations indicated.

The maximum accelerations of the train for the three class oftrack irregularity under different train speeds have been plotted inFig. 9.14. As was expected, the maximum response of the train mov-ing over irregular tracks appears to be greater for rougher surface,and is the worst for FRA class 4. The maximum response of the

Table 9.3. Track PSD model parameters.

FRA class 4 5 6

Av (m) 2.39 × 10−5 9.35 × 10−6 1.50 × 10−6

Ωr (rad/m) 2.06 × 10−2 2.06 × 10−2 2.06 × 10−2

Ωc (rad/m) 0.825 0.825 0.825



-20

-15

-10

-5

0

5

10

15

20

25

30

0 30 60 90 120 150 180 210 240 270 300 330

Position along Track (m)

Tra

ck G

eo

me

ry D

evia

tio

ns (

mm

)

FRA class 4 (max. deviation = 15.8 mm)



Fig. 9.13. The irregular track profiles for FRA classes 4, 5 and 6.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400

Train Speed (km/h)

Ma

xim

um

Acce

lera

tio

n o

f T

rain

(m

/s2)

FRA class 4

FRA class 5

FRA class 6

smooth

FRA class 4(no track)

SNCF a = 0.49 m/s2

Eurocode a = 1.0 m/s2

Fig. 9.14. Maximum acceleration responses of train under various train speeds.



train moving over irregular tracks approaches a limit value for trainspeeds v over 200 km/h. The asymptotic values for FRA classes 4,5 and 6 are estimated as 1.4, 0.9 and 0.4 m/s2, respectively. Thereason for existence of such a limit is that the exciting frequenciesinduced by track irregularities at higher speeds are filtered out by thesuspension system of each vehicle in transmission to the car body.Moreover, the maximum acceleration of the train for FRA class 6 isbelow the allowable limit of 0.49 m/s2 (= 0.05 g) imposed by France-SNCF (Grandil and Ramondenc, 1990) or Taiwan-HSR concerningpassengers’ riding comfort, while those for the FRA classes 4 and 5are not. If a less strict limit of 1.0 m/s2 is imposed, as suggested byEurocode (1995), only FRA class 4 remains unqualified.

The amplification factor and decrement ratio for the maximumcontact forces have been presented in Fig. 9.15 for the three differentclasses of track quality, which shows a trend of increase for highertrain speeds. Of interest is the fact that the decrement ratios forall the three track classes are below the limit of 0.25 set in Japan’s

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400

Train Speed (km/h)

Am

plif

ica

tio

n F

acto

r a

nd

De

cre

me

nt

Ra

tio

for

Ma

xim

um

Co

nta

ct

Fo

rce

of

tra

in

FRA class 4

FRA class 5

smooth

FRA class 6

Amplification Factor = V d,max / V s

Decrement Ratio = (V s -V d,min ) / V s

FRA class 6

FRA class 4

smooth

FRA class 5

safety limit = 0.25

Fig. 9.15. The amplification factor and decrement ratio for maximum contactforce of train.



provision for the running safety of trains (Ma and Zhu, 1998), indi-cating that the train can run safely over the bridge under the trackconditions specified for speeds up to 400 km/h.

The riding comfort of trains is an issue of great concern for groundvehicles, especially for high-speed trains. In this study, the ridingcomfort of the train is assessed by Sperling’s ride index Wz (Gargand Dukkipati, 1984) defined as:

Wz =

( nf∑i=1

W 10zi

)1/10

. (9.46)

Here, nf is the total number of the discrete frequencies of the accel-eration response of the train identified by the FFT and Wzi is thecomfort index corresponding to the ith discrete frequency, computedas

Wzi = [a3i B(fi)3]1/10 , (9.47)

where ai denotes the amplitude of the acceleration response (m/s2)of the ith frequency identified by the FFT and B(fi) a weightingfactor,

B(fi) = 0.588[

1.911f2i + (0.25f2

i )2

(1 − 0.277f2i )2 + (1.563fi − 0.0368f3

i )2

]1/2

. (9.48)

Figure 9.16 shows Sperling’s ride index computed for the train underdifferent speeds and track qualities. As can be seen, Sperling’s indexremains generally constant for trains moving at moderate to highspeeds (100 km/h ≤ v ≤ 300 km/h), and increases slightly at speedsover 300 km/h. For poorer track quality (i.e., for lower FRA class),there exists larger Sperling’s index. Compared with the limit val-ues set for different comfort levels in Fig. 9.16 (Garg and Dukkipati,1984), the following observations can be made for train speeds inthe range of 100 km/h ≤ v ≤ 300 km: (1) for smooth track andtracks of FRA class 6, the level of riding comfort of the train is con-sidered between “Just noticeable (Wz = 1)” and “Clearly noticeable(Wz = 2)”; (2) for tracks of FRA class 5, the comfort level becomesworse and falls within the limits of “Clearly noticeable (Wz = 2)”



0

0.5

1

1.5

2

2.5

3

3.5

0 50 100 150 200 250 300 350 400

Train Speed (km/h)

Sp

erl

ing

's R

ide

In

de

x f

or

Tra

in

Wz

FRA class 4

FRA class 5

FRA class 6

smooth

FRA class 4 (no track)

Strong, irregular,

but still tolerableMore pronounced

but not unpleasant

Clearly noticeable

Just noticeable

Fig. 9.16. Sperling’s ride index for train under various train speeds.

and “More pronounced but not unpleasant (Wz = 2.5)”; and (3) fortracks with FRA class 4, i.e., the worst quality, the riding comfortdeteriorates to a level between “More pronounced but not unpleasant(Wz = 2.5)” and “Strong, irregular, but still tolerable (Wz = 3)”.Note that the comfort level for the smooth track is the best levelthat can be achieved by the train under the present conditions setfor the train, track and bridge. Figure 9.17 shows the relation be-tween Sperling’s ride index Wz and the maximum deviation of thetrack irregularity, which can be described by a cubic polynomial.The maximum track deviations corresponding to the riding comfortlevels stated above are also shown in the figure.

9.10.4. Effect of the Track System

To evaluate the effect of the track system on the response of thebridge and moving train, a parallel analysis was performed with thetrack system omitted. The impact factor computed for the bridgeunit by such an analysis was also shown in Fig. 9.11 (indicated as thesolution with “no track”), together with the maximum acceleration



0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25

Maximum Deviation of Track Irregularity (mm)

Sp

erl

ing

's R

ide

In

de

x f

or

Tra

in

Wz

Strong, irregular, but still tolerable (wz=3)

More pronounced but not unpleasant (wz=2.5)

Clearly noticeable (Wz=2)

Just noticeable (Wz=1)

Dmax=5.7 mmDmax=11.9 mm

Dmax=21.8 mm

Fig. 9.17. Sperling’s ride index with relation to maximum track deviation.

and Sperling’s index of the train in Figs. 9.14 and 9.16. In general,the impact factor computed for such a case for the bridge is lessthan that for the case with the track system taken into account.Besides, the impact factor reaches its peak at a higher speed dueto the relatively higher loaded fundamental frequency of the train–bridge system. For FRA class 4, the maximum acceleration andSperling’s index of the train computed for the case with no trackare slightly larger than those for the case with track for train speedsover 200 km/h. They are nearly the same for both cases for trainspeeds below 200 km/h. It can be concluded that the presence ofthe track system can increase the train-induced impact effect on thebridge, while reducing the vibration amplitude (i.e., raising the ridingcomfort) of the traveling train.


A procedure for analyzing the vehicle–rails–bridge system was pre-sented, by which the dynamic response of each subsystem can be



computed. The procedure was then applied to investigating thesteady-state response and riding comfort of the train moving over arailway bridge comprising a series of simply-supported beams. Thefollowing are the conclusions made from the numerical studies: (1)The steady-state response of the train moving over a smooth trackdisplays periodic behaviors, of which the dominant frequencies re-late to the train speed and the bridge length. (2) The vibrations ofthe rails and bridge caused by the moving train also possess somecharacteristic frequencies related to the train speed and the vehi-cle length. (3) Resonance can occur in the interaction system ofthe train, rails, and bridge, as the vehicle length, frequencies of thebridge and train speed meet some specific relations, which can resultin dramatic amplification of the response of each component. (4) Theresonant effect on the rails and bridge responses was underestimatedby existing specifications for the high-speed range, which should beconsidered in the design of railway bridges involving flexible bridgestructures or for trains moving at high speeds. (5) The riding com-fort of the train can be significantly affected by the presence of trackirregularity. It remains nearly independent of the train speed in themoderate to high-speed range. (6) The effect of the track systemshould be taken into account in the analyzing the train–bridge sys-tem. Otherwise, one may underestimate the impact effect of thebridge and overestimate the vibration response of the train.



Chapter 10

Vehicle–Rails–Bridge Interaction —Three-Dimensional Modeling

The procedure presented in Chapters 8 and 9 is extended in thischapter to deal with the three-dimensional aspects of the vehicle–rails–bridge interaction (VRBI) system. In this chapter, the vehi-cle is idealized as an assembly of a car body, two bogies and fourwheelsets, which has a total of 27 degrees of freedom. The bridgeis assumed to carry two parallel tracks, each consisting of two railsof infinite length supported by the spring-dashpot units that areuniformly-distributed. The bridge is modeled by the central finiterail (CFR) element, left semi-infinite rail (LSR) element, and rightsemi-infinite rail (RSR) element, depending on the parts of rails con-cerned. Through elimination of the contact forces, three types ofvehicle–rails interaction (VRI) elements were derived, with due ac-count taken of the vehicle properties. The equation of motion forthe entire VRBI model was then constructed by assembling the VRIelements, ordinary rail elements and bridge elements. The presentprocedure allows us to analyze a wide range of three-dimensionalVRBI problems, including the crossing of two trains on the bridge,the variation of wheel/rail contact forces, the risk of derailment ofmoving trains, and so on.

10.1. Introduction

The dynamic response of bridge structures to the moving loadsexerted by trains has been widely investigated by researchers, as wasreviewed in Chapter 1. For the works that dealt with the train–bridge

311



interactions, the scope of study was restricted mainly to the two-dimensional aspects of the train–bridge system, aimed at analyzingthe vertical vibration of the bridge. For the relatively small amountof research works that dealt with the three-dimensional aspects ofthe train–bridge system, e.g., those by Bhatti et al. (1985), Wakuiet al. (1995), and Schupp and Jaschinski (1996), among others, onlypartial considerations were made for the dynamic effects of the rail-way bridge and moving trains, which is not sufficient if focus is to beextended to some advanced aspects of the vehicle–rails–bridge inter-action (VRBI) systems, such as the lateral and rotational or torsionalvibrations, the responses induced by two trains in crossing, the riskof derailment of moving trains, and so on. All these problems arecrucial and have to be considered in the design of high-speed railwaybridges, concerning the maneuverability of the train at high speeds.Besides, the track system directly involved in the interaction withthe moving train was either ignored or only partially considered inmost previous studies. Since the track system is a flexible mediumvibrating with the train (moving subsystem) and the bridge (sta-tionary subsystem), it can affect seriously the extent of interactionbetween the two subsystems, especially for trains moving in the high-speed range. For this reason, the track system will be included inthe analysis of the train–bridge system, with which the contact forcesbetween the wheels and rails can be computed.

In this chapter, the three-dimensional characteristics of the train–track–bridge system, together with assumptions made for modelingsuch a system, will first be summarized. A three-dimensional VRBImodel will then be constructed based on these assumptions. Next,the equations of motion for the major components of the model, i.e.,the vehicle, rails and bridge, will be formulated, based on which equa-tions of motion for the entire VRBI system are assembled. The proce-dure for performing the nonlinear dynamic analysis is also presented.The theory proposed herein will be verified through comparison ofthe results for a typical example obtained by the present 3D proce-dure with those by the 2D procedure presented in Chapter 8. Thedynamic interactions between the moving trains and railway bridgeunder various conditions of track irregularities will be investigated.


Vehicle–Rails–Bridge Interaction — Three-Dimensional Modeling 313

The risk of derailment of a train running over the bridge will alsobe investigated. The research results presented in this chapter arebased largely on the paper by Wu et al. (2001), but with substantialadditions and revisions.

10.2. Three-Dimensional Models forTrain, Track and Bridge

Figure 10.1 shows a simply-supported railway bridge traveled by atrain of speed v. In this study, the train is idealized as a series ofidentical moving vehicles, each composed of one car body, two iden-tical bogies and four identical wheelsets, as depicted in Fig. 10.2. Allthese components are assumed to be rigid. The suspension system,whether vertical or lateral, of each wheelset of the vehicle is mod-eled as a linear spring-dashpot unit. As shown in Fig. 10.3(a), thebridge is made of a box girder carrying two parallel tracks, whichallows two trains to move in the same or opposite directions. Eachof the two tracks is simplified as a set of two infinite rails lying on asingle-layer ballast foundation. The two rails of each track are iden-tical and connected by uniformly-distributed rigid sleepers. Eachset of the two rails will be represented collectively as a Bernoulli–Euler beam of constant sections. By neglecting the nonlinear effectsand interlock shear, the ballast foundation will be represented byuniformly-distributed linear spring-dashpot units.

Due to the flexibility of the ballast foundation, each track is al-lowed to move longitudinally (x direction), vertically (y direction)and laterally (z direction), as well as to rotate about the longitu-dinal axis. The two rails of each track have the same longitudinaland lateral displacements due to the constraint of rigid sleepers. Thevertical contact forces between the wheels and the two rails of eachtrack are assumed to act along the centerline of the cross sections ofthe two rails. Therefore, each individual rail has no torsional defor-mations, although the track to which the rail belongs may undergosome torsional or rotational motion. The torques produced by thelateral forces acting through the torsional center of the two rails willbe neglected. The torsional resistance of the two rails against the



L

train

bridge

central track section right track sectionleft track section

CFR element


bridge element

ballast

rail

Fig. 10.1. Simply-supported railway bridge traveled by a moving train.

ballast reaction is provided by the cross-sectional areas of the tworails rigidly-connected by sleepers. Furthermore, the four wheelsetsof each vehicle are supposed to be in full contact with the rails atall times (i.e., no jumps occur) and move with the two rails in thevertical and lateral directions. It should be added that the mass andmass moment of inertia of the sleepers are considered as part of thetwo rails of each track.

The bridge has constant sections and uniform properties, whichis idealized as a three-dimensional Bernoulli–Euler beam, as shownin Fig. 10.3(a). The mass and mass moment of inertia of the ballastlayer on the bridge are included as part of the bridge. The phys-ical parameters of the vehicle, track and bridge have been shownin the figure, with their definitions summarized in Tables 10.1 and10.2. In addition, the deviations in geometry of the track, i.e., trackirregularities, are also taken into account.

10.3. Vehicle Equations and Contact Forces

All the degrees of freedom (DOFs) permitted for the vehicle body,bogies, and wheelsets have been shown in Fig. 10.2. The vehiclebody has a total of five DOFs with respect to its center of gravity G,



G

cl

tl

sl

alpd

sd

csh

tsh

tph

pyk pyc

0r

ev

ew

eϕ

eψ

tfv

tfw

tfϕ

trv

trw

trϕ

1wv

1ww

2wv

2ww

3wv

3ww

4wv

4ww

4wv

4ww4wθ

1cv

1ch

1V

1H

1Q

2ch2H

3cv

3ch

3V

3H

3Q

4ch4H

5cv

5ch

5V

5H

5Q

6ch6H

7cv

7ch

7V

7V

7H

7H

7Q

8ch

8V

8H

8H

157

8 6 24

3

C Body

Bogie

Bogie

Wheelset

Wheelset

C Body

Fig. 10.2. Three-dimensional vehicle–rails model.



(a)

Fig. 10.3. Rail elements: (a) CFR, (b) LSR, and (c) RSR.



(b)




(c)




Table 10.1. The properties of vehicle model.


Mass of car body Mc 41.75 tMass moment of inertia of car body around x axis I∗

cx 23.2 t-m2

Mass moment of inertia of car body around y axis I∗cy 2100 t-m2

Mass moment of inertia of car body around z axis I∗cz 2080 t-m2

Mass of bogie Mt 3.04 tMass moment of inertia of bogie around x axis I∗

tx 1.58 t-m2

Mass moment of inertia of bogie around y axis I∗ty 2.34 t-m2

Mass moment of inertia of bogie around z axis I∗tz 3.93 t-m2

Mass of wheelset Mw 1.78 tMass moment of inertia of wheelset around x axis I∗

w 1.14 t-m2

Half of longitudinal distance between centers of gravityof front and rear bogies lc 8.75 m

Half of wheelbase lt 1.25 mLongitudinal distance between center of gravity of bogie

and nearest side of car body ls 3.75 mHalf of transverse distance between contact points of

wheel and rail la 0.75 mHalf of transverse distance between vertical primary

suspension systems dp 1.00 mHalf of transverse distance between vertical secondary

suspension systems ds 1.23 mVertical distance between center of gravity of car body

and lateral secondary suspension system hcs 0.75 mVertical distance between lateral secondary suspension

system and center of gravity of bogie hts 0.42 mVertical distance between center of gravity of bogie and

lateral primary suspension system htp 0.20 mStiffness of vertical primary suspension system kpy 590 kN/mDamping of vertical primary suspension system cpy 19.6 kN-s/mStiffness of lateral primary suspension system kpz 2350 kN/mDamping of lateral primary suspension system cpz 0 kN/mStiffness of vertical secondary suspension system ksy 265 kN/mDamping of vertical secondary suspension system csy 45.1 kN-s/mStiffness of lateral secondary suspension system ksz 176 kN/mDamping of lateral secondary suspension system csz 39.2 kN-s/mNominal radius of wheel r0 0.455 m

∗For SKS series 300 vehicle model, extracted from Wakui et al. (1995) with somemodifications.



Table 10.2. The properties of track and bridge.

Item Notation Value

TrackYoung’s modulus Et 210 GPaPoisson’s ratio vt 0.3Per-unit-length mass1 mt 0.587 tPer-unit-length mass moment of inertia about x axis1 I∗t 0.383 t-m2

Sectional area2 At 1.54 × 10−5 m2

Flexural moment of inertia about y axis2 Ity 1.03 × 10−5 m4

Flexural moment of inertia about z axis2 Itz 6.12 × 10−5 m4

Half of gauge of rails la 0.75 mHalf of length of sleeper ld 1.3 mTransverse distance between center lines of track

and bridge lb 2.35 mPer-unit-area vertical stiffness of ballast on bridge k∗

bv1 92.3 MN/m2

Per-unit-area vertical damping of ballast on bridge c∗bv1 22.6 MN-s/m2

Per-unit-area lateral stiffness of ballast on bridge k∗bh1 3.85 MN/m2

Per-unit-area lateral damping of ballast on bridge c∗bh1 22.6 MN-s/m2

Per-unit-area longitudinal stiffness of ballast on bridge k∗bh1 3.85 MN/m2

Per-unit-area longitudinal damping of ballast on bridge c∗bh1 22.6 MN-s/m2

Per-unit-area vertical stiffness of ballast on approach k∗bv2 92.3 MN/m2

Per-unit-area vertical damping of ballast on approach c∗bv2 22.6 MN-s/m2

Per-unit-area lateral stiffness of ballast on approach k∗bh2 3.85 MN/m2

Per-unit-area lateral damping of ballast on approach c∗bh2 22.6 MN-s/m2

Per-unit-area longitudinal stiffness of ballast onapproach k∗

bh2 3.85 MN/m2

Per-unit-area longitudinal damping of ballast onapproach c∗bh2 22.6 MN-s/m2

Sleeper space D 0.6 m

BridgeYoung’s modulus Eb 28.25 GPaPoison’s ratio vb 0.2

Per-unit-length mass3 mb 41.74 tPer-unit-length mass moment of inertia about x axis3 I∗b 495 t-m2

Sectional area Ab 7.73 m2

Torsional moment of inertia about x axis Ibx 15.65 m4

Flexural moment of inertia about y axis Iby 74.42 m4

Flexural moment of inertia about z axis Ibz 7.84 m4

Bridge length L 30 mVertical distance between bridge deck and center of

torsion h 1.2 m

1. Including the masses of the rails and sleepers.2. For two rails.3. Including the masses of the bridge and ballast.



i.e., the vertical, lateral, rolling, yawing and pitching DOFs, whichare denoted as de = 〈ve we θe ϕe ψe〉T . Similarly, both the frontand rear bogies of the vehicle are assigned five DOFs each with re-spect to their center points, denoted as df = 〈vtf wtf θtf ϕtf ψtf 〉Tand dr = 〈vtr wtr θtr ϕtr ψtr〉T , respectively. Each wheelset is as-signed three DOFs, i.e., the vertical, lateral and rolling DOFs, atthe center of the axle, denoted as dwi = 〈vwi wwi θwi〉T , wherei = 1 ∼ 4. Thus, the upper part of the 3D vehicle model, thatis, the part not in direct contact with the rails, has a total of15 DOFs, as indicated by the vector du = 〈de df dr〉T ,and the wheel part has a total of 12 DOFs, which is denoted asdw = 〈dw1 dw2 dw3 dw4〉T . In addition, there exists a to-tal of eight contact points with two rails (see also Fig. 10.2), each ofwhich has a vertical and a lateral DOF. Let vci and hci respectivelydenote the vertical and lateral displacements of the ith contact point.The total contact-point DOFs for one vehicle can be denoted as

dc = 〈 vc1 hc1 vc2 hc2 vc3 hc3 vc4 hc4

vc5 hc5 vc6 hc6 vc7 hc7 vc8 hc8 〉T . (10.1)

Correspondingly, the contact forces are as follows:

f∗c = 〈V1 H1 V2 H2 V3 H3 V4 H4

V5 H5 V6 H6 V7 H7 V8 H8 〉T , (10.2)

where Vi and Hi respectively denote the vertical and lateral contactforces of the ith contact point. Because the lateral contact forcesacting on the two wheels of each wheelset are the same (i.e., H1 = H2,H3 = H4, . . .), the contact forces f∗

c can be rewritten in a compactform as

fc = 〈V1 H1 V2 V3 H3 V4 V5 H5 V6

V7 H7 V8 〉T . (10.3)

In this connection, the vector f∗c is referred to as a complete

form.



In a time-history analysis, one is interested in the behavior of astructure during the time step from t to t + ∆t. The equations ofmotion for the vehicle at time t + ∆t with partitions for the upper(noncontact) and wheel (contact) parts can be written as[

[muu] [muw]

[mwu] [mww]

] dudw

t+∆t

+[

[cuu] [cuw]

[cwu] [cww]

] dudw

t+∆t

+[

[kuu] [kuw]

[kwu] [kww]

] dudw

t+∆t

= fuefwe

t+∆t

+[

[lu]

[lw]

]fct+∆t , (10.4)

where [muu], [muw], [mwu] and [mww] denote the components of themass matrices, [cuu], [cuw], [cwu] and [cww] the components of thedamping matrices, and [kuu], [kuw], [kwu] and [kww] the componentsof the stiffness matrices of the vehicle, with the subscripts u and w

denoting the upper and wheel parts, respectively; fue and fwedenote the force vectors acting on the upper and wheel parts; [lu]and [lw] the associated transformation matrices; and fc denotes thecontact forces acting on the four wheelsets as stated above. Since thecontact forces are acting only on the wheel part, i.e., the wheelsets ofthe vehicle, the transformation matrix for the upper part [lu] shouldbe set to [0]. The transformation matrix for the wheel part [lw] isgiven as

[lw] =

[l][l]

[l][l]

12×12

(10.5)

with

[l] =

1 0 1

0 2 0−la 2r0 la

3×3

, (10.6)



where la denotes half of the axle length of the wheelset and r0 thenominal radius of the wheel (also see Table 10.1). All of the otherpartitioned matrices and vectors involved have been listed in Ap-pendix I, where W is the static axle load for each wheelset and g

is the acceleration of gravity. In the above derivation, the vehicle isassumed to vibrate from the static equilibrium position.

The wheelset displacements dw can be related to the contact-point displacements dc of the two rails with surface irregularitiesby the constraint equation,

dw = [Γ]dc + r . (10.7)

Here, [Γ] is a constraint matrix and r is a vector used to representthe track irregularities. For the vehicle and rail models considered inthis study, the constraint matrix [Γ] is

[Γ] =

[γ][γ]

[γ][γ]

12×16

(10.8)

with

[γ] =

12

012

0

0 1 0 0

− 12la

01

2la0

3×4

(10.9)

and the vector r represents the rail irregularities at four wheelsetpositions,

r = 〈 r1 r2 r3 r4 〉T , (10.10)

where

ri =⟨ 1

2[rv1(xi) + rv2(xi)] rh(xi)

12la

[−rv1(xi) + rv2(xi)]⟩T

,

i = 1 ∼ 4 . (10.11)



In Eq. (10.8), rv1(x) and rv2(x) denote the vertical deviations of thetwo rails, rh(x) denotes the deviations in lateral alignment of the tworails, and xi is the position of the ith wheelset of the vehicle at timet + ∆t.

One key step in analysis of the present interaction problem is tosolve for the contact forces fc existing between the moving (i.e.,the vehicle) and the nonmoving (i.e., the rails) parts of the train–bridge system. This can be done by following the procedure outlinedin Chapter 8 from Eqs. (8.5) to (8.12), namely, by solving the upper-part vehicle displacement vector du from the first row of Eq. (10.4)using the Newmark finite difference scheme, and then substitutingthe displacement vector du and derivatives into the second row ofEq. (10.4). This will result in the following expression for the contactforces fc at time t + ∆t:

fct+∆t = [mc][Γ]dct+∆t + [cc][Γ]dct+∆t + [kc][Γ]dct+∆t

+ [kc]rt+∆t + pct+∆t + qct

= [mc]dct+∆t + [cc]dct+∆t + [kc]dct+∆t

+ pct+∆t + qct , (10.12)

where the contact matrices [mc], [cc], [kc] and load vectors pct+∆t,qct have been given in Eq. (8.15). Since the procedure for derivingthe contact forces fc follows exactly the same lines as those pre-sented as Eqs. (8.5)–(8.16) in Chapter 8, no attempt will be madeherein to recapitulate the related details.

The contact force vector fc as presented in Eq. (10.12) relatesonly to the contact-point displacements dc, which can be aug-mented through introduction of a transformation matrix [λ] to yieldthe contact forces f∗

c in complete form as

f∗c t+∆t = [λ]fct+∆t

= [mc]dct+∆t + [cc]dct+∆t

+ [kc]dct+∆t + pct+∆t + qct , (10.13)



where

[λ] =

[δ][δ]

[δ][δ]

16×12

(10.14)

with

[δ] =

1 0 00 1 00 0 10 1 0

4×3

, (10.15)

the contact matrices are

[mc] = [λ][mc] ,

[cc] = [λ][cc] ,

[kc] = [λ][kc] ,

(10.16)

and the load vectors associated with the contact forces are

pct+∆t = [λ]pct+∆t ,

qct = [λ]qct .(10.17)

Through expansion of Eq. (10.13), one can obtain the vertical andlateral contact forces for each contact point as

Vi,t+∆t

= pc(2i−1),t+∆t + qc(2i−1),t

+16∑

j=1

(mc(2i−1)j dcj,t+∆t + cc(2i−1)j dcj,t+∆t + kc(2i−1)jdcj,t+∆t) ,

(10.18)



Hi,t+∆t = pc(2i),t+∆t + qc(2i),t

+16∑

j=1

(mc(2i)j dcj,t+∆t + cc(2i)j dcj,t+∆t + kc(2i)jdcj,t+∆t) ,

(10.19)

where i = 1 ∼ 8, looping over all the wheels.

10.4. Equations for the Rail and Bridge Elements

Each of the two tracks can be divided into three parts, i.e., thecentral, left, and right track sections, as shown in Fig. 10.1. Further-more, with reference to the top view in Fig. 10.3, the track locatedon the right side (when viewed along the positive x direction) ofthe bridge or approach will be referred to as Track A, and the trackon the left side as Track B. Each of the two track structures in thecentral track section is modeled as a rail element supported by theuniformly-distributed spring-dashpot units and in turn by the bridgeelements. The rail element used to represent each track (which con-sists of two rails) is of length l and will be referred to as the centralfinite rail (CFR) element. Similarly, each of the two track structureson the two side approaches will be modeled as a semi-infinite rail el-ement supported by the uniformly-distributed spring-dashpot units,which will be referred to either as the left semi-infinite rail (LSR)element or the right semi-infinite rail (RSR) element. As for thecentral section, since the two tracks (i.e., Tracks A and B) each lieon one side of the cross section of the bridge, the CFR element asso-ciated with one track is somewhat different from the other. However,the rail elements for the two tracks on the two-side approaches areexactly the same, because they sit on the same stationary roadbeds.It follows that no distinction need to be made for the LSR elementsused to represent the two tracks on the left approach. The same isalso true for the RSR elements on the right approach. Because theCFR elements for the two tracks (i.e., Tracks A and B) are different,they will be derived separately.



10.4.1. Central Finite Rail (CFR) Element for

Track A

As shown in Fig. 10.3(a), there are two tracks of which each consistsof two rails connected by the sleepers. In this study, a central finiterail (CFR) element of length l and modeled as conventional beam willbe used to represent the combination effect of the two rails of eachtrack, which are treated as two beams of negligible torsional rigidityconnected by the sleepers, which are modeled as rigid bars. Withreference to the top view in Fig. 10.3(a), we shall first discuss theCFR element associated with Track A, namely, the one connected tothe right side of the three-dimensional bridge element (when viewedalong the positive x direction) by a uniformly-distributed spring-dashpot-unit (ballast) layer. The nodal DOFs of the CFR elementused to represent the combined effect of the two rails on Track A arespecified at the two ends as

dA = 〈uA1 vA1 wA1 θA1 ϕA1 ψA1

uA2 vA2 wA2 θA2 ϕA2 ψA2 〉T . (10.20)

From here on, the subscript A of each quantity will be used to denotethat the quantity is associated with Track A. Correspondingly, thenodal DOFs of the bridge element are

db = 〈ub1 vb1 wb1 θb1 ϕb1 ψb1

ub2 vb2 wb2 θb2 ϕb2 ψb2 〉T . (10.21)

By the principle of virtual work, the equation of equilibrium for theCFR element can be written as∫ l

0EtAtu

′Aδu′

Adx +∫ l

0EtItzv

′′Aδv′′Adx +

∫ l

0EtItyw

′′Aδw′′

Adx

= −∫ l

0mt(uAδuA + vAδvA + wAδwA)dx

−∫ l

0I∗t θAδθAdx −

∫ l

0ct(uAδuA + vAδvA + wAδwA)dx



−∫ l

0c∗t θAδθAdx + 2ld

∫ l

0[pc(x) + pk(x)]δuAdx

+∫ l

0

∫ lb+ld

lb−ld

[qc(x, z) + qk(x, z)][δvTA − (z − lb)δθA]dzdx

+ 2ld∫ l

0[rc(x) + rk(x)]δwAdx + 〈δdA〉fA , (10.22)

where all the physical parameters involved have been defined in Ta-ble 10.2, in particular, all the quantities with subscript t denotequantities associated with the track or rails, δ denotes a variationalquantity, ct and c∗t are the material damping coefficients for the trans-lational and torsional motions of the rails, x is the local coordinate,0 x l, (uA, vA, wA) the axial (longitudinal), vertical and lateraldisplacements, fA denotes the external forces directly acting onthe nodal points of the rail element of Track A, i.e.,

fA = 〈FAx1 FAy1 FAz1 MAx1 MAy1 MAz1

FAx2 FAy2 FAz2 MAx2 MAy2 MAz2 〉T , (10.23)

and (pc, pk), (qc, qk) and (rc, rk) the unit axial, vertical and lateralinteraction forces arising from the relative motion of the rail andbridge elements, which can be defined as

pc(x) = c∗bh1[ub(x) − uA(x)] ,

pk(x) = k∗bh1[ub(x) − uA(x)] ,

(10.24)

qc(x, z) = c∗bv1[vb(x) − zθb(x) − vA(x) + (z − lb)θA(x)] ,

qk(x, z) = k∗bv1[vb(x) − zθb(x) − vA(x) + (z − lb)θA(x)] ,

(10.25)

rc(x) = c∗bh1[wb(x) + hθb(x) − wA(x)] ,

rk(x) = k∗bh1[wb(ξ) + hθb(x) − wA(x)] .

(10.26)

The subscripts c and k in the preceding three equations indicatethat the forces are damping or stiffness-related. The displace-ments (uA, vA, wA, θA) of the rail element associated with Track A



(consisting of two rails) can be related to the nodal DOFs as

uA(x) = 〈Nu〉dA ,

vA(x) = 〈Nv〉dA ,

wA(x) = 〈Nw〉dA ,

θA(x) = 〈Nθ〉dA .

(10.27)

Similarly, the displacements (ub, vb, wb, θb) of the bridge element canbe expressed as

ub(x) = 〈Nu〉db ,

vb(x) = 〈Nv〉db ,

wb(x) = 〈Nw〉db ,

θb(x) = 〈Nθ〉db ,

(10.28)

where the interpolation vectors for the axial, vertical, lateral, andtorsional displacements are given as follows:

〈Nu〉 = 〈N1 0 0 0 0 0 N2 0 0 0 0 0 〉T ,

〈Nv〉 = 〈 0 N3 0 0 0 N4 0 N5 0 0 0 N6 〉T ,

〈Nw〉 = 〈 0 0 N3 0 −N4 0 0 0 N5 0 −N6 0 〉T ,

〈Nθ〉 = 〈 0 0 0 N1 0 0 0 0 0 N2 0 0 〉T .

(10.29)

Here, N1 and N4 are linear, and N2, N3, N5 and N6 are cubic Her-mitian functions. It should be noted that the torsional strain energywas not included in Eq. (10.22) due to the assumption of negligibletorsional deformation in each rail.

Substituting the preceding displacement fields, i.e., Eqs. (10.27)and (10.28), into the virtual work equation in Eq. (10.22) yields theequation of motion for the CFR element on Track A as follows:

[mA]dA + [cA]dA + [kA]dA = fA + [cAb]db + [kAb]db .

(10.30)



Here, the mass, damping and stiffness matrices [mA], [cA] and [kA]are

[mA] = [mA0] ,

[cA] = [cA0] + [cA1] + [cA2] + [cA3] + [cA4] ,

[kA] = [kA0] + [kA1] + [kA2] + [kA3] + [kA4] ,

(10.31)

and the interaction matrices [cAb] and [kAb] are

[cAb] = [cAb1] + [cAb2] + [cAb3] + [cAb4] + [cAb5] + [cAb6] ,

[kAb] = [kAb1] + [kAb2] + [kAb3] + [kAb4] + [kAb5] + [kAb6] ,(10.32)

where [mA0], [cA0] and [kA0] are due to the inertial, material dampingand stiffness effects of the rail element. The matrices [mA0] and [kA0]have been listed in Appendix J. The material damping matrix [cA0]will be included in the system equation, along with those for the otherelements, based on the Rayleigh damping assumption, which willjust be skipped herein. The matrices [cA1] ∼ [cA4], [cAb1] ∼ [cAb6],[kA1] ∼ [kA4] and [kAb1] ∼ [kAb6] related to the interactions betweenthe bridge and CFR elements on Track A, can be given as

[cA1] = 2ldc∗bh1[ψu] ,

[cA2] = 2ldc∗bv1[ψv ] ,

[cA3] = l1c∗bv1[ψθ] ,

[cA4] = 2ldc∗bh1[ψw] ,

(10.33)

[cAb1] = 2ldc∗bh1[ψu] ,

[cAb2] = 2ldc∗bv1[ψv] ,

[cAb3] = −2ldlbc∗bv1[ψvθ] ,

[cAb4] = l1c∗bv1[ψθ] ,

[cAb5] = 2ldc∗bh1[ψw] ,

[cAb6] = 2hldc∗bh1[ψwθ] ,

(10.34)



[kA1] = 2ldk∗bh1[ψu] ,

[kA2] = 2ldk∗bv1[ψv ] ,

[kA3] = l1k∗bv1[ψθ] ,

[kA4] = 2ldk∗bh1[ψw] ,

(10.35)

[kAb1] = 2ldk∗bh1[ψu] ,

[kAb2] = 2ldk∗bv1[ψv] ,

[kAb3] = −2ldlbk∗bv1[ψvθ] ,

[kAb4] = l1k∗bv1[ψθ] ,

[kAb5] = 2ldk∗bh1[ψw] ,

[kAb6] = 2hldk∗bh1[ψwθ] ,

(10.36)

where l1 = (l3b2 − l3b1 − 6ls · l2b )/3, lb2 = lb + ld, lb1 = lb − ld and thematrices [ψu], [ψv], [ψw], [ψθ], [ψvθ] and [ψwθ] are defined as follows:

[ψu] =∫ l

0Nu〈Nu〉dx ,

[ψv] =∫ l

0Nv〈Nv〉dx ,

[ψw] =∫ l

0Nw〈Nw〉dx ,

[ψθ] =∫ l

0Nθ〈Nθ〉dx ,

[ψvθ] =∫ l

0Nv〈Nθ〉dx ,

[ψwθ] =∫ l

0Nw〈Nθ〉dx .

(10.37)

The results for all the matrices in Eq. (10.37) can be found inAppendix J.




Track B

The CFR element for Track B, that is, the one interacting with theleft side of the cross sections of the bridge element (when viewedalong the positive x direction), is also shown in Fig. 10.3(a). Thenodal DOFs of the CFR element are denoted as follows:

dB = 〈uB1 vB1 wB1 θB1 ϕB1 ψB1

uB2 vB2 wB2 θB2 ϕB2 ψB2 〉T , (10.38)

in which the subscript B denotes Track B. Similarly, by the principleof virtual work, the equation of equilibrium for the CFR element onTrack B can be written as

∫ l

0EtAtu

′Bδu′

Bdx +∫ l

0EtItzv

′′Bδv′′Bdx +

∫ l

0EtItyw

′′Bδw′′

Bdx

= −∫ l

0mt(uBδuB + vBδvB + wBδwB)dx

−∫ l

0I∗t θBδθBdx −

∫ l

0ct(uBδuB + vBδvB + wBδwB)dx

−∫ l

0c∗t θBδθBdx + 2ld

∫ l

0[pc(x) + pk(x)]δuBdx

+∫ l

0

∫ ld−lb

−ld−lb

[qc(x, z) + qk(x, z)][δvTB − (z + lb)δθB ]dzdx

+ 2ld∫ l

0[rc(x) + rk(x)]δwBdx + 〈δdB〉fB , (10.39)

where fB denotes the external nodal forces of the rail element, i.e.,

fB = 〈FBx1 FBy1 FBz1 MBx1 MBy1 MBz1

FBx2 FBy2 FBz2 MBx2 MBy2 MBz2 〉T . (10.40)



The unit axial, vertical and lateral interaction forces (pc, pk),(qc, qk) and (rc, rk) between the rail and bridge elements are asfollows:

pc(x) = c∗bh1[ub(x) − uB(x)] ,

pk(x) = k∗bh1[ub(x) − uB(x)] ,

(10.41)

qc(x, z) = c∗bv1[vb(x) + zθb(x) − vB(x) + (z + lb)θB(x)] ,

qk(x, z) = k∗bv1[vb(x) − zθb(x) − vB(x) + (z + lb)θB(x)] ,

(10.42)

rc(x) = c∗bh1[wb(x) + hθb(x) − wB(x)] ,

rk(x) = k∗bh1[wb(x) + hθb(x) − wB(x)] .

(10.43)

The displacements (uB , vB , wB , θB) of the rail element on Track B

can be related to the element nodal DOFs as

uB = 〈Nu〉dB ,

vB = 〈Nv〉dB ,

wB = 〈Nw〉dB ,

θB = 〈Nθ〉dB .

(10.44)

Here, the interpolation functions 〈Nu〉, 〈Nv〉, 〈Nw〉 and 〈Nθ〉 are thesame as those given previously for the CFR element for Track A. Us-ing Eqs. (10.41)–(10.44), one obtains from Eq. (10.39) after some ma-nipulations the equation of motion for the CFR element of Track B

as follows:

[mB ]dB + [cB ]dB + [kB ]dB = fB + [cBb]db + [kBb]db .

(10.45)

Clearly, Eq. (10.45) is identical in form to Eq. (10.30) for the CFRelement of Track A. The system matrices in Eq. (10.45) can be



given as

[mB] = [mB0] ,

[cB ] = [cB0] + [cB1] + [cB2] + [cB3] + [cB4] ,

[kB ] = [kB0] + [kB1] + [kB2] + [kB3] + [kB4] ,

(10.46)

and the interaction matrices as

[cBb] = [cBb1] + [cBb2] + [cBb3] + [cBb4] + [cBb5] + [cBb6] ,

[kBb] = [kBb1] + [kBb2] + [kBb3] + [kBb4] + [kBb5] + [kBb6] ,(10.47)

where all the matrices on the right side are exactly the same as thecorresponding terms given in Eqs. (10.33)–(10.36), except for thematrices [cBb3] and [kBb3], which are given as

[cBb3] = −[cAb3] ,

[kBb3] = −[kAb3] .(10.48)

As can be seen, the difference between the CFR elements for Tracks A

and B originates mainly from interaction with the torsional defor-mation of the bridge element.

10.4.3. The Bridge Element

By the principle of virtual work, along with the displacement fieldsdefined previously in Eqs. (10.27) and (10.28), the equation of motionfor the bridge element can be derived as

[mb]db + [cb]db + [kb]db= fb + [cbA]dA + [kbA]dA

+ [cbB ]dB + [kbB ]dB , (10.49)



where fb denotes the external nodal forces,

fb = 〈Fbx1 Fby1 Fbz1 Mbx1 Mby1 Mbz1

Fbx2 Fby2 Fbz2 Mbx2 Mby2 Mbz2 〉T , (10.50)

and the mass, damping and stiffness matrices [mb], [cb] and [kb] aredefined as follows:

[mb] = [mb0] ,

[cb] = [cb0] + [cb1] + [cb2] + [cb3] + [cb4] + [cb5] + [cb6]

+ [cb7] + [cb8] + [cb9] + [c′b1] + [c′b2] + [c′b3] + [c′b4]

+ [c′b5] + [c′b6] + [c′b7] + [c′b8] + [c′b9] ,

[kb] = [kb0] + [kb1] + [kb2] + [kb3] + [kb4] + [kb5] + [kb6]

+ [kb7] + [kb8] + [kb9] + [k′b1] + [k′

b2] + [k′b3] + [k′

b4]

+ [k′b5] + [k′

b6] + [k′b7] + [k′

b8] + [k′b9] ,

(10.51)

where [mb0] and [kb0] can be found in Appendix J, and the matrices[cb1] ∼ [cb9], [kb1] ∼ [kb9] and [c′b1] ∼ [c′b9], [k′

b1] ∼ [k′b9] arise from the

interaction of Tracks A and B with the bridge, which can be given as

[cb1] = [c′b1] = 2ldc∗bh1[ψu] ,

[cb2] = [c′b2] = 2ldc∗bv1[ψv ] ,

[cb3] = [c′b3] = l2c∗bv1[ψθ] ,

[cb4] = [c′b4] = 2ldc∗bh1[ψw] ,

[cb5] = [c′b5] = 2h2ldc∗bh1[ψθ] ,

[cb6] = −[c′b6] = −2ldlbc∗bv1[ψvθ] ,

[cb7] = −[c′b7] = −2ldlbc∗bv1[ψθv] ,

[cb8] = [c′b8] = 2hldc∗bh1[ψwθ] ,

[cb9] = [c′b9] = 2hldc∗bh1[ψθw] ,

(10.52)



and

[kb1] = [k′b1] = 2ldk∗

bh1[ψu] ,

[kb2] = [k′b2] = 2ldk∗

bv1[ψv ] ,

[kb3] = [k′b3] = l2k

∗bv1[ψθ] ,

[kb4] = [k′b4] = 2ldk∗

bh1[ψw] ,

[kb5] = [k′b5] = 2h2ldk

∗bh1[ψθ] ,

[kb6] = −[k′b6] = −2ldlbk∗

bv1[ψvθ] ,

[kb7] = −[k′b7] = −2ldlbk∗

bv1[ψθv] ,

[kb8] = [k′b8] = 2hldk

∗bh1[ψwθ] ,

[kb9] = [k′b9] = 2hldk

∗bh1[ψθw] ,

(10.53)

where l2 = (l3b2 − l3b1)/3, [ψθv] = [ψvθ ]T and [ψθw] = [ψwθ]T , and[ψu], [ψv], [ψw], [ψθ], [ψvθ] and [ψwθ] are the same as those definedin Eq. (10.37). The matrices [cbA], [kbA], [cbB ] and [kbB ] originatefrom the interactions between the bridge and the two tracks, whichare identical to the matrices [cAb], [kAb], [cBb] and [kBb], respectively,i.e.,

[cbA] = [cAb] ,

[kbA] = [kAb] ,

[cbB ] = [cBb] ,

[kbB ] = [kBb] .

(10.54)

From the preceding equations, it is clear that the lateral and torsionalvibrations of the bridge are coupled, as implied by the terms involvingthe matrices [ψθw] and [ψwθ] in Eqs. (10.52) and (10.53), which canbe attributed to the fact that the torsional center of the box-girderbridge is located at a distance below the bridge deck on which thetwo tracks are lying. However, the vertical and torsional vibrations of



the bridge do not interact each other, although there also exist termsinvolving the matrices [ψvθ] and [ψθv] in Eqs. (10.52) and (10.53),due to the fact that all these terms will cancel each other, as can beobserved from the following relations: [cb6] = −[c′b6], [cb7] = −[c′b7],[kb6] = −[k′

b6], and [kb7] = −[k′b7].

10.4.4. Left Semi-Infinite Rail (LSR) Element for

Track A

As shown in Fig. 10.3(b), the two tracks (each consisting of two rails)extend to infinity on the left side. In this study, each of the two trackswill be represented by a left semi-infinite rail (LSR) element, whichis a semi-infinite beam element used to represent the combinationeffect of the two rails assumed to be of negligible torsional rigiditybut connected by rigid bars (i.e., sleepers). The LSR element hasonly a single node situated at the middle point between the startends of the two beams. The nodal displacements of the LSR elementassociated with Track A may be denoted as

dAl = 〈uA2 vA2 wA2 θA2 ϕA2 ψA2 〉T . (10.55)

Correspondingly, the nodal forces of the element are

fAl = 〈FAx2 FAy2 FAz2 MAx2 MAy2 MAz2 〉T . (10.56)

By the principle of virtual work, one can write

∫ 0

−∞EtAtu

′Aδu′

Adx +∫ 0

−∞EtItzv


∫ 0

−∞EtItyw

′′Aδw′′

Adx

= −∫ 0

−∞mt(uAδuA + vAδvA + wAδwA)dx

−∫ 0

−∞I∗t θAδθAdx −

∫ 0

−∞ct(uAδuA + vAδvA + wAδwA)dx

−∫ 0

−∞c∗t θAδθAdx − 2ld

∫ 0

−∞(c∗bh2uA + k∗

bh2uA)δuAdx



−∫ 0

−∞

∫ lb+ld

lb−ld

[qc(x, z) + qk(x, z)][δvA − (z − lb)δθA]dzdx

− 2ld∫ 0

−∞(c∗bh2wA + k∗

bh2wA)δwAdx + 〈δdAl〉fAl , (10.57)

where a quantity preceded by δ denotes a virtual displacement, x isthe local coordinate, −∞ < x 0, and the interaction forces qc(x, z),qk(x, z) are defined as

qc(x, z) = c∗bv2[vA(x) − (z − lb)θA(x)] ,

qk(x, z) = k∗bv2[vA(x) − (z − lb)θA(x)] .

(10.58)

The axial, vertical, lateral and torsional displacements of the elementcan also be related to the nodal displacements as

uA = 〈Nu〉dAl ,

vA = 〈Nv〉dAl ,

wA = 〈Nw〉dAl ,

θA = 〈Nθ〉dAl ,

(10.59)

where the interpolation vectors are

〈Nu〉 = 〈N1 0 0 0 0 0 〉T ,

〈Nv〉 = 〈 0 N3 0 0 0 N4 〉T ,

〈Nw〉 = 〈 0 0 N5 0 −N6 0 〉T ,

〈Nθ〉 = 〈 0 0 0 N2 0 0 〉T .

(10.60)

The interpolation functions, i.e., N1 ∼ N6, can be obtained from thestatic solution to the problem of two rigidly-connected beams restingon the Winkler foundation subjected to a unit force or moment,



that is,

N1 = eλux ,

N2 = eλθx ,

N3 = eλvx(cos λvx − sin λvx) ,

N4 =1λv

eλvx sin λvx ,

N5 = eλwx(cos λwx − sinλwx) ,

N6 =1

λweλwx sin λwx ,

(10.61)

where λu, λv, λw and λθ denote the longitudinal, vertical, lateral, andtorsional characteristic numbers of the twin beam-Winkler founda-tion system,

λu =√

2ldk∗bh2

EtAt, λv = 4

√ldk

∗bv2

2EtItz,

λw = 4

√ldk

∗bh2

2EtIty, λθ =

√2l3dk

∗bv2

3EtItz.

(10.62)

By using Eq. (10.61) and the definition of the displacement fields,Eq. (10.57) can be manipulated to yield the equation of motion forthe LSR element on Track A,

[ml]dAl + [cl]dAl + [kl]dAl = fAl , (10.63)

where the mass, damping and stiffness matrices [ml], [cl] and [kl] ofthe LSR element can be computed as

[ml] = [ml0] ,

[cl] = [cl0] + [cl1] + [cl2] + [cl3] + [cl4] ,

[kl] = [kl0] + [kl1] + [kl2] + [kl3] + [kl4] .

(10.64)



Here, the matrices [ml0] and [kl0] have been listed in Appendix J,and the other matrices are given as follows:

[cl1] = 2ldc∗bh2[ψu] ,

[cl2] = 2ldc∗bv2[ψv] ,

[cl3] = l1c∗bv2[ψθ] ,

[cl4] = 2ldc∗bh2[ψw] ,

(10.65)

[kl1] = 2ldk∗bh2[ψu] ,

[kl2] = 2ldk∗bv2[ψv] ,

[kl3] = l1k∗bv2[ψθ] ,

[kl4] = 2ldk∗bh2[ψw] .

(10.66)

The matrices [ψu], [ψv ], [ψθ] and [ψw] in the preceding equation havealso been listed in Appendix J.

10.4.5. Right Semi-Infinite Rail (RSR) Element for

Track A

Similar to the LSR element for Track A discussed in Sec. 10.4.4, theright semi-infinite rail (RSR) element is a semi-infinite beam elementused to simulate the combination effect of two rails with negligibletorsional rigidity connected by rigid bars (i.e., sleepers), having onenode situated at the middle point between the start ends of the tworails, as shown in Fig. 10.3(c). The nodal displacements for the RSRelement on Track A are

dAr = 〈uA1 vA1 wA1 θA1 ϕA1 ψA1 〉T . (10.67)


fAr = 〈FAx1 FAy1 FAz1 MAx1 MAy1 MAz1 〉T . (10.68)



The axial, vertical, lateral and torsional displacements of the elementcan be related to the nodal DOFs as:

uA = 〈Nu〉dAr ,

vA = 〈Nv〉dAr ,

wA = 〈Nw〉dAr ,

θA = 〈Nθ〉dAr ,

(10.69)

where the interpolation vectors 〈Nu〉, 〈Nv〉, 〈Nw〉 and 〈Nθ〉 are iden-tical in form to those for the LSR element. For the present case,the interpolation functions can also be determined from the staticsolution as

N1 = e−λux ,

N2 = e−λθx ,

N3 = e−λvx(cos λvx + sin λvx) ,

N4 =1λv

e−λvx sin λvx ,

N5 = e−λwx(cos λwx + sin λwx) ,

N6 =1

λwe−λwx sin λwx ,

(10.70)

where λu, λv, λw and λθ are the same as those given in Eq. (10.62).Following the same procedure as that for the LSR element, one canderive the equation of motion for the RSR element as:

[mr]dAr + [cr]dAr + [kr]dAr = fAr , (10.71)

where the mass, damping and stiffness matrices [mr], [cr] and [kr]are

[mr] = [mr0] ,

[cr] = [cr0] + [cr1] + [cr2] + [cr3] + [cr4] ,

[kr] = [kr0] + [kr1] + [kr2] + [kr3] + [kr4] .

(10.72)



Here, [mr0] and [kr0] have been given in Appendix J, and the matrices[cr1] ∼ [cr4] and [kr1] ∼ [kr4] are the same as those for the LSRelement, that is,

[cr1] = [cl1] ,

[cr2] = [cl2] ,

[cr3] = [cl3] ,

[cr4] = [cl4] ,

(10.73)

[kr1] = [kl1] ,

[kr2] = [kl2] ,

[kr3] = [kl3] ,

[kr4] = [kl4] .

(10.74)


Track B

The LSR element for Track B shown in Fig. 10.3(b) is exactly thesame as that for Track A, except that the DOFs should be replacedby

dBl = 〈uB2 vB2 wB2 θB2 ϕB2 ψB2 〉T . (10.75)


fBl = 〈FBx2 FBy2 FBz2 MBx2 MBy2 MBz2 〉T , (10.76)

The equation of motion for the LSR element on Track B is

[ml]dBl + [cl]dBl + [kl]dBl = fBl , (10.77)

where the matrices [ml], [cl] and [kl] have been given in Eq. (10.64).




Track B

The RSR element for Track B is also the same as that for Track A,as shown in Fig. 10.3(c), except that the nodal DOFs should bereplaced by

dBr = 〈uB1 vB1 wB1 θB1 ϕB1 ψB1 〉T . (10.78)


fBr = 〈FBx1 FBy1 FBz1 MBx1 MBy1 MBz1 〉T . (10.79)

The equation of motion for the present element is

[mr]dBr + [cr]dBr + [kr]dBr = fBr , (10.80)

where [mr], [cr] and [kr] are the same as those given in Eq. (10.72).Note that in a step-by-step time-history analysis, the equations

of motion for the CFR, LSR, RSR and bridge elements formulatedabove should be interpreted as those established for the deformedposition of the system at time t + ∆t.

10.5. VRI Element Considering Vertical and LateralContact Forces

Assume that at time t+∆t, the four wheelsets of a vehicle are actingsimultaneously at the e1, e2, e3 and e4th rail elements of Track A

or B. The rail elements and the wheelsets acting over them willbe collectively referred to as the vehicle–rails interaction (VRI) ele-ments, as they are directly interacting with each other. Consider theeith element that is acted upon by the vertical and lateral compo-nents of the (2i − 1)th and (2i)th contact forces, i.e., V(2i−1),t+∆t,H(2i−1),t+∆t, V2i,t+∆t, and H2i,t+∆t. The equation of motion forthe eith rail element of Track A at time t + ∆t may be written



as follows:

[mAi]dAit+∆t + [cAi]dAit+∆t + [kAi]dAit+∆t

= fAit+∆t + εi([cAb]dbi + [kAb]dbi) − fAcit+∆t ,

(10.81)

where [mAi] = [mA], [cAi] = [cA], [kAi] = [kA], dAi = dA, andεi = 1 for the case with the contact forces acting on the CFR element;[mAi] = [ml], [cAi] = [cl], [kAi] = [kl], dAi = dAl, and εi = 0 forthe LSR element; [mAi] = [mr], [cAi] = [cr], [kAi] = [kr], dAi =dAr, and εi = 0 for the RSR element; and fAci denotes thevector of equivalent nodal forces resulting from the action of the(2i − 1)th and (2i)th vertical and lateral contact forces V(2i−1),t+∆t,H(2i−1),t+∆t, V2i,t+∆t, and H2i,t+∆t:

fAcit+∆t = NRviV(2i−1),t+∆t + NR

wiH(2i−1),t+∆t

+ NLviV2i,t+∆t + NL

wiH2i,t+∆t , (10.82)

where NRv and NR

w denote the interpolation vectors for the verti-cal and lateral displacements of the right rail of the eith rail element(see Fig. 10.3, when viewed along the positive x direction), whichactually consists of two rails. These interpolation vectors vary ac-cording to the type of elements, i.e., CFR, RSR or LSR elements,to which the contact forces are acting. Similarly, NL

v and NLw

denote the interpolation vectors for the vertical and lateral displace-ments of the left rail of the eith element, which are also dependenton the type of elements. All the interpolation vectors mentioned hereare somewhat different from those for the rail element. For instance,the interpolation vectors NR

v , NRw , NL

v and NLw for the CFR

element are:

NRv = 〈0 N3 0 − laN1 0 N4 0 N5 0 − laN2 0 N6〉T ,

NRw = 〈0 0 N3 0 − N4 0 0 0 N5 0 − N6 0〉T ,

NLv = 〈0 N3 0 laN1 0 N4 0 N5 0 laN2 0 N6〉T ,

NLw = NR

w ,

(10.83)



and those for the LSR or RSR elements are

NRv = 〈 0 N3 0 −laN2 0 N4 〉T ,

NRw = 〈 0 0 N5 0 −N6 0 〉T ,

NLv = 〈 0 N3 0 laN2 0 N4 〉T ,

NLw = NR

w .

(10.84)

In Eqs. (10.83) and (10.84), the interpolation functions N1 ∼ N6

involved are identical to those defined for the CFR, LSR or RSRelements.

In Eq. (10.82), the subscript i associated with the vectorsNR

vi, NRwi, NL

vi and NLwi is used to indicate that they

should be evaluated at the contact position of the ith wheelset,i.e.,

NRvi = NR

v (xi) ,

NRwi = NR

w (xi) ,

NLvi = NL

v (xi) ,

NLwi = NL

w (xi) ,

(10.85)

where xi is the local coordinate of the position of the ith wheelseton the eith element. By using Eqs. (10.82), (10.18) and (10.19),Eq. (10.81) for the eith rail element of Track A can be rewritten asfollows:

[mAi]dAit+∆t + [cAi]dAit+∆t + [kAi]dAit+∆t

= fAit+∆t + εi([cAb]dbit+∆t + [kAb]dbit+∆t)

−4∑

j=1

([m∗cij ]drjt+∆t + [c∗cij]drjt+∆t + [k∗

cij ]drjt+∆t)

−p∗cit+∆t − q∗cit , (10.86)



where the asterisked matrices represent the linking action transmit-ted through the car body by the ejth element (under the jth wheelload) on the eith element (under the ith wheel load) and the inter-action between the right and left rails of the same track through thefour wheelsets of the vehicle considered,

[m∗cij ] =

4∑k=1

[Nk

i (

4∑l=1

mc[4(i−1)+k][4(j−1)+l]〈N lj〉)]

,

[c∗cij ] =4∑

k=1

[Nk

i (

4∑l=1

cc[4(i−1)+k][4(j−1)+l]〈N lj〉)]

,

[k∗cij ] =

4∑k=1

[Nk

i (

4∑l=1

kc[4(i−1)+k][4(j−1)+l]〈N lj〉)]

,

(10.87)

and the equivalent nodal loads are

p∗cit+∆t =4∑

k=1

(Nki pc[4(i−1)+k],t+∆t) ,

q∗cit =4∑

k=1

(Nki qc[4(i−1)+k],t) ,

(10.88)

where Nβα is defined as

Nβα =

NRvα = NR

v (ξα) , for β = 1 ,

NRwα = NR

w (ξα) , for β = 2 ,

NLvα = NL

v (ξα) , for β = 3 ,

NLwα = NL

w (ξα) , for β = 4 .

(10.89)

The equation of motion as given in Eq. (10.86) is the condensed equa-tion of motion for the VRI element of Track A, as all the relevantvehicle DOFs have been eliminated. Similarly, the condensed equa-tion of motion for the VRI element of Track B can be written as



follows:

[mBi]dBit+∆t + [cBi]dBit+∆t + [kBi]dBit+∆t

= fBit+∆t + εi([cBb]dbit+∆t + [kBb]dbit+∆t)

−4∑

j=1

([m∗cij ]dBjt+∆t + [c∗cij ]dBjt+∆t

+ [k∗cij]dBjt+∆t) − p∗cit+∆t − q∗cit , (10.90)

where [mBi] = [mB ], [cBi] = [cB ], [kBi] = [kB ], dBi = dB,and εi = 1 for the case with the contact forces acting on the CFRelement; [mBi] = [ml], [cBi] = [cl], [kBi] = [kl], dBi = dBl, andεi = 0 for the LSR element; [mBi] = [mr], [cBi] = [cr], [kBi] = [kr],dBi = dBr, and εi = 0 for the the RSR element, and [m∗

cij],[c∗cij ], [k∗

cij ], p∗cit+∆t, and q∗cit are the same as those defined inEqs. (10.87) and (10.88). All the asterisked matrices and load vec-tors involved in Eqs. (10.86) and (10.90) are time-dependent andshould be updated at each time step in an incremental analysis. Thecondensation process described above should be repeated until theinteraction effects of all the vehicles moving on Track A or B (orboth) have been included.

10.6. VRI Element Considering General ContactForces

For the case where the longitudinal (x-direction) contact forces orig-inate from acceleration or deceleration of the vehicles are considered,the contact force fAcit+∆t in Eq. (10.82) for Track A should bemodified as

fAcit+∆t

= NRviV(2i−1),t+∆t + NR

wiH(2i−1),t+∆t + NRuiQ(2i−1),t+∆t

+ NLviV(2i),t+∆t + NL

wiH(2i),t+∆t + NLuiQ(2i),t+∆t ,

(10.91)



where Q(2i−1),t+∆t and Q(2i),t+∆t denote the longitudinal contactforces exerted on the two rails by the ith wheelset, which can berelated to the vertical contact forces V(2i−1),t+∆t and V(2i),t+∆t interms of the frictional coefficient µi as

Q(2i−1),t+∆t = µiV(2i−1),t+∆t ,

Q(2i),t+∆t = µiV(2i),t+∆t .(10.92)

In Eq. (10.91), Nu denotes the interpolation vector for the axialdisplacement of the eith element, which varies according to the typeof rail elements, i.e., CFR, RSR or LSR elements, and NR

ui =NR

u (xi) and NLui = NL

u (xi) are evaluated at the position ofthe ith wheelset.

Considering the more general expression in Eq. (10.91) for thecontact forces, along with Eq. (10.92), one can derive a condensedequation of motion for the eith rail element of Track A or B that isidentical in form to Eq. (10.86) or (10.90), but with the asteriskedmatrices given as

[m∗cij ] =

4∑k=1

[Nk

i (

4∑l=1

mc[4(i−1)+k][4(j−1)+l]〈N lj〉)]

+2∑

m=1

[µiNm

i ′(

4∑l=1

mc[4(i−1)+m][4(j−1)+l]〈N lj〉)]

,

[c∗cij ] =4∑

k=1

[Nk

i (

4∑l=1

cc[4(i−1)+k][4(j−1)+l]〈N lj〉)]

+2∑

m=1

[µiNm

i ′(

4∑l=1

cc[4(i−1)+m][4(j−1)+l]〈N lj〉)]

,

[k∗cij ] =

4∑k=1

[Nk

i (

4∑l=1

kc[4(i−1)+k][4(j−1)+l]〈N lj〉)]

+2∑

m=1

[µiNm

i ′(

4∑l=1

kc[4(i−1)+m][4(j−1)+l]〈N lj〉)]

,

(10.93)



and the following for the nodal loads,

p∗cit+∆t =4∑

k=1

(Nki pc[4(i−1)+k],t+∆t)

+2∑

m=1

(µiNmi ′pc[4(i−1)+m],t+∆t) ,

q∗cit =4∑

k=1

(Nki qc[4(i−1)+k],t)

+2∑

m=1

(µiNmi ′qc[4(i−1)+m],t) ,

(10.94)

where Nβα has been defined in Eq. (10.89) and Nβ

α′ can be givenas follows:

Nβα′ =

NRuα = NR

u (ξα) , for β = 1 ,

NLuα = NL

u (ξα) , for β = 2 .(10.95)


In simulating the vehicle–rails–bridge interaction system, the parts ofthe rails of each track that are directly acted upon by the wheel loadsshould be modeled by the VRI elements. However, for the remainingparts of the rails that are not directly under the action of wheelloads, they should be modeled by the ordinary rail elements. Byassembling all the VRI elements, rail elements and bridge elementsfor the system considered, the structural equations at time t + ∆t

can be established:

[M ]Dt+∆t + [C]Dt+∆t + [K]Dt+∆t

= Ft+∆t − P ∗c t+∆t − Q∗

ct , (10.96)

where [M ], [C] and [K] are the mass, damping and stiffness ma-trices; D denotes the nodal DOFs of the entire system, D =



〈DA DB Db〉T , in particular, DA, DB and Db denotethe DOFs associated with Track A, Track B, and the bridge; F de-notes the corresponding external loads, F = 〈FA FB Fb〉T ,with FA, FB, Fb indicating the forces pertaining to Track A,Track B, and the bridge; and P ∗

c t+∆t and Q∗ct are the equivalent

contact forces in global coordinates.In establishing the system matrices, one first constructs the ma-

trices [M0], [C0] and [K0] for the railway bridge that is free of anyvehicle actions, that is,

[M0] =

[MA] 0 0

0 [MB ] 0

0 0 [Mb]

=

[ml] + [mr] +∑

[mA] 0 0

0 [ml] + [mr] +∑

[mB ] 0

0 0∑

[mb]

,

(10.97)

[C0] = [C00] + [C0b] = [C00] +

[CA] 0 −[CAb]

0 [CB] −[CBb]

−[CbA] −[CbB] [Cb]

= [C00] +

(4∑

i=1

[cli]

)+

(4∑

i=1

[cri]

)0 −

∑ 6∑i=1

[cAbi]

+

(∑ 4∑i=1

[cAi]

)

0

(4∑

i=1

[cli]

)+

(4∑

i=1

[cri]

)−∑ 6∑

i=1

[cBbi]

+

(∑ 4∑i=1

[cBi]

)

−∑ 6∑

i=1

[cbAi] −∑ 6∑

i=1

[cbBi]∑ 9∑

i=1

([cbi]+[c′bi])

,

(10.98)



[K0] =

(4∑

i=1

[kli]

)+

(4∑

i=1

[kri]

)0 −

∑ 6∑i=1

[kAbi]

+

(∑ 4∑i=1

[kAi]

)

0

(4∑

i=1

[kli]

)+

(4∑

i=1

[kri]

)−∑ 6∑

i=1

[kBbi]

+

(∑ 4∑i=1

[kBi]

)

−∑ 6∑

i=1

[kbAi] −∑ 6∑

i=1

[kbBi]∑ 9∑

i=1

([kbi]+[k′bi])

,

(10.99)

and then add to them the interaction effects resulting from the mov-ing vehicles via the VRI elements, as represented by the asteriskedterms given in Eqs. (10.87) and (10.93), to form the system matrices[M ], [C] and [K], as given below:

[M ] =

[MA] + [M∗c ] 0 0

0 [MB] + [M∗c ] 0

0 0 [Mb]

=

[MA] +

nA∑k=1

4∑

i=1

4∑j=1

[m∗cij ]

k

0 0

0 [MB] +

nB∑k=1

4∑

i=1

4∑j=1

[m∗cij ]

k

0

0 0 [Mb]

,

(10.100)



[C] = [C00] +

[CA] + [C∗c ] 0 −[CAb]

0 [CB] + [C∗c ] −[CBb]

−[CbA] −[CbB ] [Cb]

= [C00] +

[CA]+

nA∑k=1

4∑

i=1

4∑j=1

[c∗cij ]

k

0 −[CAb]

0 [CB]+

nB∑k=1

4∑

i=1

4∑j=1

[c∗cij ]

k

−[CBb]

−[CbA] −[CbB ] [Cb]

,

(10.101)

[K] =

[KA] + [K∗c ] 0 −[KAb]

0 [KB] + [K∗c ] −[KBb]

−[KbA] −[KbB] [Kb]

=

[KA] +

nA∑k=1

4∑

i=1

4∑j=1

[k∗cij ]

k

0 −[KAb]

0 [KB] +

nB∑k=1

4∑

i=1

4∑j=1

[k∗cij]

k

−[KBb]

−[KbA] −[KbB] [Kb]

,

(10.102)

where nA and nB denote the number of the vehicles comprising thetrains moving on Tracks A and B, respectively. Similarly, the equiv-alent contact forces P ∗

c t+∆t and Q∗ct are

P ∗c t+∆t =

nA∑k=1

(4∑

i=1

p∗cit+∆t

)k

nB∑k=1

(4∑

i=1

p∗cit+∆t

)k

0

, (10.103)



Q∗ct =

nA∑k=1

(4∑

i=1

q∗cit

)k

nB∑k=1

(4∑

i=1

q∗cit

)k

0

. (10.104)

Note that the subscript k, which indicates the kth vehicle of the trainmoving on Track A (or B), should be looped over from 1 to nA (ornB) in order to account for the vehicles effects.

At each time step, it is necessary to check whether a rail elementbecomes a VRI element and vice versa, and to update the entries ofthe system matrices and vectors, concerning the contribution of theasterisked terms or components in Eqs. (10.100)–(10.104), for theDOFs that are directly affected by the vehicle actions, according tothe change in position of the contact points. One advantage of thepresent approach is that the total number of DOFs of the VRBI sys-tem remains identical to that of the original railway bridge, while thesymmetry property of the original system is fully preserved. In addi-tion, the damping matrix [C00] of the railway bridge in Eq. (10.101)can be determined based on the Rayleigh damping assumption as

[C00] = α0[M0] + α1[K0] , (10.105)

where [M0] and [K0] are the mass and stiffness matrices of the railwaybridge, respectively, and the two coefficients α0 and α1 are

α0 =2ξω1ω2

ω1 + ω2,

α1 =2ξ

ω1 + ω2,

(10.106)

where ξ is the damping ratio, ω1 and ω2 are the first and secondfrequencies of natural vibration of the railway bridge. The New-mark β method with β = 0.25 and γ = 0.5 will be employed for



Start

Construct [M0],[C0b] and [K0], Eq. (10.97)–(10.99).

Input data for vehicles and railway bridge.

1. Perform eigenvalue analysis. 2. Determine [C00] using Eqs. (10.105) and (10.106). 3. Form [C0] = [C00] + [C0b].

1. Construct [mc],[cc] and [kc] using Eq. (8.14) of Chapter 8.

2. Form [ ]cm ,[ ]cc andck using Eq. (10.16).

t'= t +t

Compute Nvand Nh for each wheelset.

Form [ ]c t tp

+∆ and [ ]c t

q using Eq. (10.17).

1. Construct [m*cij],[c

*cij],[k

*cij],p*

cit+t

and q*cit, Eqs. (10.93) and (10.94).

2. Assemble [M*c],[C

*c],[K

*c],P*

ct+t

and Q*ct, Eqs. (10.100)–(10.104).

3. Construct global system matrices: [M]=[M0]+[M*

c], [C]=[C0]+[C*c] & [K]

=[K0]+[K*c], Eqs. (10.100)–(10.104).

Solve D∆ from condensed system equation: [M] D t+t+[C] D t+t+[K] D t+t =Ft+t

+P*ct+t+Q*

ct by Newmark’s method.

1. Obtain D t+t, D t+t and D t+t from

D∆ , and compute ud t+t, ud t+t

and ud t+t using Eqs. (10.4) and (8.7).

2. Compute Vi,t+t and Hi,t+t using Eqs. (10.18) and (10.19).

0u u ud d d= = =

0D D D= = =

Stop

Next time step

Fig. 10.4. Procedure for time-history analysis.

solving the system equations presented in Eq. (10.96). The proce-dure for incremental analysis of the three-dimensional vehicle–rails–bridge interaction system is summarized in the flow chart given inFig. 10.4.

10.8. Simulation of Track Irregularities

Track irregularities consist of deviations of rails from ideal geometryof track layout. As shown in Fig. 10.5, four types of track irregularity



vr

hr

cr2

gr

)( profilevertical

tyirregualrielevation

tyirregularialignment

)( levelcross

tyirregularitionsupereleva

tyirregularigauge

geometryideal

geometryideal

gaugehalfstandard

Fig. 10.5. Four types of track irregularity.



can be distinguished, i.e., elevation, alignment, superelevation andgauge irregularities (Fryba, 1996), which are caused mainly by wear,initial installation errors, degradation of support materials, improperclearances, bridge support or pier settlement and their combinations.Only the deviations in elevation, alignment and superelevation areconsidered in this study, which can be expressed as stationary pro-cesses in space, i.e., as random functions in terms of the longitudinalcoordinate x. In railway engineering practice, the track irregularityis frequently characterized by the one-sided power spectral density(PSD) function of the track geometry. The PSD functions used inthe study for the elevation irregularity (vertical profile), alignmentirregularity and superelevation irregularity (cross level) are given asfollows (Fries and Coffey, 1990):

Sv,a(Ω) =AvΩ2

c

(Ω2 + Ω2r)(Ω2 + Ω2

c)for elevation andalignment irregularities ,

Sc(Ω) =(AvΩ2

c/la)Ω2

(Ω2 + Ω2r)(Ω2 + Ω2

c)(Ω2 + Ω2s)

for superelevationirregularity ,

(10.107)

where Ω = 1/Lr denotes the spatial frequency (Hz) and Lr

is the length of the irregularity (m). Table 10.3 contains thevalues for the coefficients involved in Eqs. (10.107), which areequivalent to Classes 4, 5 and 6 of track classification used by the

Table 10.3. Track PSD model parameters.

Quality (FRA class) Very poor (4) Poor (5) Moderate (6)

Av(m) 2.39 × 10−5 9.35 × 10−6 1.50 × 10−6

Ωs(rad/m) 1.130 0.821 0.438Ωr(rad/m) 2.06 × 10−2 2.06 × 10−2 2.06 × 10−2

Ωc(rad/m) 0.825 0.825 0.825



Federal Railroad Administration (FRA) (Fries and Coffey, 1990).The track classes refer to track designations that range from 1 to6, with class 6 indicating the best and class 1 the worst. How-ever, the PSD function cannot be directly used in time-domainanalysis because of its frequency-based nature. To overcome thisproblem, the spectral representation method was implemented togenerate the vertical profile and alignment irregularity of the railsfrom the PSD functions as described in Eqs. (10.107). The rail pro-files generated will be included in the time-history analysis in thischapter.

By applying the spectral representation method, the deviations inthe vertical profile, horizontal alignment, and cross level (or superel-evation), i.e., rv(x), rh(x) and rc(x), respectively, of the two rails ofeach track can be written along the longitudinal axis x as (Claus andSchiehlen, 1998)

rv(x) =√

2N−1∑n=0

An cos(Ωnx + αn) ,

rh(x) =√

2N−1∑n=0

Bn cos(Ωnx + βn) ,

rc(x) =√

2N−1∑n=0

Cn cos(Ωnx + γn) ,

(10.108)

where N denotes the total number of discrete spatial frequen-cies considered, and Ωn is the nth discrete frequency, which iscomputed as

Ωn = n∆Ω = n(Ωu − Ωl)

N, (10.109)

where Ωu and Ωl respectively denote the uppermost and lowest fre-quencies considered, and n = 1, 2, . . . , N − 1. The coefficients An,



Bn and Cn in Eq. (10.108) are defined as

A0 = B0 = C0 = 0 ,

A1 = B1 =

√(1π

Sv,a(∆Ω) +46π

Sv,a(0))

∆Ω ,

C1 =

√(1π

Sc(∆Ω) +46π

Sc(0))

∆Ω ,

A2 = B2 =

√(1π

Sv,a(2∆Ω) +16π

Sv,a(0))

∆Ω ,

C2 =

√(1π

Sc(2∆Ω) +16π

Sc(0))

∆Ω ,

An = Bn =

√(1π

Sv,a(n∆Ω))

∆Ω ,

Cn =

√(1π

Sc(n∆Ω))

∆Ω ,

(10.110)

for n = 3, 4, . . . , N−1. The independent random phase angles αn, βn

and γn (n = 1, 2, . . . , N − 1) are uniformly-distributed in the range[0, 2π]. Note that the results generated in this way may not satisfythe requirements for the specified deviations in track geometry, andthus normalization on the final results is necessary. In this study, theresults computed for rv(x), rh(x) and rc(x) for FRA track Classes 4,5 and 6 are normalized so that their maximum deviations equal themaximum tolerable deviations specified in Table 10.4 for very poor,poor and moderate tracks according to the standards for high-speedtracks (Esveld, 1989). After the normalization, the vertical profileand alignment irregularities for the right and left rails of the track



Table 10.4. Maximum tolerable deviations for various railirregularities.

Quality (FRA class) Very poor (4) Poor (5) Moderate (6)

rv,max(mm) 4.05 3.38 2.70rh,max(mm) 5.10 4.25 3.40rc,max(mm) 1.50 1.25 1.00

can be computed as

rvr(x) = rv(x) − rc(x) ,

rvl(x) = rv(x) + rc(x) ,

rhr(x) = rhl(x) = rh(x) .

(10.111)

In the simulation, the following are assumed: Ωl = 0.0209 rad/m,Ωu = 12.5664 rad/m and N = 3540. The normalized vertical profileand alignment irregularities for the right and left rails of the track forFRA Classes 4–6 were shown in Figs. 10.6(a)–10.6(c), respectively.

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60 70 80 90 10

Position Along Rails (m)

Tra

ck G

eom

etr

y V

ariations (m

m)

0

vertical profile for right rail

vertical profile for left rail

alignment irregularity for two rails

moderate track quality

(FRA class 6)

(a)

Fig. 10.6. Track irregularity for: (a) class 6, (b) class 5, and (c) class 4.



-10

-8

-6

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60 70 80 90 10


Tra

ck G

eom

etr

y V

ariations (m

m)

0




poor track quality

(FRA class 5)

(b)

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60 70 80 90 10


Tra

ck G

eom

etr

y V

ariations (m

m)

0




very poor track quality

(FRA class 4)

(c)




0=deviation 0=deviation0=deviation 0=deviationlengthsampling

datumprofile

irregular

antianti anti anti

)(aCase

(a)

0=deviation 0=deviation0≠deviation0≠deviationlengthsampling

datumprofile

irregular

anti anti

)(bCase

(b)

Fig. 10.7. Generation of continuous irregular profile from finite length with: (a)zero deviations and (b) nonzero deviations.

Note that if the length for the irregular track needed in analysisexceeds the sampling length, which is selected as 100 m in this study,the same irregularities should be used repeatedly in certain manneruntil the entire length of the track is fully covered, as illustrated inFig. 10.7.

10.9. Verification of the Proposed Theory andProcedure

In this section, a typical vehicle–rails–bridge interaction problem willbe analyzed both by the present 3D procedure and the 2D procedurepresented in Chapter 8. The results obtained by the former will



L

Fig. 10.8. Simply-supported bridge subjected to a moving vehicle.

then be compared with those by the latter to validate the modeland formulation developed in this chapter. Figure 10.8 shows asimply-supported single-track bridge of length L = 30 m traveled bya moving vehicle. For comparison, only a single track located alongthe centerline of the bridge is considered in this example. The dataadopted for the vehicle are extracted from those for the Shinkansen(SKS) train (Wakui et al., 1995), as listed in Table 10.1. We chooseto use this car model since it is the only one of which the funda-mental properties are readily available to us. The rails are assumedto be of the UIC-60 type and the sleepers to be made of concrete.With the properties of the rails and sleepers given, the data specifiedfor the track have been listed in Table 10.2. The bridge is made ofprestressed concrete and has constant cross sections of uniform prop-erties. The data adopted for the bridge were also listed in Table 10.2.

In the 2D analysis, the following data, equivalent to those for the3D model, are used for the vehicle: Mc = 41.75 t, Ic = 2080 t-m4,Mt = 3.04 t, It = 3.93 t-m4, Mw = 1.78 t, kp = 1180 kN/m, cp =39.2 kN-s/m, ks = 530 kN/m, cs = 90.2 kN-s/m, lc = 8.75 m,lt = 1.25 m, and ls = 3.75 m; the following for the track (i.e., rails)and ballast: Et = 210 GPa, vt = 0.3, It = 6.12 × 10−5 m4, mt =0.587 t/m, At = 1.5374 × 10−5 m2, kbv1 = kbv2 = 240 MPa andcbv1 = cbv2 = 58.8 kN-s/m2; and the following for the bridge: Eb =28.25 GPa, vb = 0.2, Ib = 7.839 m4, Ab = 8.730 m2, mb = 41.74 t/mand L = 30 m. In each analysis, the vehicle is assumed to pass



through the bridge from x = 0 to 100 m with a constant speedv = 100 m/s and the bridge is modeled as 10 beam elements.

The first and second frequencies of the railway bridge simulatedby the 3D model have been obtained respectively as 25.09 and100.34 rad/s through an independent eigenvalue analysis, which areidentical to those obtained by the 2D model. The results for themidspan vertical response of the bridge obtained by the two modelshave been plotted in Figs. 10.9 and 10.10, and those of the trackin Figs. 10.11 and 10.12. The acceleration response of the verticaland pitching motions of the vehicle was also plotted in Figs. 10.13and 10.14, respectively. From these figures, it is observed that theresponses obtained by the two models agree very well. As a result,the reliability of the present formulation and procedure in dealingwith the 3D vehicle–rails–bridge interaction problems is confirmed.

In these figures, the responses identified as Track A for a double-track bridge, which were obtained using the present procedure, as-suming the same properties and lb = 2.35 m (see Fig. 10.3 for def-inition) for each track are also shown. As can be seen, the resultsobtained for the case of a double-track bridge are very close to those

-1.2E-03

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

0 0.5 1 1.5 2 2.5


Bridge M

idspan V

ert

ical D

ispla

cem

ent (

m)

3

3D (single track)

3D (double tracks - Track A)

2D

Fig. 10.9. Midspan vertical displacement of bridge.



-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3


Brid

ge

Mid

sp

an

Ve

rtic

al A

cce

lera

tio

n

(m/s

^2

)

3D (single track)


2D

Fig. 10.10. Midspan vertical acceleration of bridge.

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

0 0.5 1 1.5 2 2.5


Tra

ck M

idspan V

ert

ical D

ispla

cem

ent (

m)

3

3D (single track)


2D

Fig. 10.11. Midspan vertical displacement of track.



-8

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2 2.5


Tra

ck M

idsp

an

Ve

rtic

al A

cce

lera

tio

n

(m/s

^2

)

3

3D (single track)


2D

Fig. 10.12. Midspan vertical acceleration of track.

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 0.5 1 1.5 2 2.5 3


Ve

hic

le B

od

y V

ert

ica

l A

cce

lera

tio

n (m

/s^2

)

3D (single track)


2D

Fig. 10.13. Vertical acceleration of vehicle.



-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 0.5 1 1.5 2 2.5


Vehic

le B

ody P

itchin

g A

ccele

ration

(ra

d/s

^2

)

3

3D (single track)


2D

Fig. 10.14. Rotational acceleration (pitching) of vehicle.

for the case of a single-track bridge, an indication of the reliabilityof the present model and procedure in dealing with the interactionsbetween the double-track bridge and running vehicles.

10.10. Dynamic Characteristics of Train–Rails–BridgeSystems

In this section, the three-dimensional modeling scheme developed inprevious sections for the vehicle–rails–bridge interaction system willbe employed to study the dynamic interactions between the railwaybridge and moving trains under various conditions.

10.10.1. Properties of the Railway Vehicles and

Bridge

The data adopted for the train car have been listed in Table 10.1,which are modified from those for the Series 300 rail cars used inJapan’s Shinkansen system (Wakui et al., 1995). The rails are as-sumed to be of UIC-60 type and the sleepers made of monolithic



concrete blocks. The bridge is a prestressed concrete bridge of con-stant cross sections and uniform properties. The properties of therails, sleepers, ballast layer and bridge have been made available inTable 10.2.

10.10.2. Natural Frequencies of the Railway Vehicles

and Bridge

The natural frequencies and vibration modes of the vehicle obtainedby performing an eigenvalue analysis were listed in Table 10.5. Itcan be seen from the first and fifth modes that the lateral and rollingmotions of the vehicle are coupled with each other, as implied in themodeling of vehicle in Sec. 10.3. In addition, the first five modes aremainly related to the motion of the vehicle body, indicating that thevehicle body will be more sensitive to low-frequency excitations thanto high-frequency excitations, which can be attributed primarily tothe isolation effect of the suspension systems installed on the vehi-cle. The first five natural frequencies of the railway bridge were alsoshown in Table 10.5, along with the attributes of the correspond-ing vibration modes. As can be seen, the modes corresponding tolower frequencies (i.e., the first and second frequencies) are typicalof vertical vibrations, due to the relatively small vertical rigidity ofthe bridge, compared with the lateral and torsional rigidities. As aresult, it can be expected that the vertical vibration of the bridgeplays a role more important than the lateral and torsional vibrationsunder the action of the moving trains.

10.10.3. Dynamic Interactions Between the Train and

Bridge

In this subsection, the dynamic interactions between the train andthe railway bridge will be investigated. The train is assumed to con-sist of 15 identical vehicles and is allowed to pass the bridge throughTrack A (see Fig. 10.3). Two running cases with train speeds ofv = 60 m/s (216 km/h) and 100 m/s (360 km/h) are considered. Ineach case, the train starts at position x0 = −50 m relative to thebridge and stops at position xf = 430 m. Track irregularity is not



Table 10.5. Natural frequencies and vibration modes for vehicle andbridge.

Vehicle

Mode 1 2 3 4 5

f (Hz) 0.504 0.723 0.775 0.895 1.423v∗

e 0 0.9666 0 0 0we −0.2090 0 0 0 −0.2090θe 0.9448 0 0 0 0.9448ϕe 0 0 0.7079 0 0ψe 0 0 0 0.3919 0vtf 0 0.1812 0 0.6505 0wtf −0.0015 0 −0.2971 0 −0.0015θtf 0.1783 0 −0.4015 0 0.1783ϕtf 0 0 0 0 0ψtf 0 0 0 0 0vtr 0 0.1812 0 −0.6505 0wtr −0.0015 0 0.2971 0 −0.0015θtr 0.1783 0 0.4015 0 0.1783ϕtr 0 0 0 0 0ψtr 0 0 0 0 0

Railway bridge

Mode 1 2 3 4 5

f (Hz) 3.97 15.86 20.16 20.31 20.75ω (rad/s2) (24.93) (99.64) (126.69) (127.62) (130.37)

Attribute Vertical vertical longitudinallateral- lateral-torsional torsional

∗The symbols denote the degrees of freedom of the car body and bogiesof the vehicle (see Sec. 10.3).

considered here, namely, smooth and straight track geometry is as-sumed. The railway bridge is modeled as 10 elements and the timeincrement is selected to be ∆t = 0.005 s.

The midspan vertical, lateral and torsional displacements of thecenterline of the bridge were plotted in Figs. 10.15(a)–10.15(c). Themidspan vertical, lateral and torsional displacements of Track A wereplotted in Figs. 10.16(a)–10.16(c). As can be seen, the bridge and thetrack vibrate periodically as the vehicles pass sequentially throughthe bridge. The displacement responses of the track are nearly the



-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

-3 0 3 6 9 12


Bri

dg

e M

idsp

an

Ve

rtic

al D

isp

lace

me

nt

(m

)

15

v=100 m/s

v=60 m/s

(a)

-1.0E-08

0.0E+00

1.0E-08

2.0E-08

3.0E-08

4.0E-08

5.0E-08

6.0E-08

7.0E-08

8.0E-08

9.0E-08

-3 0 3 6 9 12


Bri

dg

e M

idsp

an

La

tera

l D

isp

lace

me

nt

(m

)

15

v=100 m/s

v=60 m/s

(b)

Fig. 10.15. Midspan displacements of the bridge: (a) vertical, (b) lateral, and(c) torsional.



-2.0E-06

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

1.4E-05

-3 0 3 6 9 12


Bri

dg

e M

idsp

an

To

rsio

na

l A

ng

le

(ra

d)

15

v=100 m/s

v=60 m/s

(c)


-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

-3 0 3 6 9 12


Tra

ck M

idsp

an

Ve

rtic

al D

isp

lace

me

nt

(m

)

15

v=100 m/s (track A)

v=60 m/s (track A)

(a)

Fig. 10.16. Midspan displacements of the track: (a) vertical, (b) lateral, and (c)torsional.



-1.0E-06

1.0E-06

3.0E-06

5.0E-06

7.0E-06

9.0E-06

1.1E-05

1.3E-05

1.5E-05

-3 0 3 6 9 12


Tra

ck M

idsp

an

La

tera

l D

isp

lace

me

nt

(m)

15

v=100 m/s (track A)

v=60 m/s (track A)

(b)

-2.0E-06

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

1.2E-05

-3 0 3 6 9 12


Tra

ck M

idsp

an

To

rsio

na

l A

ng

le

(ra

d)

15

v=100 m/s (track A)

v=60 m/s (track A)

(c)




same as that of the bridge, except for the lateral displacement, whichis much larger than that of the bridge (the latter is virtually zero).This can be attributed mainly to the relatively larger vertical, butsmaller lateral stiffnesses of the ballast. In contrast, the accelerationof Track A, in particular, the vertical acceleration (not shown here),is much larger than that of the bridge due to the direct action of thevehicular loads. In Figs. 10.17(a)–10.17(f), the acceleration responsesof the first and the fifteenth (last) vehicles were presented, revealingthat the last vehicle tends to vibrate more violently than the first one,particularly for the vertical and pitching motions. The implication isthat the dynamic response of the last car of a train should be givenmuch more attention concerning the safety and riding comfort of thetrain moving over a bridge.

In general, the vertical responses of the bridge and track for v =100 m/s are much larger than those for v = 60 m/s, mainly owing tothe occurrence of resonance between the train and the railway bridge,as was discussed in Chapter 9. The resonance phenomenon is furtherconfirmed by the fact that the amplitude of the vertical vibration forv = 100 m/s increases with the number of vehicles passing throughthe bridge. The lateral and torsional responses for the same trainspeed, however, do not display similar resonance phenomenon. Itshould be noted that the lateral and torsional responses of the bridgefor v = 60 m/s are slightly larger than those for v = 100 m/s dueto the occurrence of secondary lateral and torsional resonances atthat speed. In general, the acceleration responses (especially thevertical and pitching responses) of the first and the fifteenth vehiclesfor v = 100 m/s are larger than those for v = 60 m/s, also due tothe vertical train–railway bridge resonance at v = 100 m/s. Thedynamic characteristics of the train and bridge under different trainspeeds will be studied later.

10.10.4. Train–Rails–Bridge Interaction Considering

Track Irregularities

It is desirable to study the influence of track irregularities, i.e.,vertical profile, alignment irregularity, and cross level, on the



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Ve

hic

le V

ert

ica

l A

cce

lera

tio

n

(m/s

^2

)

15

v=100 m/s (1st car)

v=100 m/s (15th car)

v=60 m/s (1st car)

v=60 m/s (15th car)

(a)

-3.0E-04

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Ve

hic

le L

ate

ral A

cce

lera

tio

n

(m/s

^2

)

15

v=100 m/s (1st car)


v=60 m/s (1st car)

v=60 m/s (15th car)

(b)

Fig. 10.17. Acceleration responses of the vehicles: (a) vertical, (b) lateral, (c)rolling, (d) yawing, and (e) pitching.



-5.0E-04

-4.0E-04

-3.0E-04

-2.0E-04

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3.0E-04

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Ve

hic

le R

olli

ng

Acce

lera

tio

n

(ra

d/s

^2

)

15

v=100 m/s (1st car)


v=60 m/s (1st car)

v=60 m/s (15th car)

(c)

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-4.0E-05

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-3 0 3 6 9 12


Ve

hic

le Y

aw

ing

Acce

lera

tio

n

(ra

d/s

^2

)

15

v=100 m/s (1st car)


v=60 m/s (1st car)

v=60 m/s (15th car)

(d)




-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

-3 0 3 6 9 12


Ve

hic

le P

itch

ing

Acce

lera

tio

n

(ra

d/s

^2

)

15

v=100 m/s (1st car)


v=60 m/s (1st car)

v=60 m/s (15th car)

(e)


train–rails–bridge interactions. Furthermore, for the purpose of safeoperation and track maintenance, it is necessary to investigate thedynamic behaviors of trains and railway bridges under various trackconditions. The deviations in ideal geometry of the two rails of thetrack that have been generated in Figs. 10.6(a)–10.6(c) consideringsimultaneously the three types of irregularity (i.e., vertical profile,alignment irregularity, and cross level) for three different track qual-ities (FRA Classes 4, 5 and 6) will be adopted. The conditions forthe time-history analysis in this subsection are the same as those ofthe preceding subsection.

The time-history responses of the bridge, Track A, and the fif-teenth vehicle of the train for the three classes of track were shown inFigs. 10.18–10.20, respectively, along with those for case with smoothand straight track geometry. From these figures, it is observed thatthe vertical profile irregularity has only marginal influence on theresponses of the bridge and the track. On the other hand, the align-ment irregularity can significantly aggravate the bridge and track



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Bri

dg

e M

idsp

an

Ve

rtic

al D

isp

lace

me

nt

(m

) very poor (FRA class 4)

poor (FRA class 5)

moderate (FRA class 6)

ideal geometry

(a)

-1.0E-06

-8.0E-07

-6.0E-07

-4.0E-07

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0.0E+00

2.0E-07

4.0E-07

6.0E-07

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1.2E-06

1.4E-06

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Bri

dg

e M

idsp

an

La

tera

l D

isp

lace

me

nt

(m

) very poor (FRA class 4)

poor (FRA class 5)


ideal geometry

(b)

Fig. 10.18. Bridge displacements for different track qualities: (a) vertical, (b)lateral, and (c) torsional.



-1.0E-06

0.0E+00

1.0E-06

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9.0E-06

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Bri

dg

e M

idsp

an

To

rsio

na

l A

ng

le

(ra

d)

very poor (FRA class 4)

poor (FRA class 5)


ideal geometry

(c)


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-3 -1 1 3 5 7 9 11 13


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sp

an

Ve

rtic

al D

isp

lace

me

nt

of

Tra

ck A

(m)

15


poor (FRA class 5)


ideal geometry

(a)

Fig. 10.19. Track displacements for different track qualities: (a) vertical, (b)lateral, and (c) torsional.



-6.0E-04

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3.0E-04

-3 -1 1 3 5 7 9 11 13


Mid

span L

ate

ral D

ispla

cem

ent of T

rack A

(m)

15


poor (FRA class 5)


ideal geometry

(b)

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Mid

sp

an

To

rsio

na

l A

ng

le o

f T

rack A

(ra

d)


poor (FRA class 5)


ideal geometry

(c)




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15

th C

ar

Ve

rtic

al A

cce

lera

tio

n (m

/s^2

)


poor (FRA class 5)


ideal geometry

(a)

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15

th C

ar

La

tera

l A

cce

lera

tio

n

(m/s

^2

)


poor (FRA class 5)


ideal geometry

(b)

Fig. 10.20. Vehicle responses for different track qualities: (a) vertical, (b) lateral,and (c) rolling.



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th C

ar

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llin

g A

cce

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n

(ra

d/s

^2

)


poor (FRA class 5)


ideal geometry

(c)


lateral vibrations, and the cross-level irregularity can increase signif-icantly the torsional response of the track. As for the vehicle, theirregular vertical profiles can drastically increase the level of verticaland pitching vibrations, and the alignment irregularity can greatlyamplify the lateral, rolling and yawing motions. The cross level canalso influence the lateral, rolling and yawing vibrations of the vehicle,but the degree of influence is less than that of the alignment irregu-larity. In general, the vehicles appear to be much more sensitive totrack irregularities than the track and bridge.

In addition, the vibrations of the train–rails–bridge system for atrack with better quality are much smaller than those for a trackwith poorer quality. The dynamic characteristics of vibrations forthe three track classes are similar in trend, but different in magni-tude, due to the fact that the same property of spatial frequency isimplied by the three classes of irregularity. Figure 10.21 shows thevariation in the contact forces between the rails and the two frontand two rear wheels of the 15th car (see Fig. 10.2 for numbering of



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Conta

ct F

orc

es for

1st W

heel of 15th

Car

(kN

)


poor (FRA class 5)


ideal geometry

vertical contact force

lateral contact force

(a)

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0

10

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30

40

50

60

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90

-3 -1 1 3 5 7 9 11 13


Co

nta

ct

Fo

rce

s f

or

2n

d W

he

el o

f 1

5th

Ca

r

(kN

)

15


poor (FRA class 5)


ideal geometry



(b)

Fig. 10.21. Contact forces for the front and rear wheels of 15th car due to varioustrack irregularities: (a) 1st wheel, (b) 2nd wheel, (c) 7th wheel, and (d) 8th wheel.



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90

-3 -1 1 3 5 7 9 11 13 15


Conta

ct F

orc

es for

7th

Wheel of 15th

Car

(kN

)


poor (FRA class 5)


ideal geometry



(c)

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Conta

ct F

orc

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Wheel of 15th

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(kN

)


poor (FRA class 5)


ideal geometry



(d)




the wheels). As can be seen, the vertical contact forces vary lessrapidly compared with the lateral contact forces. Both the verticaland lateral contact forces vary more drastically as the vehicle is actingon the bridge. In Table 10.6, the maximum, mean, and standarddeviation of the vertical and lateral contact forces for the four wheelsof the vehicle under the three track conditions were listed. In thetable the measured values for the lateral force (Ma and Zhu, 1998)are also shown. It can be seen that the maximum values obtainedby the present analysis for the axle lateral force agree well with themeasured ones.

Table 10.6. Maximum, mean, and standard deviation for vertical andlateral contact forces.

Vertical contact force∗ (kN)

Quality Very poor Poor Moderate(FRA class) (4) (5) (6)

Item∗∗ Z µ σ ζ µ σ ζ µ σ

1st Wheelset 148.6 135.2 1.2 148.6 135.2 1.1 148.7 135.2 1.02nd Wheelset 144.1 135.2 1.2 144.2 135.2 1.0 144.3 135.1 0.93rd Wheelset 150.7 135.2 1.2 150.5 135.2 1.1 150.4 135.2 1.04th Wheelset 145.8 135.1 1.1 145.7 135.1 1.0 145.8 135.1 0.9

Lateral contact force∗ (kN)

Quality Very poor Poor Moderate(FRA class) (4) (5) (6)

Item∗∗ Z µ σ ζ µ σ ζ µ σ

1st Wheelset 23.8 6.7 4.1 19.8 5.6 3.4 15.9 4.5 2.82nd Wheelset 21.6 7.2 4.4 18.0 6.0 3.7 14.4 4.8 2.93rd Wheelset 26.3 7.1 4.7 21.9 5.9 3.9 17.5 4.7 3.14th Wheelset 26.0 7.8 5.0 21.7 6.5 4.1 17.3 5.2 3.3

MeasuredSKS (v = 260 km/h) = 0.2P = 0.2 × 134.8 = 27 kN.

values TGV (v = 515.3 km/h) = 32 kN for front wheelset

= 35 kN for rear wheelset

∗v = 360 km/h.∗∗ζ = maximum, µ = mean, σ = standard deviation.



10.11. Dynamic Effects Induced by Trains at DifferentSpeeds

In reality, the train can move over the approach and bridge at differ-ent speeds. There exists a need to investigate the dynamic responsesof the train–rails–bridge systems under different train speeds, as theycan be quite different, especially when the resonance phenomenon isto be considered. The dynamic effects to be investigated in this sec-tion include the impact factor for the midspan displacement of thebridge and the maximum acceleration of the train. The impact factorI used here is the same as the one defined in Eq. (1.1), i.e.,

I =Rd(x) − Rs(x)

Rs(x), (10.112)

where Rd(x) and Rs(x) denote the maximum dynamic and staticresponses, respectively, of the bridge at cross-section x caused bythe moving train. Because of the lack of a lateral static deflection(no lateral static axle loads acting on the bridge), the impact effectfor the lateral vibration of the bridge cannot be investigated usingthe definition of Eq. (10.112). For this reason, only the maximumresponse will be investigated for the lateral displacement. In addi-tion, the same nondimensional speed parameter S as the one used inprevious chapters will be used to denote the speed of train, namely,

S =πv

ω1L, (10.113)

where ω1 denotes the first natural frequency, L the length of thebridge and v the train speed. For a specific bridge, the speed pa-rameter S is proportional to the train speed. The value of S con-sidered in the present study varies from S = 0 (v = 0 km/h) to 0.7(v = 600 km/h). The other assumptions are identical to those ofSec. 10.10.

The impact responses for the midspan of the bridge were plottedin Fig. 10.22 with respect to the speed parameter S for four classes oftrack quality, namely, the FRA Classes 4, 5, 6 and ideal track geom-etry. As can be seen, the impact factor for the vertical displacementreaches a peak value at S = 0.425 (v = 364 km/h), indicating the



0

0.5

1

1.5

2

2.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

Nondimensional Speed Parameter S

Impact F

acto

r fo

r B

ridge M

idspan V

ert

ical

Dis

pla

ce

me

nt


poor (FRA class 5)


ideal geometry

(a)

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7


Bri

dg

e M

idsp

an

Ma

xim

um

La

tera

l

Dis

pla

ce

me

nt

(x1

0-6

m)


poor (FRA class 5)


ideal geometry

(b)

Fig. 10.22. Impact factors for bridge midspan displacement due to different trackqualitites: (a) vertical, (b) lateral, and (c) torsional.



0

0.05

0.1

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0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7


Imp

act

Fa

cto

r fo

r B

rid

ge

Mid

sp

an

Torsio

na

l

An

gle


poor (FRA class 5)


ideal geometry

(c)


occurrence of resonance between the train and the bridge. Note thatthe impact factor is nearly independent of the track quality. Themaximum lateral displacement of the bridge has a peak value atS = 0.525 (v = 450 km/h) for all the three FRA classes, which van-ishes for the case of ideal track. This is mainly due to resonance in thelateral vibration, caused by the coincidence of any of the driving fre-quencies of the train with any of the frequencies implied by the trackirregularity. Unlike the vertical displacement, substantial differenceexists between the lateral displacements for the four classes of trackquality considered. In particular, nearly zero lateral displacement isobserved for the case with ideal track geometry.

The impact factor for the midspan torsional angle of the bridgeincreases generally with the increase in S, with rather small differ-ence existing between the four track classes. In addition, two localmaxima can be observed for the impact factor at S = 0.525 and0.575, which can be attributed to the occurrence of resonance in tor-sion between the train and the bridge. The maximum impact factors



for the midspan vertical and torsional displacements of the bridgeare 2.13 and 0.23, respectively, while the maximum midspan lateraldisplacement is 0.006 mm.

The maximum acceleration of the train with respect to the speedparameter S was plotted in Fig. 10.23 for the four classes of trackquality. As can be seen, the maximum acceleration of the train forthe track with irregularities appears to be much larger than that forthe ideal track. Moreover, much larger difference can be observed forthe maximum lateral, rolling and yawing accelerations of the trainbetween the irregular cases and the ideal case than for the maximumvertical and pitching accelerations, due to the fact that nearly nolateral, rolling and yawing vibrations are induced on the train as itmoves over a smooth and straight track.

Note that owing to the relatively large responses induced by theirregularities, the resonance effect on the vertical and pitching re-sponses becomes almost invisible for the irregular tracks, except for

0

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Ve

hic

le M

axim

um

Ve

rtic

al A

cce

lera

tio

n

(m/s

^2)


poor (FRA class 5)


ideal geometry

(a)

Fig. 10.23. Maximum vehicle acceleration for different train speeds: (a) vertical,(b) lateral, (c) rolling, (d) yawing, and (e) pitching.



0

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ideal geometry

(b)

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um

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llin

g A

cce

lera

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n

(ra

d/s

^2

)


poor (FRA class 5)


ideal geometry

(c)




0

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hic

le M

axim

um

Ya

win

g A

cce

lera

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(ra

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(d)

0

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Ve

hic

le M

axim

um

Pitch

ing

Acce

lera

tio

n

(ra

d/s

^2

)


poor (FRA class 5)


ideal geometry

(e)




Table 10.7. Asymptotic limits and maximum values of vehicleacceleration.

Vehicle Vertical Lateral Rolling Yawing Pitchingacceleration (m/s2) (m/s2) (rad/s2) (rad/s2) (rad/s2)

Asymptotic0.22 0.57 1.22 — 0.04Limita

Maximum0.16 0.53 1.09 0.07 0.03Valueb

Tolerable0.98 0.49 — — —Valuec

av = 0 ∼ 600 km/h (very poor track).bv = 0 ∼ 360 km/h (very poor track).cAdopted by Taiwan HSR. For vertical acceleration, the following valuesare used by UIC, SKS, and ICE: 1, 1.96, 0.49 m/s2, respectively.

the yawing motion, the maximum accelerations of the train appear tohave an asymptotic limit over the whole range of speeds considered.The asymptotic limits have been listed in Table 10.7, along with themaximum accelerations of the train in the speed range S < 0.42(= 360 km/h) considered by the Taiwan High Speed Railways. Itcan be seen that the maximum lateral acceleration of the train forthe very poor track (FRA Class 4) exceeds the limit of 0.49 m/s2, butthat for the poor track (FRA Class 5) is below the limit, indicatingthat the track should be maintained frequently to ensure that themaximum deviations are less than those for the poor quality, i.e.,rv,max = 3.38 mm, rh,max = 4.25 mm and rc,max = 1.25 mm.

10.12. Response Induced by Two Trains in Crossing

For a two-way railroad bridge, two trains on two different tracks maycross each other on the bridge with the same or different speeds. Thecrossing movement of the trains can result in drastically larger verti-cal vibrations for the trains and bridge than those by a single train,which may be harmful to the riding quality of the trains themselves,as well as to the maintenance of the track structures. Three cases willbe analyzed to investigate the effects induced by the crossing of two



trains: (a) vA = 100 m/s, vB = 100 m/s, xA0 = 0 m, xAf = 430 m,xB0 = 30 m, xBf = −400 m, (b) vA = 100 m/s, vB = 100 m/s,xA0 = 0 m, xAf = 430 m, xB0 = 220 m, xBf = −400 m, and (c)vA = 100 m/s, xA0 = 0 m, xAf = 430 m (single train), where vA,xA0 and xAf respectively denote the speed, starting and stoppingpositions of the train on Track A, and vB , xB0 and xBf denote thecorresponding quantities for the train on Track B. The train onTrack A is assumed to move along the positive x direction, whichwill be referred to as train A and the train on Track B along thenegative x direction, which will be referred to as train B. Each ofthe two trains consists of 15 identical cars. Case (a) is conceived forstudying the train and bridge responses caused by two trains crossingeach other on the midspan of the bridge (v = 100 m/s). Case (b)is used to study the system responses induced when the mid-portion(i.e., the 8th car) of train A moves to the center of the bridge, whiletrain B starts to enter the bridge (v = 100 m/s). Case (c) is con-ceived only for the purpose of comparison (v = 100 m/s). In thefollowing, the crossing movement of Case (a) will be referred to assymmetric crossing movement (in the view point of the bridge), andthose encountered in Cases (b) as asymmetric crossing movement.

The results computed for the midpoint response of the bridgewere shown in Fig. 10.24. As can be seen, the vertical response ofthe bridge to the symmetric crossing of the two trains (Case (a))is larger than that to the passage of train A alone (Case (c)). Themaximum displacements of the bridge for Case (a) and Case (c) are7.8 and 4.2 mm, respectively. Clearly, the maximum bridge responseto the passage of two trains need not be two times larger as thatto the passage of a single train. Notice that the vertical responseof the bridge for Case (b) becomes smaller than that for Case (c)after train B enters the bridge, i.e., for vt/L 6.3. This can beattributed primarily to the cancellation of the vehicular loads underthe asymmetric crossing of the trains in Case (b). Meanwhile, thelateral displacements of the bridge for the three cases are negligiblysmall.

The midspan torsional response of the bridge for Case (a) is rathersmall due to cancellation of the torsional moments induced by the



(a)

(b)

Fig. 10.24. Bridge responses due to crossing of two trains: (a) vertical, (b)lateral, and (c) torsional.



(c)


two series of loads, as can be seen from Fig. 10.24(c). The midspantorsional vibrations of the bridge in Cases (b) and (c) are, however,rather large compared with that of Case (a). Furthermore, the tor-sional vibration in Case (b) changes its equilibrium position fromthe deformed one by a single train (θb = 6.510−6 rad) to the unde-formed one (θb = 0 rad), and oscillates more drastically than thatin Case (c), after the entrance of train B into the bridge, i.e., forvt/L 6.3.

The response of the 8th car of train A and that of the 1st carof train B are selected for investigation, as they are the typical carsof the two trains crossing on the bridge. From the vehicle responsesgiven in Fig. 10.25(a), one observes that the vertical accelerationof a train under symmetric crossing is larger than that for a sin-gle train. Moreover, a train that first travels over the bridge underthe condition of asymmetric crossing (e.g., train A in Case (b)) vi-brates less severely than it does when traveling alone over the bridge(see Case (c)). On the contrary, the maximum response of a train



-0.25

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-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

8th

Car

Vert

ical A

ccele

ration (m

/s^2

)

case (a)

case (b)

case (c)

Train A

(a)

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

1st C

ar

Vert

ical A

ccele

ration (m

/s^2

)

case (a)

case (b)

case (c)

Train B

(b)

Fig. 10.25. Vehicle accelerations due to crossing of two trains: (a) and (b) verti-cal, (c) and (d) lateral, (e) and (f) rolling, (g) and (h) yawing, (i) and (j) pitching.



-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

8th

Car

Late

ral A

ccele

ration (m

/s^2

) case (a)

case (b)

case (c)

Train A

(c)

-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

1st C

ar

Late

ral A

ccele

ration (m

/s^2

) case (a)

case (b)

case (c)

Train B

(d)




-5.0E-04

-4.0E-04

-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

0 0.2 0.4 0.6 0.8 1 1.2

Time (s)

8th

Car

Rolli

ng A

ccele

ration (r

ad/s

^2)

case (a)

case (b)

case (c)

Train A

1.4

(e)

-5.0E-04

-4.0E-04

-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

1st C

ar

Rolli

ng A

ccele

ration (r

ad/s

^2)

case (a)

case (b)

case (c)

Train B

(f)




-8.0E-05

-6.0E-05

-4.0E-05

-2.0E-05

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

8th

Car

Yaw

ing A

ccele

ration (r

ad/s

^2)

case (a)

case (b)

case (c)

Train A

(g)

-8.0E-05

-6.0E-05

-4.0E-05

-2.0E-05

0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

1st C

ar

Yaw

ing A

ccele

ration (r

ad/s

^2)

case (a)

case (b)

case (c)

Train B

(h)




-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

8th

Car

Pitchin

g A

ccele

ration (r

ad/s

^2)

case (a)

case (b)

case (c)

Train A

(i)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

1st C

ar

Pitchin

g A

ccele

ration (r

ad/s

^2)

case (a)

case (b)

case (c)

Train B

(j)




that travels over the bridge lately under asymmetric crossing (e.g.,train B in Case (b)) is larger than the case when it passes aloneover the bridge (see Fig. 10.25(b)). The maximum lateral acceler-ation of a train induced by symmetric or asymmetric crossing doesnot differ markedly from that by the passage of a single train (seeFigs. 10.25(c) and 10.25(d)). Similar phenomenon can also be ob-served for the rolling acceleration of the trains (see Figs. 10.25(e)and 10.25(f)). However, the maximum response of the yawing ac-celeration of the train under the condition of asymmetric crossingis larger than those caused by the other two types of train move-ment (see Figs. 10.25(g) and 10.25(h)). Least yawing response willbe induced on the trains under symmetric crossing. As for the pitch-ing acceleration, the behaviors of the train are similar to those ofthe vertical acceleration, namely, trains A and B have the largestresponses when the two trains cross each other symmetrically andasymmetrically, respectively, over the bridge (see Figs. 10.25(i) and10.25(j)).

10.13. Criteria for Derailment and Safety Assessmentof Trains

The running safety of trains has been of great concern in railwayengineering for a long time, particularly due to the development ofhigh-speed railways and the need to upgrade existing railways. Sev-eral mechanisms that can result in the derailment of a running trainhave been identified through analytical and experimental investiga-tions, based on which some indices have been proposed for evaluatingthe possibility or risk of derailment of trains. In this section, five sim-ple and practical criteria will be employed for assessing the runningsafety of trains traveling over bridges.

The first index is the axle load decrement ratio (PD). This indexis defined as the ratio of the decrement in vertical axle load of awheelset to the vertical static axle load of the same wheelset, i.e.,

PD =∆Q

Qs=

Qs − Q

Qs, (10.114)



where Qs denotes the axle static load of a wheelset and Q the dy-namic vertical axle load (0 < Q < Qs). Large PD values indicate thatthe axle load or the vertical contact force acting on the wheelset issubstantially reduced, which is detrimental to the lateral stability ofthe wheelset. Therefore, a limit should be placed on the value of thePD index to prevent the wheelset from the occurrence of derailment.An upper limit of 0.25 on the PD value was used in Japan’s specifica-tions for the design of high-speed railways (Ma and Zhu, 1998), whichwill also be adopted herein. Note that in this study the axle staticload Qs is set equal to W (see Sec. 10.3) and Q = [V(2i−1) + V(2i)],where [V(2i−1) + V(2i)] > 0 and i = 1,2, 3 or 4 (for 4 wheelsets ofa railway car). If [V(2i−1) + V(2i)] < 0, which indicates that the ithwheelset will jump and the train has the potential for derailment.

The second index adopted is the single wheel lateral to verticalforce ratio (SYQ). This index is used as a measure of proximity toa flange climb derailment situation for a single wheel (not wheelset),which is defined as

SYQ =Y

Q, (10.115)

where Y and Q respectively denote the lateral and vertical contactforces acting through the right or left wheel of a wheelset. A value of1.2 will be adopted as the upper bound of the SYQ ratio (Elkins andCarter, 1993), over which the train is said to have the potential ofderailment. Here, Vi and Hi (i = 1, 8) should be used in computingthe SYQ value for each wheel (or contact point), that is, Q = Vi andY = |Hi|, for Vi > 0. If Vi < 0 and Hi = 0, the wheel will jump;however, if Vi < 0 and Hi = 0, the wheel has the potential to exhibitflange climb derailment.

The third index is the wheelset lateral to vertical force ratio (YQ).This index is similar to the SYQ index, but is considered for the entirewheelset (which consists of two wheels), rather than a single wheel.The YQ ratio is, in general, more realistic than the SYQ ratio, andhas been used by many authorities and high-speed rail lines in theirspecifications, for example, the UIC, ICE, SKS and Mainland China.



The definition of the YQ index is

YQ =Y

Q, (10.116)

where Y and Q respectively denote the lateral and vertical contactforces acting on a wheelset: Q = [V(2i−1) + V(2i)] > 0 and Y =|H(2i−1) + H(2i)|, i = 1, 2, 3 or 4. If Q = [V(2i−1) + V(2i)] < 0, oneor two wheels of the ith wheelset encounter the jump condition, andthe train is at a high degree of risk of derailment. According to theexisting specifications, the value of the YQ ratio must not exceed 0.8to ensure the safety against derailment (Ma and Zhu, 1998).

Another index adopted herein is the bogie-side lateral to verticalforce ratio (BYQ). This index is associated with gauge spreading orrail roll over derailment. Derailments of this type can occur whenthe gauge faces of the two rails are spread apart sufficiently for therim face of one wheel to drop inside of the gauge face of the rail withwhich it is in contact (Elkins and Carter, 1993). Moreover, this typeof derailment is closely related to the combined effect of all wheelson one side of a bogie. Accordingly, the BYQ criterion is defined asthe ratio of the sum of the lateral forces on all the wheels on one sideof a bogie to the sum of all the vertical loads on the same wheels,namely,

BYQ =Y

Q, (10.117)

where Y and Q represent the lateral and vertical contact forces actingon one rail (right or left) by the two wheels on one side of a bogie,respectively, that is, Q = [V(4j−3) + V(4j−1)], Y = |H(4j−3) + H(4j−1)|for right rail and Q = [V(4j−2) + V(4j)], Y = |H(4j−2) + H(4j)| for leftrail, where j = 1 (for front bogie) and 2 (for rear bogie). It shouldbe noted that (1) Q = [V(4j−3) +V(4j−1)] (or Q = [V(4j−2) +V(4j)]) forV(4j−3), V(4j−1) > 0 (or V(4j−2), V(4j) > 0) and Y = |H(4j−3)+H(4j−1)|(or Y = |H(4j−2)+H(4j)|), (2) Q = V(4j−3) for V(4j−3) > 0, V(4j−1) 0and Y = |H(4j−3)|, and (3) Q = V(4j−1) for V(4j−1) > 0, V(4j−3) 0and Y = |H(4j−1)|. The BYQ ratio should not exceed 0.6 in order tobe safe from a rail roll over derailment (Elkins and Carter, 1993).



The last index adopted herein is the lateral track force (Y ). Alimit is placed on the maximum lateral force exerted by an axle onthe track in order to minimize the risk of track panel shift. It shouldbe noted that track panel shift has become increasingly importantwith increased train speeds and popular use of continuously weldedrails. The maximum allowable lateral axle force Ylim (kN) specifiedby Prud’homme can be given as follows (Elkins and Carter, 1993):

Ylim = α

(10 +

Qs

3

)(kN) , (10.118)

where Qs is the static axle force (kN) = W , and α is a modificationfactor; α = 1.0 in general and α = 0.85 for poorer quality track.The maximum allowable lateral axle force for the present study isobtained as 55 kN by Eq. (10.118) for W = 135 kN and α = 1.0.

The relations between the maximum values of the above five in-dices and the speed parameter S under different track conditionshave been plotted in Figs. 10.26(a)–10.26(e). As can be seen, largervalues of the indices have been predicated for tracks with poorer qual-ity, implying that a bad-maintained or deteriorating track structurehas a higher possibility of derailment, except for the PD ratio, theindices exhibit some local peaks at certain speeds, primarily due tothe lateral train–rails–bridge resonance occurring at these speeds, asevidenced by Fig. 10.26(e) for the lateral force Y . Moreover, largerdifference exists in the values of the indices between different trackqualities at the resonant speeds.

The maximum SYQ ratio for the train exceeds the limit of 1.2and the maximum YQ ratio approaches the allowable value of 0.8 atsome resonant speeds for the poorest track quality considered (FRAClass 4). Such an observation indicates that the resonance occurringbetween the train, track and bridge can greatly aggravate the runninginstability of the train, particularly when the train moves on a poortrack structure.

In general, the index values for higher resonant speeds (S 0.5)are larger than those for lower resonant speeds (S < 0.5). Althoughthe values for the SYQ ratio at some speeds are larger than theallowable one, it does not necessarily mean that derailment will



0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


Maxim

um

PD

Ratio very poor (FRA class 4)

poor (FRA class 5)


ideal geometry

allowable value = 0.25

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


Ma

xim

um

SY

Q R

atio


poor (FRA class 5)


ideal geometry


(b)

Fig. 10.26. Maximum index values with relation to the speed parameter: (a)PD ratio, (b) SYQ ratio, (c) YQ ratio, (d) BYQ ratio, and (e) lateral force Y.



0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


Ma

xim

um

YQ

Ra

tio


poor (FRA class 5)


ideal geometry allowable value = 0.8

(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


Ma

xim

um

BY

Q R

atio


poor (FRA class 5)


ideal geometry


(d)




0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


Ma

xim

um

La

tera

l F

orc

e Y

(k

N)


poor (FRA class 5)


ideal geometry allowable value = 55 kN

(e)


immediately occur on the train at these speeds. One reason for thisis that the allowable value of 1.2 has been proved to be a conser-vative estimate by relevant investigations. Another reason is thateven though a very large lateral force is exerted on a wheel of thetrain, it does require some duration of time for the wheel to derail,which therefore may not occur in practice. However, it is true thatthe train would be at a higher risk of derailment when traveling ona poor-quality track at the resonant speeds. Based on the resultsobtained for the five criteria, it is concluded that the safety (or sta-bility) of the train passing over the bridge with various speeds forthe four track qualities considered is acceptable. However, the traindeserves special attention when it moves over tracks of bad qualities.Besides, the train considered herein shows higher potential for thewheel climb derailment than for other types of derailment, as impliedby comparatively large SYQ values.




A three-dimensional VRBI model for analyzing the train–rails–bridgeinteractions was established. The vehicle is modeled as an assemblyof a car body, two bogies and four wheelsets, which has a total of27 DOFs. Such a model allows us to consider the coupling effect be-tween the lateral and rolling vibrations, which may be caused by thedifference in elevation of the center of gravity of the car body, bogieand wheelsets, and the linking action of any two wheels connectedby a rigid axle. The track structure is idealized as a continuous twinrails system of infinite length supported by spring-dashpot units.The use of the twin rails model enables us to minimize the numberof DOFs used in modeling of the track, while taking into accountthe constraint imposed by the sleepers on the two rails. In additionto the vertical vibration, the lateral and torsional responses of thetrain–rails–bridge system can be obtained simultaneously using thepresent VRBI model, which are useful for evaluation of the runningsafety of trains over the bridge. The procedure developed for ana-lyzing the 3D VRBI system was verified in the numerical study of atypical example.

The procedure developed above was applied to analysis of theVRBI systems in the three-dimensional sense, with due account takenof the random track irregularities. The results indicate that reso-nance can occur in the lateral and torsional vibrations of the bridgeat some specific speeds, similar to that encountered in vertical vibra-tions. Besides, the presence of track irregularity can greatly increasethe response of the train, track and bridge. Moreover, the extent ofinfluence is much larger for the train than for the track and bridge.There is no doubt that track irregularity should be taken into ac-count in the design and maintenance of railway bridges concerningthe running safety of the train.

For a two-way railroad bridge, it is necessary to consider the im-pact effects caused by the crossing movement of two trains on thebridge, in addition to those by a single moving train. In general, thevertical vibration of the bridge appears to be more violent under thecrossing of two trains, but the lateral and torsional responses may



be increased or reduced, depending on the way the two trains crosseach other. The vertical and pitching vibrations of a train will beaggravated when it crosses with another train on the bridge. Such aphenomenon is more severe for trains moving at high speeds.

Finally, the possibility of derailment of the train was assessedthrough the use of five common indices. The results of computationfor these five indices show that the train will be at a higher risk ofderailment as it travels on a bad-conditioned track, mainly due tothe relatively large lateral forces induced between the wheelset ofthe train and the rails. By comparison of the maximum index valuescomputed with the tolerable limits, it is concluded that the traincan safely pass through the bridge under the conditions specified inthe present study, i.e., those assumed for the track irregularity, trainmodel, track and bridge properties, and so on, over a wide range ofspeeds.



Chapter 11

Stability of Trains Moving overBridges Shaken by Earthquakes

The dynamic stability of trains moving over bridges shaken by earth-quakes is studied in this chapter. Unlike the dynamic analysis ofstructures subjected only to an earthquake motion, the dynamicanalysis of a bridge subjected to a moving train and earthquakemotion requires not only information on the acceleration, but alsovelocity and displacement of the ground motion. Four typical earth-quakes, including the 1999 Chi–Chi Earthquake, were adopted asthe input excitations, each of which was normalized to have a mod-erate intensity. The results indicate that a train initially resting onthe bridge can stay safely when subjected to any of the four earth-quakes, on the condition that no inelastic deformations occur on thebridge and track structures. The characteristics of the vertical com-ponent of ground motions can affect significantly the stability of thetrain–rails–bridge system. As a preliminary attempt, safety, possi-ble instability and instability regions will be established for a trainrunning over a bridge for each of the four earthquakes consideredusing a three-phase plot, from which the maximum allowable speedfor the train to run safely under the specific ground acceleration canbe obtained.

11.1. Introduction

Seismic resistance of bridge structures is an issue of great concern inearthquake-prone regions. As for railway bridges, it is possible thatthe bridge itself may remain safe during an earthquake, but may

409



not be safe enough for the trains to move over it due to excessivevibrations of the sustaining bridge. Evidently, the safety of the mov-ing trains over a bridge shaken by earthquakes is a subject of greatconcern for railway engineers, especially in those countries that areearthquake-prone. This problem is becoming more important due tothe increasing use of elevated bridges as the supporting structures forrailways in metropolitan areas, and the advancement in locomotivetechnologies for enhancing the driving power of railway trains. Tothe knowledge of the authors, however, rather limited research workshave been conducted on such a subject.

In the study by Miura (1996), emphasis was placed on theearthquake-induced displacement of tracks and structures, as well asthe damage of trains caused by earthquake excitations, rather thanon the dynamic stability of trains during an earthquake. Miyamotoet al. (1997) investigated analytically the running safety of railwayvehicles under the action of earthquakes using a three-dimensionalsimplified vehicle model, where sine waves are used as the input exci-tation and the vehicle is assumed to remain stationary on the track.Ma and Zhu (1998) studied the response of high-speed trains andcontinuous rigid-frame bridges under three different ground motionsby the random vibration theory. In their work, only the lateral forceswere obtained, while the track system was not taken into account.It follows that the index values computed for evaluation of safety(or derailment) of the train may not be accurate enough. More-over, the assessment of running safety of the train was made only forground motions with specific intensities, which may not be sufficientfor devising any safety guidelines that are useful for the safe opera-tion of trains, as a train may be attacked by earthquakes of variousintensities.

In this chapter, the stability of a train initially resting onor traveling over a bridge under different seismic excitations willbe investigated using the train and bridge models developed inChapter 10 with inclusion of the earthquake-induced effect. As apreliminary effort toward assessment of the safety of moving trainsshaken by earthquakes, the bridge is assumed to remain fully elas-tic during the earthquake, while some simple criteria, such as the


Stability of Trains Moving over Bridges Shaken by Earthquakes 411

derailment index, are adopted for evaluating the safety of the mov-ing train. It is realized that margin does exist for improving the trainand bridge models, say, to include the effect of inelastic deformations,while the derailment of a moving train may have to be evaluated ona statistics basis, rather than on some simplified threshold values.For this reason, the present formulation should be regarded as theone that serves to lay out the basic framework of the problem con-sidered. Even though the results have been computed for severaldifferent cases, they should be regarded as a qualitative description,rather than as a quantitative assessment. The materials presented inthis chapter are based mainly on the paper by Yang and Wu (2002),but with significant revisions and supplements.

11.2. Analysis Model for Train–Rails–Bridge System

Figure 11.1 shows a train traveling with speed v over a simply-supported bridge shaken by an earthquake. The train is modeledin the same way as the one described in Chapter 10, namely, we as-sume the train to consist of a series of separate identical cars, of which

L

train

bridge

central track section right track sectionleft track section

CFR element


vertical

lateral

ground motion

ground motion

bridge element

ballast

rail

Fig. 11.1. A train traveling over a simply-supported bridge shaken by earth-quakes.



each is composed of one car body, two bogies and four wheelsets, asshown in Fig. 11.2. The total number of degrees of freedom (DOFs)implied by each vehicle is 27. Such a model enables us to simulatethe vertical, lateral, rolling, yawing and pitching motions of the carbody, as well as the vertical and lateral contact forces between therails and wheels.

C Body

Bogie

Wheelset

1V3V

5V7V Sleeper

Rail

Ballast

Bridge or Soil Roadbed( )

(a)

1H

2H

3H

4H

5H

6H

7H

8H ATrack

BTrack

Sleeper

1

26

7

8 4

35

Rails

1st wheelset

2nd wheelset

3rd wheelset4th wheelset

(b)

Fig. 11.2. Train car, track and bridge models: (a) side view, (b) top view, and(c) rear view.



7V 7H8V 8

H

Bogie

Wheelset

C Body

BTrack ATrack

BallastSleeper

Rails

Bridge

(c)


The bridge is assumed to be made of a box girder of uniform crosssections that can carry two parallel tracks, i.e., Tracks A and B. Eachof the two tracks is simulated as a set of infinite, continuous twinrails lying on a single-layer ballast foundation. The bridge girder, byitself, is modeled as a three-dimensional Bernoulli–Euler beam (seeFig. 11.2). As can be seen, the train car, rails and bridge form atrain–bridge interaction system, or more specifically, a vehicle–rails–bridge interaction (VRBI) system. For the sake of simplicity, eachof the subsystems, i.e., the vehicle, track and bridge, is assumed toremain fully elastic during the earthquake excitation.

Again, the fundamental data provided by Wakui et al. (1995)for the Series 300 car model of Japan’s Shinkansen (SKS) high-speedtrain will be adopted in this chapter. We choose to use this model,simply because this is the only vehicle model of which the mechanicaldata were made available to us. The rails are assumed to be of theUIC-60 type. The fundamental data of the vehicle, track and bridgemodels that are required in analysis have been listed in Tables 10.1.



and 10.2, which are also available in Wu et al. (2001). It should benoted that all the conclusions made in this chapter remain strictlyvalid only for the particular bridge and vehicle models, as well as theassumptions, adopted in the study.

11.3. Railway–Bridge System with Ground Motions

In this section, focus will be placed on inclusion of the ground motioneffects in the equations of motion for each component of the railway–bridge system. The methodology is similar to the one presented inChapter 10. In particular, let us consider the railway–bridge systemshown in Fig. 11.1, where three types of elements are identified for therails, i.e., the central finite rail (CFR), left semi-infinite rail (LSR)and right semi-infinite rail (RSR) elements. The bridge girder ismodeled by the conventional 3D beam elements, each of which has atotal of 12 DOFs at the two nodes, with three translations and threerotations at each node. The equations of motion for the CFR, LSR,RSR and bridge elements that have been presented in Chapter 10will be extended to include the earthquake-induced effects. All thequantities associated with Track A, Track B, and the bridge will bedenoted by symbols with subscripts “A”, “B” and “b”, respectively.


Track A

The two rails of each track are assumed to be identical and con-nected by rigid sleepers that are uniformly-distributed. Just as inSec. 10.4.1, the two rails will be considered together and representedby a single line of rail elements, also known as the central finiterail (CFR) elements, lying on the ballast layer that is modeled byuniformly-distributed spring-dashpot units. The CFR element isnothing but a conventional 3D beam element, of which the nodaldisplacement vector can be given as follows:

dA = 〈uA1 vA1 wA1 θA1 ϕA1 ψA1

uA2 vA2 wA2 θA2 ϕA2 ψA2 〉T , (11.1)



where the subscript A indicates that the quantity is associated withTrack A. Correspondingly, the nodal force vector fA is

fA = 〈FAx1 FAy1 FAz1 MAx1 MAy1 MAz1

FAx2 FAy2 FAz2 MAx2 MAy2 MAz2 〉T . (11.2)

By the principle of virtual work, the equation of equilibrium for theCFR element located on Track A can be written as

∫ l

0EtAtu

′Aδu′

Adx +∫ l

0EtItzv


∫ l

0EtItyw

′′Aδw′′

Adx

= −∫ l

0mt(uAδuA + vAδvA + wAδwA)dx −

∫ l

0I∗t θAδθAdx

−∫ l

0ct(uAδuA + vAδvA + wAδwA)dx

−∫ l

0c∗t θAδθAdx + 2ld

∫ l

0[pc(x) + pk(x)]δuAdx

+∫ l

0

∫ lb+ld

lb−ld

[qc(x, z) + qk(x, z)][δvA − (z − lb)δθA]dzdx

+ 2ld

∫ l

0[rc(x) + rk(x)]δwAdx + 〈δdA〉fA , (11.3)

where all the quantities with subscript t indicates that they are as-sociated with the track or rails, Et is Young’s modulus, At is thecross-sectional area, Ity, Itz are the moment of inertia about y, z

axes, mt is the per-unit-length mass (including the mass of sleep-ers), I∗t the per-unit-length mass moment of inertia about x axis(including the contribution of sleepers), ct, c∗t the visco-damping co-efficients, lb the distance between the center line of Track A and thebridge, ld the half-length of sleepers, l the length of the rail element;dA denotes the nodal displacement of the rail element, fA thecorresponding external loads; uA, vA, wA the displacements alongx, y, z-axes; and θA the rotation about x-axis of the rail element.



The interaction forces arising from the relative motion of the rail andbridge elements, i.e., (pc, pk), (qc, qk) and (rc, rk), can be defined as

pc(x) = c∗bh1(ub(x) − uA(x)) ,

pk(x) = k∗bh1(ub(x) − uA(x)) ,

qc(x, z) = c∗bv1(vb(x) − zθb(x) − vA(x) + (z − lb)θA(x)) ,

qk(x, z) = k∗bv1(vb(x) − zθb(x) − vA(x) + (z − lb)θA(x)) ,

rc(x) = c∗bh1(wb(x) + hθb(x) − wA(x)) ,

rk(x) = k∗bh1(wb(x) + hθb(ξ) − wA(x)) ,

(11.4)

where c∗bh1, k∗bh1 and c∗bv1, k∗

bv1, respectively, denote the unit-area hor-izontal and vertical damping and stiffness coefficients of the ballaston the bridge; ub, vb, wb the displacements along the three axes; andθb the rotation about x axis of the bridge element; h is the verticaldistance between the deck and the center of torsion of the bridge sec-tion. The longitudinal and lateral damping and stiffness coefficientsof the ballast are assumed to be the same, denoted as c∗bh1 and k∗

bh1.The displacement fields of the rail and bridge elements can be ex-pressed in terms of the nodal DOFs using linear or cubic Hermitianinterpolation functions (Paz, 1985). Substituting these displacementfields into Eq. (11.3) yields the equation of motion for the CFR ele-ment composed of twin rails for Track A that is free of any groundmotions as

[mA]dA + [cA]dA + [kA]dA= fA + [cAb]db + [kAb]db , (11.5)

where [mA], [cA] and [kA] denote the mass, damping and stiffnessmatrices of the rail element, dA is the nodal displacement of the railelement, as given in Eq. (11.1), and db is the nodal displacementvector of the underlying bridge element,

db = 〈ub1 vb1 wb1 θb1 ϕb1 ψb1

ub2 vb2 wb2 θb2 ϕb2 ψb2 〉T , (11.6)



The damping and stiffness matrices on the right-hand side ofEq. (11.5), i.e., [cAb] and [kAb], represent the interaction effects causedby the relative motion of the two rails and bridge elements throughthe ballast layer simulated as uniformly-distributed spring-dashpotunits. Details for the matrices and vectors involved in Eq. (11.5) areavailable in Sec. 10.4.1.

For a railway bridge subjected to a ground motion, both the dis-placement vectors dA and db appearing in Eq. (11.5) should beinterpreted as the total or absolute displacements of the rail andbridge elements, respectively. The total displacements of the bridgeelement db can be divided into two parts:

db = dnb + dr

b , (11.7)

where dnb denotes the natural deformations and dr

b the rigid dis-placements of the bridge element due to the ground motion. Thelatter can be determined as

drb = [R]ug , (11.8)

where [R] denotes the transformation matrix and ug the supportdisplacements of the bridge due to the ground motion, which areprescribed in general. Based on the assumption that the supportmotions occur synchronously and that no rotations are induced bythe ground motions on the bridge supports, the transformation ma-trix [R] can be given as

[R] =[

[r]

[r]

],

[r] =[

[I]3×3

[0]3×3

],

(11.9)

where [I] is a 3 × 3 unit matrix and [0] a zero matrix. The supportdisplacements ug are assumed to be three-dimensional,

ug = 〈ugx ugy ugz 〉T , (11.10)

where ugx, ugy andugz denote the displacement components alongthe three axes.



Substituting Eqs. (11.7)–(11.10) into Eq. (11.5), one obtains theequation of motion for the CFR element with due account taken ofthe ground motion as

[mA]dA + [cA]dA + [kA]dA= f t

A + [cAb]dnb + [kAb]dn

b , (11.11)

where f tA denotes the total equivalent nodal forces of the element

under the ground motion,

f tA = fA + f c

A + fkA

= fA + [cAb][R]ug + [kAb][R]ug . (11.12)

As can be seen, the parts of the equivalent nodal forces induced byground motions, i.e., f c

A and fkA, relate to the displacement and

velocity, but not acceleration, of the ground or supports of the bridge.


Track B

Let the nodal displacement vector of the central finite rail (CFR)element on Track B be denoted as

dB = 〈uB1 vB1 wB1 θB1 ϕB1 ψB1

uB2 vB2 wB2 θB2 ϕB2 ψB2 〉T , (11.13)

and the associated nodal force vector be denoted as

fB = 〈FBx1 FBy1 FBz1 MBx1 MBy1 MBz1

FBx2 FBy2 FBz2 MBx2 MBy2 MBz2 〉T . (11.14)

By following the procedure presented in Sec. 11.3.1, the equation ofmotion of the CFR element located on Track B with due account forthe ground motions can be given as

[mB]dB + [cB ]dB + [kB ]dB= f t

B + [cBb]dnb + [kBb]dn

b , (11.15)



where

f tB = fB + f c

B + fkB

= fB + [cBb][R]ug + [kBb][R]ug . (11.16)

Based on the assumption that no rocking motion are induced by theearthquake on the bridge supports, the extra equivalent nodal forcesinduced by the earthquake on the CFR element for both tracks shouldbe the same, namely,

f cB = f c

A ,

fkB = fk

A .(11.17)

The same results can be obtained for the two tracks through ma-nipulation of the triple products underlined in Eq. (11.12) involvingthe matrices [cAb] and [kAb] and in Eq. (11.16) involving the matrices[cBb] and [kBb].

11.3.3. Bridge Element

The bridge element is also regarded as a three-dimensional solid beamelement, which has a total of 12 DOFs, as indicated in Eq. (11.6).Following the same procedure as that for the CFR element above,one can derive the equation of motion for the bridge element, withdue account taken of the interaction with the rail elements of the twotracks through the ballast layer, as follows:

[mb]dnb + [cb]dn

b + [kb]dnb

= f tb + [cbA]dA + [kbA]dA

+ [cbB ]dB + [kbB ]dB , (11.18)

where dnb denotes the natural deformations of the bridge element,

dA and dB the nodal displacements of the associated rail ele-ments of Tracks A and B, respectively, [mb], [cb] and [kb] the mass,damping and stiffness matrices of the beam element, and [cbA], [kbA]and [cbB ], [kbB ] the damping and stiffness effects resulting from theinteraction with the rail elements of Tracks A and B, respectively,



through the ballast layer. The system matrices [mb], [cb] and [kb]are identical to those given in Eq. (10.51). Due to the interactionbetween the rail and bridge elements, the matrices [cbA] and [kbA]are identical to [cAb] and [kAb], respectively, that is, [cbA] = [cAb] and[kbA] = [kAb]. Similarly, [cbB ] and [kbB ] are identical to [cBb] and[kBb], respectively. All the interaction matrices [cAb], [kAb] and [cBb],[kBb] have been made available in Eqs. (10.32) and (10.47). The totalforces f t

b of the bridge element are

f tb = fb + fm

b + f cb + fk

b = fb − [mb][R]ug − ([cbA] + [cbB ])[R]ug

− ([kbA] + [kbB ])[R]ug . (11.19)

Here, fmb represents the equivalent inertia forces due to the rigid

motions of the bridge element caused by the ground motion, andf c

b and fkb the equivalent damping and restoring forces due to the

restraint effect of the ballast layer beneath Tracks A and B relativeto the bridge element under the ground motion. Evidently, the partsof nodal forces induced by the ground motion on the bridge elementrelate not only to the acceleration, but also to the displacement andvelocity of the ground.


Tracks A and B

Both the left semi-infinite rail (LSR) and right semi-infinite rail(RSR) elements have one side with infinite boundary, which thereforehas only a single node at one end. As was described in Sec. 10.4.4,the infinite boundary of both the LSR and RSR elements is repre-sented by functions of decaying nature. Unlike the CFR element thatis lying on the bridge girder, both the LSR and RSR elements arelying on the embankments and therefore are directly affected by theground motion. The equation of motion for the LSR element used tosimulate Track A under the ground motion can be written as follows:

[ml]dAl + [cl]dAl + [kl]dAl = f tAl , (11.20)



where dAl denotes the nodal displacements of the element, whichconsists of three translations and three rotations at the starting node,

dAl = 〈uA2 vA2 wA2 θA2 ϕA2 ψA2 〉T , (11.21)

and [ml], [cl] and [kl] denote the mass, damping and stiffness matricesof the LSR element, which are identical to those given in Eq. (10.64).The total nodal forces f t

Al of the element, which consist of the loadsdirectly acting on the node and the equivalent nodal forces due toground motion, can be given as

f tAl = fAl + f c

Al + fkAl , (11.22)

where fAl denotes the loads directly acting on the nodes,

fAl = 〈FAx2 FAy2 FAz2 MAx2 MAy2 MAz2 〉T (11.23)

and f cAl and fk

Al the forces induced by the ground motion

f cAl = 2ldc∗bh2

∫ 0

−∞ugxNudx + 2ldc∗bv2

∫ 0

−∞ugyNvdx

+ 2ldc∗bh2

∫ 0

−∞ugzNwdx ,

fkAl = 2ldk∗

bh2

∫ 0

−∞ugxNudx + 2ldk∗

bv2

∫ 0

−∞ugyNvdx

+ 2ldk∗bh2

∫ 0

−∞ugzNwdx ,

(11.24)

where Nu, Nv and Nw denote the interpolation vectors for theLSR element of Track A as defined in Sec. 10.4.4 and c∗bh2, k∗

bh2 andc∗bv2, k∗

bv2 the damping and stiffness coefficients of the ballast per-unit-area on the soil roadbed along the horizontal (h) and vertical(v) directions. Supposing that the ground displacement and velocitydo not vary along the track, Eq. (11.24) can be rewritten,

f cAl = [cgl]ug = [ cglx cgly cglz ]ug ,

fkAl = [kgl]ug = [ kglx kgly kglz ]ug ,

(11.25)



where

cglx =⟨

2ldc∗bh2

λu0 0 0 0 0

⟩T

,

cgly =⟨

02ldc∗bv2

λv0 0 0

−2ldc∗bv2

2λ2v

⟩T

,

cglz =⟨

0 02ldc∗bh2

λw0

2ldc∗bh2

2λ2w

0⟩T

,

(11.26)

kglx =⟨

2ldk∗bh2

λu0 0 0 0 0

⟩T

,

kgly =⟨

02ldk∗

bv2

λv0 0 0

−2ldk∗bv2

2λ2v

⟩T

,

kglz =⟨

0 02ldk∗

bh2

λw0

2ldk∗bh2

2λ2w

0⟩T

,

(11.27)

where λu, λv, λw denote the longitudinal, vertical and lateral charac-teristic numbers of the beam-Winkler foundation system, which havebeen given in Eqs. (10.62).

Similarly, the equation of motion for the LSR element used torepresent Track B under the ground motion can be derived as

[ml]dBl + [cl]dBl + [kl]dBl = f tBl , (11.28)

where the total nodal loads acting on the element are

f tBl = fBl + f c

Bl + fkBl . (11.29)

Here, the external nodal force vector fBl has been given inEq. (10.76), i.e.,

fBl = 〈FBx2 FBy2 FBz2 MBx2 MBy2 MBz2 〉T (11.30)

and the earthquake-induced forces are

f cBl = f c

Al ,

fkBl = fk

Al .(11.31)




Tracks A and B

By following the same procedure for derivation of the LSR element,the equation of motion for the RSR element of Track A consideringthe ground motion can be derived,

[mr]dAr + [cr]dAr + [kr]dAr = f tAr , (11.32)

where dAr denotes the nodal displacement vector,

dAr = 〈uA1 vA1 wA1 θA1 ϕA1 ψA1 〉T (11.33)

and [mr], [cr] and [kr] the mass, damping and stiffness matrices ofthe RSR element, which have been made available in Eq. (10.72).The total nodal forces f t

Ar of the element are

f tAr = fAr + f c

Ar + fkAr , (11.34)

where the external load vector fAr is

fAr = 〈FAx1 FAy1 FAz1 MAx1 MAy1 MAz1 〉T (11.35)

and the earthquake-induced forces are

f cAr = [cgr]ug = [ cgrx cgry cgrz ]ug ,

fkAr = [kgr]ug = [ kgrx kgry kgrz ]ug .

(11.36)

Assuming that the earthquake motion is synchronous for the twoends of the bridge, one therefore has the following:

cgrx = cglx ,

cgry =⟨

02ldc∗bv2

λv0 0 0

2ldc∗bv2

2λ2v

⟩T

,

cgrz =⟨

0 02ldc∗bh2

λw0

−2ldc∗bh2

2λ2w

0⟩T

,

(11.37)



kgrx = kglx ,

kgry =⟨

02ldk∗

bv2

λv0 0 0

2ldk∗bv2

2λ2v

⟩T

,

kgrz =⟨

0 02ldk∗

bh2

λw0

−2ldk∗bh2

2λ2w

0⟩T

.

(11.38)

As can be observed from the derivations presented in this sec-tion for the CFR, LSR, RSR and bridge elements, the considera-tion of support or ground motions results only in inclusion of someearthquake-induced forces in the nodal force vectors, while the mass,damping and stiffness matrices remain exactly the same as those forthe case with no ground motions. Because of this, the earthquake-induced effects can be easily included in existing vehicle–bridge in-teraction analysis programs with no change on the system matrices.The only thing that should be taken into account is to expand thevectors of nodal forces to include the nodal forces induced by theearthquake.

11.4. Method of Analysis

In the preceding section, focus has been placed on derivation of theequations of motion for each component of the railway–bridge sys-tem. By assembling the element matrices and vectors for all thecomponents involved, the global matrices and vectors, as well as theequations of motion, for the railway–bridge or supporting systemcan be established. The railway–bridge system represents only thenonmoving subsystem of the train–bridge system. The other sub-system is the moving train, which consists of a number of vehicles.In this study, the equations of motion for the moving train are con-structed using the procedure presented in Sec. 10.3. The supportingand moving subsystems interact with each other through the con-tact points existing between the rails and rolling wheels, as the trainmoves. Clearly, the equations of motion for the two subsystems arecoupled and time-dependent.



As shown in Fig. 11.2, each of the train cars considered has eightwheels, i.e., i = 1 ∼ 8. Each wheel has a vertical and a lateral contactforce. Thus, there is a total of 16 contact forces, i.e., j = 1 ∼ 16.Using the Newmark-type finite-difference formulas, together with theconstraint relation for relating the wheelset displacements to the raildisplacements, one can first solve the equations of motion for eachvehicle to obtain the vertical and lateral contact forces, i.e., Vi andHi, in terms of the contact-point displacements dcj and vehicle forcespc and qc as in Eqs. (10.18) and (10.19), that is,

Vi,t+∆t = pc(2i−1),t+∆t + qc(2i−1),t

+16∑

j=1

(mc(2i−1)j dcj,t+∆t + cc(2i−1)j dcj,t+∆t

+ kc(2i−1)jdcj,t+∆t) , (11.39)

Hi,t+∆t = pc(2i),t+∆t + qc(2i),t

+16∑

j=1

(mc(2i)j dcj,t+∆t + cc(2i)j dcj,t+∆t

+ kc(2i)jdcj,t+∆t) , (11.40)

where the subscripts i = 1 ∼ 8, j = 1 ∼ 16, and the matrices[mc], [cc], [kc] and vectors pc, qc relate to the physical propertiesand wheel-load effects of the vehicle, as defined in Eqs. (10.16) and(10.17).

By using the dynamic condensation technique developed in Chap-ter 8, the DOFs of each of the moving vehicles can be condensed intothe associated rail element(s) in contact to form the vehicle–rails in-teraction (VRI) element(s). By assembling all the VRI elements, theordinary rail elements and the bridge elements, the equations of mo-tion for the entire railway–bridge system can be established, whichappear as second-order differential equations. The Newmark β inte-gration method can then be called for to solve the system equationsfor each time step. Such an approach enables us to compute the dy-namic responses of all the components involved in the train–bridge



system, including the vehicle response, contact forces and bridge re-sponse. As was mentioned previously, the equations of motion forthe rail elements and bridge element considering the ground motionare identical to those for the case with no ground motions, exceptthat the earthquake-induced forces should be included in the nodalforces for the present case. Consequently, the procedure presented inSec. 10.7 for constructing the system matrices and the one in Sec. 8.7for performing the time-history analysis remain applicable herein foranalysis of the train–bridge system shaken by earthquakes, if theearthquake-induced forces, which are functions of the acceleration,velocity and displacement of the ground motion, are duly includedand updated at each incremental step.

11.5. Description of Input Earthquake Records

Four sets of ground accelerations induced by earthquakes are selectedas the source of vibration. The first two sets are those for the 1940El Centro and 1994 Northridge Earthquakes. The last two sets, i.e.,TAP003 and TCU068, recorded at the free-field stations of Taipeiand Taichung, respectively, during the 1999 Chi–Chi Earthquake inTaiwan, are used to simulate the far field and near fault excitations.The records for the El Centro and Northridge Earthquakes containonly a horizontal component of vibrations, while those for the Chi–Chi Earthquake contain both the EW horizontal and vertical com-ponents. In this study, the horizontal excitation of the earthquake isapplied in the lateral (z) direction of the bridge for the sake of eval-uating the stability of passing trains. Whenever the lateral groundmotion is scaled down in terms of the PGA, the vertical ground mo-tion is scaled down as well in a proportional manner.

Table 11.1 shows some key data for the EW and vertical motionsof the TAP003 and TCU068 ground motions. As can be seen, thePGA values of the TCU068 Station are much larger than those ofthe TAP003 Station, since the former was recorded at a station muchcloser to the fault than the latter. For the TCU068 Station, thevertical PGA appears to be as large as the lateral (EW) PGA dueto the near-fault effect. In contrast, for the TAP003 Station, the



Table 11.1. PGA, PGV, and PGD for TAP003 andTCU068 ground motions.

Ground motion TAP003 TCU068

Direction EW Vertical EW Vertical

PGA (gal) 127.45 43.29 501.87 519.72PGV (cm/s) 31.33 8.61 156.91 142.41PGD (cm) 15.35 4.15 83.92 102.34

vertical PGA is only one-third of the lateral (EW) PGA, which istypical of the far-field ground motion. The EW and vertical groundaccelerations of the TAP003 Station were plotted in Figs. 11.3(a) and11.4(a), respectively, for a duration of 70 s. For comparison, the EWand vertical ground accelerations of the TCU068 Station were plottedin Figs. 11.5(a) and 11.6(a), respectively, with the same duration.The impulse-type vertical acceleration shown in Fig. 11.6(a) is typicalof a near-fault earthquake.

-150

-100

-50

0

50

100

150

0 10 20 30 40 50 60 7

Time (s)

Gro

und S

urf

ace A

ccele

ration (g

al)

0

Chi-Chi Earthquake

TAP003 EW

PGA=127.45 gal

(a)

Fig. 11.3. Histogram of lateral (EW) motion of TAP003 Station: (a) accelera-tion, (b) velocity, and (c) displacement.



-40

-30

-20

-10

0

10

20

30

0 10 20 30 40 50 60 7

Time (s)

Gro

un

d S

urf

ace

Ve

locity

(cm

/s^2

)

0

Chi-Chi Earthquake

TAP003 EW

PGV=31.33 cm/s

(b)

-15

-10

-5

0

5

10

15

20

0 10 20 30 40 50 60 7

Time (s)

Gro

und S

urf

ace D

ispla

cem

ent (

cm

)

0

Chi-Chi Earthquake

TAP003 EW

PGD=15.35 cm

(c)




-50

-40

-30

-20

-10

0

10

20

30

40

0 10 20 30 40 50 60 7

Time (s)

Gro

und S

urf

ace A

ccele

ration (g

al)

0

Chi-Chi Earthquake

TAP003 Vertical

PGA=43.29 gal

(a)

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 10 20 30 40 50 60 7

Time (s)

Gro

un

d S

urf

ace

Ve

locity

(cm

/s)

0

Chi-Chi Earthquake

TAP003 Vertical

PGV=8.61 cm/s

(b)

Fig. 11.4. Histogram of vertical motion of TAP003 Station: (a) acceleration,(b) velocity, and (c) displacement.



-5

-4

-3

-2

-1

0

1

2

3

4

5

0 10 20 30 40 50 60

Time (s)

Gro

und S

urf

ace D

ispla

cem

ent (

cm

)

70

Chi-Chi Earthquake

TAP003 Vertical

PGD=4.15 cm

(c)


-600

-400

-200

0

200

400

600

0 10 20 30 40 50 60 7

Time (s)

Gro

und S

urf

ace A

ccele

ration (g

al)

0

Chi-Chi Earthquake

TCU068 EW

PGA=501.87 gal

(a)

Fig. 11.5. Histogram of lateral (EW) motion of TCU068 Station: (a) accelera-tion, (b) velocity, and (c) displacement.



-200

-150

-100

-50

0

50

100

0 10 20 30 40 50 60

Time (s)

Gro

un

d S

urf

ace

Ve

locity (c

m/s

)

70

Chi-Chi Earthquake

TCU068 EW

PGV=156.91 cm/s

(b)

-80

-60

-40

-20

0

20

40

60

80

100

0 10 20 30 40 50 60

Time (s)

Gro

und S

urf

ace D

ispla

cem

ent (

cm

)

70

Chi-Chi Earthquake

TCU068 EW

PGD=83.92 cm

(c)




-400

-300

-200

-100

0

100

200

300

400

500

600

0 10 20 30 40 50 60

Time (s)

Gro

und S

urf

ace A

ccele

ration (g

al)

70

Chi-Chi Earthquake

TCU068 Vertical

PGA=519.72 gal

(a)

-100

-50

0

50

100

150

200

0 10 20 30 40 50 60 7

Time (s)

Gro

und S

urf

ace V

elo

city (c

m/s

)

0

Chi-Chi Earthquake

TCU068 Vertical

PGV=142.41 cm/s

(b)

Fig. 11.6. Histogram of vertical motion of TCU068 Station: (a) acceleration,(b) velocity, and (c) displacement.



-120

-100

-80

-60

-40

-20

0

20

40

60

80

0 10 20 30 40 50 60 7

Time (s)

Gro

und S

urf

ace D

ispla

cem

ent (c

m)

0

Chi-Chi Earthquake

TCU068 Vertical

PGD=102.34 cm

(c)


As was stated previously, the seismic analysis of a train–railwaysystem requires not only information on ground acceleration, butalso ground displacement and velocity as part of the input source.Since only ground accelerations were recorded for most earthquakes,the ground velocity will be obtained through integration of the ac-celeration record with base-line correction and Ormsby filtering foreliminating the lower frequencies. Similarly, the ground displacementwill be obtained from the velocity history with Ormsby filtering. Thehistograms of the ground motions computed in this way for the EWand vertical components of the far-field station TAP003 have beenplotted as parts (b) and (c) in Figs. 11.3 and 11.4, and those of thenear-fault station TCU068 in Figs. 11.5 and 11.6. The histogramsfor the El Centro and Northridge Earthquakes are not shown sincethey are well known. In these figures the peak ground acceleration(PGA), peak ground velocity (PGV) and peak ground displacement(PGD) are also indicated.



The ballast inserted between the rails and roadbed may displaycertain nonlinear behavior under severe ground motions, especiallyin the horizontal direction, which may affect the stability of mov-ing trains, but has not been fully explored. According to ProvisionDS-804 of Germany (Deutsche Bundesbahn, 1993), the horizontalresistance-relative displacement relation of the ballast can be approx-imately regarded as linear if the relative displacement of the ballastwith respect to the bridge in the horizontal direction is less than2 mm. For the four ground motions considered in this chapter, themaximum relative displacement between the rails and the bridge inthe lateral direction caused by the ground motion is less than 2 mmif the PGA is less than 80 gal, as shown in Fig. 11.7. Furthermore, itis quite possible that a well-designed bridge remains linearly elasticwhen subjected to ground motions with a PGA of up to 80 gal. Forthe reasons stated, as well as for simplification, the lateral PGAs ofall the four ground motions will be limited to 80 gal, in order not toviolate the assumption of linearity for tracks and structures.

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100

Lateral PGA of Ground Motion (gal)

Ma

xim

um

Re

lative

Dis

pla

ce

me

nt

be

twe

en

Ra

ils a

nd

Bri

dg

e

(mm

)

120

El Centro

Northridge

TAP003

TCU068

PGA = 81 gal

Fig. 11.7. Maximum relative displacement of ballast under the action of the fourground motions.



11.6. Train Resting on Railway Bridge underEarthquake

Throughout the numerical studies, the train car is assumed to beof the SKS Series 300 type and will be represented by the modelbriefly summarized in Sec. 11.2, or given in more details in Sec. 10.3,together with the fundamental properties listed in Table 10.1. Thetime increment ∆t used in the incremental analysis is 0.005 s, whichis smaller than that used for recording the ground motions. It impliesa frequency of f = 1/∆t = 200 Hz, higher than those implied by thewheels and rails. As a first test of the theory derived, we shall studythe dynamic stability of a train car resting on a simply-supportedbridge of 30 m in length shaken by the four earthquakes considered.The car is assumed to remain stationary on Track A of the bridgewith its first wheelset located at the position x = 25 m prior to theearthquake (see Fig. 11.8). As was stated previously, all the fourground motions are normalized to have a lateral PGA of 80 gal.

L

linecenter

mx 25=mx 5=

Fig. 11.8. Train car resting on railway bridge under earthquake excitation.



11.6.1. Responses of Bridge and Train Car

Both the bridge and train car are to vibrate from their static equi-librium (deformed) positions when the earthquake occurs. The re-sponses computed from the bridge and train car subjected to thelateral (EW) and vertical excitations of the TAP003 Station wereplotted in Figs. 11.9 and 11.10, respectively, along with those due toa lateral (EW) excitation of the same station only. As the car hasbeen parked symmetrically on the bridge (see Fig. 11.8), for which noyawing or pitching vibrations will be induced on the car, the resultsfor the yawing and pitching responses were zero and just skippedfrom Figs. 11.9 and 11.10.

The following observations can be made from Fig. 11.9 for thebridge: (1) The absolute vertical response of the bridge is primarilydetermined by the vertical component of the ground motion, but thebridge displacement is very small compared with the ground displace-ment shown in Fig. 11.4(c). (2) The absolute lateral response of the

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Bridge M

idspan A

bsolu

te V

ert

ical

Dis

pla

cem

ent (

m)

lateral (EW)

lateral (EW) + vertical

Chi-Chi Earthquake

TAP003

(a)

Fig. 11.9. Midspan responses of bridge carrying a train car at rest under theTAP003 excitation: (a) vertical, (b) lateral, and (c) torsional.



-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Brid

ge

Mid

sa

pn

Ab

so

lute

La

tera

l

Dis

pla

ce

me

nt

(m

)

lateral (EW)


Chi-Chi Earthquake

TAP003

(b)

-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Bridge M

idspan T

ors

ional A

ngle

(r

ad)

lateral (EW)


Chi-Chi Earthquake

TAP003

(c)




-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Vehic

le V

ert

ical A

ccele

ration (m

/s^2

)

lateral (EW)

lateral (EW) + verticalChi-Chi Earthquake

TAP003

(a)

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ve

hic

le L

ate

ral A

cce

lera

tio

n (m

/s^2

)

lateral (EW)


Chi-Chi Earthquake

TAP003

(b)

Fig. 11.10. Responses of train car resting on bridge under the TAP003 excita-tion: (a) vertical, (b) lateral, and (c) rolling.



-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Vehic

le R

olli

ng A

ccele

ration

(rad/s

^2)

lateral (EW)


Chi-Chi Earthquake

TAP003

(c)


bridge is dominated by the lateral (EW) component of the groundmotion, but its magnitude is very small compared with that of theground. (3) The torsional response of the bridge depends primarilyon the lateral component of the ground motion due to the coupledlateral-torsional vibration of the bridge.

The following observations can be drawn from Fig. 11.10 for thetrain car: (1) The vertical acceleration of the cars is induced mainlyby the vertical component of the ground motion, while the lateraland rolling accelerations of the car are induced primarily by the lat-eral component of the ground motion. (2) The maximum verticalacceleration of the car induced (≈0.28 m/s2) is less than that ofthe track (≈1.57 m/s2), which can be attributed to the isolationeffect of the vertical suspension systems of the car. (3) The maxi-mum lateral acceleration of the car induced (≈1.18 m/s2) is nearlyas large as that of the track (≈1.24 m/s2), indicating that the lateralsuspension systems of the car are not effective in isolating the carbody from the lateral vibration of the track. The peak values for the



track responses have not been shown here, which are available in Wu(2000).

The responses of the bridge and train car to the TCU068 excita-tions are all much larger than those to the TAP003 excitations, asshown in Figs. 11.11 and 11.12, due to greater intensity of the former.The initial pulse phenomenon is observed in all the acceleration re-sponses of the bridge induced by the TCU068 excitations. It is worthnoting that the lateral and rolling accelerations of the train car alsodisplay the initial pulse behavior, which were not observed in thecase by the TAP003 excitations. Here, the maximum vertical andlateral accelerations of the car are equal to 3.92 m/s2 and 3.21 m/s2,respectively. Correspondingly, the maximum accelerations for thetracks are 7.42 m/s2 and 4.8 m/s2 (Wu, 2000). Again, it is indicatedthat the vertical suspension system of the car has a better efficiencyin isolating the car body from the vibration of the track, comparedwith the lateral suspension system.

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Brid

ge

Mid

sp

an

Ab

so

lute

Ve

rtic

al

Dis

pla

ce

me

nt

(m

)

lateral (EW)


Chi-Chi Earthquake

TCU068

(a)

Fig. 11.11. Midspan responses of bridge carrying a train car at rest under theTCU068 excitation: (a) vertical, (b) lateral, and (c) torsional.



-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Time (s)

Bridge M

idspan A

bsolu

te L

ate

ral

Dis

pla

cem

ent (

m)

50

lateral (EW)


Chi-Chi Earthquake

TCU068

(b)

-2.0E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Brid

ge

Mid

sp

an

To

rsio

na

l A

ng

le

(ra

d)

lateral (EW)


Chi-Chi Earthquake

TCU068

(c)




-3

-2

-1

0

1

2

3

4

5

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Vehic

le V

ert

ical A

ccele

ration (m

/s^2

)

lateral (EW)


Chi-Chi Earthquake

TCU068

(a)

-4

-3

-2

-1

0

1

2

3

4

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Vehic

le L

ate

ral A

ccele

ration (m

/s^2

)

lateral (EW)


Chi-Chi Earthquake

TCU068

(b)

Fig. 11.12. Responses of train car resting on bridge under the TCU068 excita-tion: (a) vertical, (b) lateral, and (c) rolling.



-3

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ve

hic

le R

olli

ng

Acce

lera

tio

n (r

ad

/s^2

)

lateral (EW)


Chi-Chi Earthquake

TCU068

(c)


11.6.2. Contact Forces between Wheels and Rails

The vertical and lateral contact forces induced by the four excitationsbetween the wheelsets and rails have been shown in Figs. 11.13(a)–10.13(d), in which only the contact forces for the first and secondwheels of the first wheelset of the car shown in Fig. 11.2(b) werepresented. As can be seen, the vertical contact forces of the twowheels oscillate and cross each other during the earthquake. Theoscillation appears to be most severe for the TAP003 excitation (seeFig. 11.13(c)), where the vertical contact force of the first wheelreaches a maximum of 77.6 kN and a minimum of 57.4 kN, and thatof the second wheel reaches a maximum of 77.3 kN and a minimumof 58.0 kN, implying a fluctuation of 30% and 29%, respectively,with respect to the static axle load of 67 kN sustained by one wheel.The crossing behavior of the two wheels of the first wheelset in eachfigure is an indication of the occurrence of rolling motion during theearthquake, which is harmful to the stability of the train car. Factors



-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Co

nta

ct

Fo

rce

s o

f th

e 1

st

Wh

ee

lse

t (

kN

)

1st wheel

2nd wheel

El Centro Earthquake

Normalized PGA = 80 gal

Vertical contact force

Lateral contact force

V 1

max = 71.1 kN

V 2

min = 63.6 kN

V 2

max = 70.1 kN

V 1

min = 64.7 kN

H 1max = H

2max = 1.8 kN

(a)

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Co

nta

ct

Fo

rce

s o

f th

e 1

st

Wh

ee

lse

t (

kN

)

1st wheel

2nd wheel

Northridge Earthquake Oxnard




H 1max = H

2max = 2.9 kN

V 1

max = 72.5 kN

V 1

min = 63.8 kN

V 2

max = 71.2 kN

V 2

min = 63.3 kN

(b)

Fig. 11.13. Contact forces of the 1st wheelset of the train car under earthquake:(a) El Centro, (b) Northridge, (c) TAP003, and (d) TCU068.



-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Co

nta

ct

Fo

rce

s o

f th

e 1

st

Wh

ee

lse

t (

kN

)

1st wheel

2nd wheel

Chi-Chi Earthquake TAP003




V 1

max = 77.6 kN

V 2min = 58.0 kN

H 1max = H

2max = 4.0 kN

V 1min = 57.4 kN

V 2

max = 77.3 kN

(c)

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Co

nta

ct

Fo

rce

s o

f th

e 1

st

Wh

ee

lse

t (

kN

)

1st wheel

2nd wheel

Chi-Chi Earthquake TCU068




V 2

max = 74.7 kN

V 1

max = 73.9 kN

V 2

min = 61.4 kN

V 1

min = 60.1 kN

H 1max = H

2max = 3.1 kN

(d)




that may affect the contact forces of the car during the earthquakeinclude the suspension stiffness of the car, the lateral–torsional stiff-ness of the bridge, the intensity (PGA) and frequency intent of theearthquake. It should be added that the fourth wheelset shows thesame behavior as that of the first wheelset due to symmetry, andthat similar, but smaller, rolling responses exist for the second andthird wheelsets, which are not shown.

11.6.3. Maximum YQ Ratio for Wheelsets in

Earthquake

As shown in Fig. 11.2, a typical train car consists of four wheelsets,and each wheelset consists of two wheels, i.e., the left and rightwheels. All the criteria presented in Sec. 10.13 can be used to assessthe risk of derailment of each wheel, each wheelset or the entire traincar. For instance, the SYQ ratio (single wheel lateral to vertical forceratio) is good for a single wheel, and the YQ ratio (wheelset lateralto vertical force ratio) can be used for a single wheelset. Both theSYQ and YQ ratios have been used as the indices by Wu (2000) inassessing the risk of derailment of train cars subjected to the samefour earthquakes as in this study.

For the present purposes, we shall adopt only the YQ ratio in eval-uating the stability of a train car resting on railway bridges shakenby earthquakes. This parameter can be defined as follows:

(Y Q)i =∣∣∣∣H(2i−1)

V(2i−1)

∣∣∣∣+∣∣∣∣H(2i)

V(2i)

∣∣∣∣ , (11.41)

where i = 1, 2, 3 or 4 for each of the four wheelsets, and Hj andVj , with j = 1 ∼ 8, respectively, denote the lateral and verticalcontact forces acting on a wheel (see Fig. 11.2(b)). It should benoted that the definition of the YQ ratio adopted here (Eq. (11.41))is somewhat different from that given in Eq. (10.116). If Vj 0, thejth wheel encounters the jump condition, and the wheelset to whichthe jth wheel belongs is at a high risk of derailment. To preventthe wheelset from derailment, an upper limit must be set on the YQratio. However, the determination of the stability limit is not easy,



which requires more analytical and experimental investigations. Inthis study, a value of YQ = 1.5 is set for the wheelset to remain stable,with no risk of derailment (Elkins and Carter, 1993). According toElkins and Carter (1993) and Ma and Zhu (1998), a wheel will be safeand free of derailment (wheel climb), if the lateral (H) to vertical (V )force ratio (SYQ ratio) for the wheel does not exceed 1.0, namely, ifSYQ = |H/V | 1.0.

Based on the above considerations, there exists a transition zonefrom YQ = 1.5 to 2.0, where a wheelset will be at a high risk ofderailment. This transition zone will be referred to as possible de-railment zone in this study. For the case when the YQ value com-puted is larger than 2.0 or when V 0, which implies the occurrenceof derailment or jump with the wheelset, it will be regarded as thecondition of derailment. Accordingly, the YQ value will be automat-ically set to 2.0, to distinguish it from the other two cases in thefigures. Note that the actual derailment of a wheelset depends notonly on the magnitude of the YQ ratio of the wheelset, but also onthe lasting time of the YQ ratio exceeding the safety limit. Thus, itis conservative to assess the derailment risk of a wheelset using onlythe YQ ratio, as is done in this study, and to assess the stabilityof a train (car) using the maximum YQ ratios computed for all thewheelsets of the train (car).

Figure 11.14 shows the time-history plot of the maximum YQ ra-tio for the four wheelsets of the train car under the four ground mo-tions. As can be seen, the peak YQ ratio computed for the TAP003excitation is the largest among the four excitations considered, inconsistence with the observation made in Sec. 11.6.2 for the contactforces. In addition, the YQ ratios computed for the TAP003 andTCU068 excitations are generally larger than those for the El Cen-tro and Northridge Earthquakes. It can be observed that all theYQ ratios computed for all the four excitations fall well below thesafety limit of 1.5, indicating that the four wheelsets of the train cardo not exhibit any risk of instability under the four ground motionsspecified. Therefore, it is concluded that a train initially at rest onthe bridge will remain stable for the four earthquakes with a PGA of80 gal, as long as the bridge remains fully elastic. Here, one should



-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20 25 30 35 40 45 50

Time (s)

YQ

Ra

tio

fo

r T

rain

Ca

r

El Centro

Northridge

TAP003

TCU068

Normalized PGA = 80 gal YQ = 0.129

YQ = 0.094YQ = 0.087

YQ = 0.053

Fig. 11.14. Time history of the YQ ratio for the train car.

not forget that in reality the PGA level can be much higher for anear-fault motion than for a far-field motion.

11.6.4. Stability of an Idle Train under Earthquakes

of Various Intensities

The railway engineers may be interested in under what PGA level ofexcitation, an initially idle train car begins to lose its stability. Forthe case of lateral excitations only, the maximum YQ ratios com-puted for the wheelsets of the train car under the four target groundmotions with various PGAs have been drawn in Fig. 11.15. As canbe seen, the maximum YQ ratios computed for the four excitationsincrease in proportion to the lateral PGA of the ground motion. Be-sides, the maximum YQ ratios for all range of the PGAs considered,i.e., up to 80 gal, are much less than the allowable limit of 1.5, in-dicating that all the wheelsets of the train car will not exhibit anyinstability or derailment risk under the ground motions specified ifthe lateral PGA does not exceed 80 gal. Of interest is the fact thatthe YQ ratio for the TAP003 motion is the largest among the four



0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50 60 70 8

Lateral PGA of Ground Motion (gal)

Ma

xim

um

YQ

Ra

tio

fo

r T

rain

Ca

r

0

El Centro

Northridge

TAP003 (lateral)

TCU068 (lateral)

TAP003 (lateral+vertical)

TCU068 (lateral+vertical)

safety limit = 1.5

Fig. 11.15. Maximum YQ ratio for the train car versus lateral PGA of the groundmotion.

ground motions of the same PGA, implying that the TAP003 motionis more detrimental to stability of the train car than the other three,given the same PGA level.

For the case of simultaneous lateral and vertical excitations, themaximum YQ ratios computed for the train car under the TAP003and TCU068 ground motions have also been shown in Fig. 11.15, withthe PGA of the vertical motion assumed to be proportional to thatof the lateral one. As can be seen, the maximum YQ ratio computedof the train car under the action of both the lateral and verticalmotions appears to be nearly the same as that under the lateralmotion only, indicating that the vertical ground motion has littleinfluence on the stability of trains initially resting on a railway bridge.An interpretation for this is that the vertical damping mechanism ofthe car has been given sufficient time to dissipate the vibrationalenergy, since the car is just “staying” on the bridge. The same is nottrue for cars traveling over the bridge, for which the acting time isvery short.



11.7. Trains Moving over Railway Bridges underEarthquakes

In this section, we shall study the stability of a train traveling over abridge shaken by earthquakes. The train is assumed to consist of 15identical cars, which will pass through the bridge at constant speeds.

11.7.1. Responses of Bridge and Train Car

Both the TAP003 and TCU068 ground motions are used as the inputexcitations. For each of the two ground motions, the train is assumedto pass through the bridge at the speeds of 60 m/s (= 216 km/h) and30 m/s (= 108 km/h). Because the time taken by the passage of thetrain over the bridge is much shorter than the duration of the actingtime of the earthquake, the extreme state for the maximum responseor instability of the train to occur cannot be easily identified. Thismakes the present investigation rather difficult. Here, we supposethat either the lateral or vertical PGA of the ground motion occursat the supports of the bridge at the instant when the train begins toenter the bridge. As to whether the lateral or vertical PGA shouldbe used as the reference for determining the ground motions dependson the characteristics of the geological conditions in the vicinity ofthe bridge considered. For the present purposes, we shall select theinstant at which the lateral PGA occurs as the instant for the trainto enter the bridge for the TAP003 motion, and the instant at whichthe vertical PGA occurs for the TCU068 motion.

The midspan responses of the bridge subjected simultaneouslyto the passage of the train and the TAP003 excitation have beenplotted with respect to the nondimensional time parameter vt/L inFig. 11.16. As was expected, the response of the bridge is inducedmainly by the earthquake excitation, that is, the vertical and lat-eral/torsional responses are caused mainly by the vertical and lateralcomponents of the earthquake, respectively. The effect of moving ve-hicles on the bridge response is quite small, as can be confirmed bythe fact that the results computed for the two cases v = 30 m/sand 60 m/s coincide generally with each other, if the time parameter



-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1


Bri

dg

e M

idsp

an

Ve

rtic

al D

isp

lace

me

nt

(m)

4

v=60 m/s (lateral (EW))

v=60 m/s (lateral (EW) + vertical)



Chi-Chi Earthquake

TAP003

(a)

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1


Bri

dg

e M

idsp

an

La

tera

l D

isp

lace

me

nt

(m

)

4





Chi-Chi Earthquake

TAP003

(b)

Fig. 11.16. Midspan responses of bridge to moving train under the TAP003excitation: (a) vertical, (b) lateral, and (c) torsional.



-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14


Bri

dg

e M

idsp

an

To

rsio

na

l D

isp

lace

me

nt

(ra

d)





Chi-Chi Earthquake

TAP003

(c)


vt/L for the horizontal coordinate is transformed to real time t. Thismeans that the effect of moving trains can be neglected in computingthe bridge response to earthquake excitations.

In Fig. 11.17, the responses of the first (leading) car of the trainunder the same earthquake excitations have been plotted with re-spect to the nondimensional time parameter vt/L. By comparingthe results obtained for bi-directional excitations with those for uni-directional (EW) excitations, it is observed that the vertical andpitching accelerations of the first train car are greatly enhanced bythe presence of vertical excitation. In contrast, the lateral, rolling,and yawing motions of the train car are generated primarily by thelateral component of excitation and are independent of vertical ex-citation. Furthermore, the lateral, rolling, and yawing accelerationsfor the higher train speed (v = 60 m/s) are larger than those forthe lower train speed (v = 30 m/s). The same is not true for thevertical and pitching accelerations, where the responses for the twoaccelerations for the higher speed (v = 60 m/s) are contrarily lessthan those for the lower speed (v = 30 m/s).



-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-1 0 1 2 3 4 5


1st

Ca

r V

ert

ica

l A

cce

lera

tio

n

(m/s

^2

)





Chi-Chi Earthquake

TAP003

(a)

-5

-4

-3

-2

-1

0

1

2

3

4

-1 0 1 2 3 4


1st

Ca

r L

ate

ral A

cce

lera

tio

n

(m/s

^2

)

5





Chi-Chi Earthquake

TAP003

(b)

Fig. 11.17. Response of train car moving over bridge under the TAP003excitation: (a) vertical, (b) lateral, (c) rolling, (d) yawing, and (e) pitching.



-8

-6

-4

-2

0

2

4

6

8

-1 0 1 2 3 4 5


1st

Ca

r R

olli

ng

Acce

lera

tio

n

(ra

d/s

^2

) v=60 m/s (lateral (EW))




Chi-Chi Earthquake

TAP003

(c)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 0 1 2 3 4 5


1st

Ca

r Y

aw

ing

Acce

lera

tio

n

(ra

d/s

^2

)





Chi-Chi Earthquake

TAP003

(d)




-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-1 0 1 2 3 4 5


1st

Ca

r P

itch

ing

Acce

lera

tio

n

(ra

d/s

^2

)





Chi-Chi Earthquake

TAP003

(e)


As for the TCU068 excitation, the results computed for the bridgeunder a train moving at speeds v = 60 and 30 m/s have been plot-ted with respect to the nondimensional time parameter vt/L inFig. 11.18, and those for the first car of the train in Fig. 11.19.Clearly, the behaviors of the bridge and train car under the TCU068excitation show a trend similar to those for the TAP003 excitation.There is no surprise that the responses of the bridge and vehicle tothe TCU068 excitation are much larger than those to the TAP003 ex-citation. As can be seen from Fig. 11.19, under the TCU068 groundmotion, all the acceleration responses of the vehicle for the higherspeed (v = 60 m/s) appear to be much larger than those for thelower speed (v = 30 m/s), which is somewhat different from thatobserved for the TAP003 ground motion. It should be noted thatfor the present case with a rather strong earthquake, the results pre-sented in Figs. 11.18 and 11.19 should not be regarded as a realisticrepresentation of the actual behaviors of the train and bridge, owingto the fact that the bridge is assumed to remain as a linearly elastic



-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1


Bri

dg

e M

idsp

an

Ve

rtic

al D

isp

lace

me

nt

(m)

4





Chi-Chi Earthquake

TCU068

(a)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1


Bri

dg

e M

idsp

an

La

tera

l D

isp

lace

me

nt

(m

)

4





Chi-Chi Earthquake

TCU068

(b)

Fig. 11.18. Midspan bridge responses to the TCU068 excitation: (a) vertical,(b) lateral, and (c) torsional.



-2.0E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

3.0E-03

3.5E-03

4.0E-03

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14


Brid

ge

Mid

sp

an

To

rsio

na

l D

isp

lace

me

nt

(ra

d)





Chi-Chi Earthquake

TCU068

(c)


-20

-15

-10

-5

0

5

10

15

20

25

30

-1 0 1 2 3 4 5


1st

Ca

r V

ert

ica

l A

cce

lera

tio

n

(m/s

^2

)





Chi-Chi Earthquake

TCU068

(a)

Fig. 11.19. Vehicle responses to the TCU068 excitation: (a) vertical, (b) lateral,(c) rolling, (d) yawing, and (e) pitching.



-25

-20

-15

-10

-5

0

5

10

15

-1 0 1 2 3 4 5


1st

Ca

r L

ate

ral A

cce

lera

tio

n

(m/s

^2

)





Chi-Chi Earthquake

TCU068

(b)

-50

-40

-30

-20

-10

0

10

20

30

40

-1 0 1 2 3 4 5

Nondimesional Time (vt/L )

1st

Ca

r R

olli

ng

Acce

lera

tio

n

(ra

d/s

^2

)





Chi-Chi Earthquake

TCU068

(c)




-5

-4

-3

-2

-1

0

1

2

3

4

-1 0 1 2 3 4 5


1st

Ca

r Y

aw

ing

Acce

lera

tio

n

(ra

d/s

^2

)





Chi-Chi Earthquake

TCU068

(d)

-4

-3

-2

-1

0

1

2

3

4

5

-1 0 1 2 3 4 5


1st

Ca

r P

itch

ing

Acce

lera

tio

n

(ra

d/s

^2

) v=60 m/s (lateral (EW))




Chi-Chi Earthquake

TCU068

(e)




structure, regardless of the PGA level of the earthquake. In fact, thebridge will exhibit large inelastic deformations or even collapse undera severe earthquake, such as the one (TCU068) considered herein, orother ground motions with lateral PGA greater than 0.4 g.

11.7.2. Maximum YQ Ratio for Moving Trains in

Earthquake

For the case of uni-directional (i.e., lateral) excitation, the train isassumed to pass through the bridge at a speed of 200 km/h (=55.6 m/s). The bridge supports are also assumed to be excited lat-erally by the earthquake at the specified PGA level exactly at theinstant when the train enters the bridge. The maximum YQ ratioscomputed for the whole train moving over the bridge subjected to thefour earthquake excitations mentioned in the preceding section withPGAs of 80 and 25 gal have been plotted with respect to the nondi-mensional time parameter vt/L in Figs. 11.20(a)–11.20(d). As can beseen from the case with PGA = 80 gal, the maximum YQ ratio forTCU068 exceeds evidently the stability limit of 1.5, indicating thatderailment may occur in near fault areas, while the maximum YQratios computed for the other three excitations are far below (for ElCentro and Northridge) or nearly equal to (for TAP003) the stabilitylimit. In contrast, for the PGA of 25 gal, the maximum YQ ratioscomputed for the four ground motions are well below (for El Centro,Northridge and TAP003) or just slightly larger than (for TCU068)the stability limit of 1.5, indicating that generally no derailment mayoccur with the train under the action of the four ground motions forthe PGA value specified.

11.7.3. Stability Assessment of Moving Trains in

Earthquake

In earthquake-prone regions, it is important to have some feeling re-garding the level of stability of a train running over bridges undersome specific earthquake excitations. For uni-directional (i.e., lat-eral) excitations, the maximum YQ ratios computed for the wholetrain under the El Centro excitation with different PGAs (up to



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 0 2 4 6 8 10 12 14


Ma

xim

um

YQ

Ra

tio

fo

r T

rain

lateral PGA = 25 gal


El Centro Earthquake

Train Speed = 200 km/h

safety limit = 1.5

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 0 2 4 6 8 10 12 14


Ma

xim

um

YQ

Ra

tio

of

Tra

in



safety limit = 1.5

Northridge Earthquake


(b)

Fig. 11.20. YQ ratio for the 1st wheelset of the 1st car of the train underearthquake: (a) El Centro, (b) Northridge, (c) TAP003, and (d) TCU068.



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 0 2 4 6 8 10 12 14


Ma

xim

um

YQ

Ra

tio

of

Tra

in



safety limit = 1.5

Chi-Chi Earthquake TAP003


(c)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 0 2 4 6 8 10 12 14


Ma

xim

um

YQ

Ra

tio

of

Tra

in



safety limit = 1.5

Chi-Chi Earthquake TCU068


(d)




80 gal) have been plotted with respect to the train speed (up to200 km/h) in Fig. 11.21. As can be seen, the values of 1.5 and 2.0were also imposed (see the vertical axis of Fig. 11.21(a)) to charac-terize the safety (with maximum YQ ratio 1.5), possible instability(with 1.5 < maximum YQ ratio 2.0) and instability (with maxi-mum YQ ratio > 2.0) regions for the train. Similarly, the maximumYQ ratios computed for the Northridge, TAP003 and TCU068 ex-citations with different PGAs have been plotted with respect to thetrain speed in Figs. 11.21–11.24, with the different regions of sta-bility identified in part (b) of the figures. In all these figures, thePGA values of 8 and 25 gal have been referred to as seismic zonesof Levels 3 and 4, respectively, according to the classification by theCentral Weather Bureau of Taiwan.

The following observations can be made from Figs. 11.21–11.24:(1) The maximum allowable speeds for stable running of the trainare higher for smaller PGAs and lower for larger PGAs. (2) Theranges for stable running of the train under TAP003 and TCU068 aremuch narrower than those for the other two excitations, indicatingthat the train has a higher risk of derailment or instability whentraveling over the bridge under the Chi–Chi Earthquake. (3) Thetrain can move safely under El Centro and Northridge Earthquakesfor all the speeds and PGAs considered. (4) If the lateral PGA ofTAP003 is less than 30 gal, the train can cross the bridge safely at aspeed of up to 200 km/h. The same is true for TCU068 if the PGAis less than 18 gal. (5) The train can move safely over the bridgewithout encountering any instability or derailment under TAP003and TCU068 with a lateral PGA of up to 80 gal, if the train speedis kept below 122 and 85 km/h, respectively.

For bi-directional (i.e., lateral and vertical) excitations, the re-sults obtained for the TAP003 and TCU068 ground motions havebeen plotted in Figs. 11.25 and 11.26, respectively. As can be seen,for the TAP003 excitation, the maximum allowable speeds for thetrain under bi-directional excitations are nearly the same as thosefor uni-directional excitation. In contrast, for the TCU068 motion,the maximum allowable speeds for the train under bi-directionalexcitations are significantly less than those under uni-directional



El Centro EarthquakeLateral

(a)

0 50 100 150 200

Train Speed (km/h)

0

20

40

60

80

Late

ral P

GA

(g

al)

El Centro EarthquakeLateral

Level 4(25 gal)

Level 5(80 gal)

Level 3(8 gal)

safety

(b)

Fig. 11.21. Risk of derailment for El Centro Earthquake: (a) three-phase plotand (b) safety boundary.



Northridge EarthquakeLateral

(a)

0 50 100 150 200

Train Speed (km/h)

0

20

40

60

80

La

tera

l P

GA

(g

al)

Northridge EarthquakeLateral

Level 5(80 gal)

Level 4(25 gal)

Level 3(8 gal)

safety

(b)

Fig. 11.22. Risk of derailment for Northridge Earthquake: (a) three-phase plotand (b) safety boundary.



Chi-Chi EarthquakeTAP003 Lateral

(a)

0 50 100 150 200

Train Speed (km/h)

0

20

40

60

80

La

tera

l P

GA

(g

al)

Chi-Chi EarthquakeTAP003 Lateral

Level 5(80 gal)

Level 4(25 gal)

Level 3(8 gal)

safety

possibleinstability

instability

(b)

Fig. 11.23. Risk of derailment for TAP003 ground motion (lateral only):(a) three-phase plot and (b) safety boundary.



Chi-Chi EarthquakeTCU068 Lateral

(a)

0 50 100 150 200

Train Speed (km/h)

0

20

40

60

80

La

tera

l P

GA

(g

al)

instability

possibleinstability

safety

Level 4(25 gal)

Level 5(80 gal)

Level 3(8 gal)

Chi-Chi EarthquakeTCU068 Lateral

(b)

Fig. 11.24. Risk of derailment for TCU068 ground motion (lateral only):(a) three-phase plot and (b) safety boundary.



Chi-Chi EarthquakeTAP003 Lateral+Vertical

(a)

0 50 100 150 200

Train Speed (km/h)

0

20

40

60

80

Late

ral P

GA

(g

al)

instability

Level 5(80 gal)

Level 4(25 gal)

Level 3(8 gal)

possibleinstability

safety

Chi-Chi EarthquakeTAP003 Lateral+Vertical

(b)

Fig. 11.25. Risk of derailment for TAP003 ground motion (lateral + vertical):(a) three-phase plot and (b) safety boundary.



Chi-Chi EarthquakeTCU068 Lateral+Vertical

(a)

0 50 100 150 200

Train Speed (km/h)

0

20

40

60

80

La

tera

l P

GA

(g

al)

Level 5(80 gal)

Level 4(25 gal)

Level 3(8 gal)

instability

possibleinstability

Chi-Chi EarthquakeTCU068 Lateral+Vertical

safety

(b)

Fig. 11.26. Risk of derailment for TCU068 ground motion (lateral + vertical):(a) three-phase plot and (b) safety boundary.



excitation. This can be attributed to the fact that the vertical PGAfor the far-field (TAP003) motion is very small, i.e., 27 gal, comparedwith the lateral PGA of 80 gal (normalized), while that for the near-fault (TCU068) motion is rather high, i.e., 83 gal, compared withthe same lateral PGA. In general, the presence of vertical excitationcan drastically reduce the stability region or the maximum allowablespeed for the train under a specific intensity (PGA).

It should be noted that if the maximum YQ ratios for all thewheelsets of the train fall into the possible instability or instabil-ity regions, it does not necessarily mean that the train will reallyencounter overall derailment or instability. One reason for this isthat the stability limits imposed on the YQ ratio are generally con-servative due to consideration of safety. Another reason is that eventhough the YQ ratio of a single wheelset may exceed the safety limit,it still requires some time for derailment to develop, due to the link-ing action of the other parts of the car or the whole train. Furtherstudy is required in this regard to investigate the development of de-railment and the mechanism involved taking into account the linkingeffect of all the train cars.


The dynamic stability of trains moving over bridges that are simul-taneously shaken by earthquakes was investigated. The equationsof motion for the train–rails–bridge system presented in Chapter 10were first generalized to include the effects of ground excitations,resulting in some nodal force terms related not only to the accelera-tion, but also velocity and displacement of the ground motion. Sincethe generalized system equations are identical in form to those withno ground motion, the analysis procedure established previously fordealing with the train–bridge interactions can directly be adopted,with modifications made only for the nodal forces to include theearthquake-induced effects.

The following conclusions remain strictly valid only for the con-ditions set in the numerical studies: (1) A train car initially restingon the bridge remains safe under the four ground motions specified,



up to a PGA level of 80 gal, assuming that no inelastic deformationoccurs on the bridge and ballast. (2) As for the train to move safelyover the bridge, the speed should be kept below the allowable speedcomputed for each excitation based on the YQ criterion. (3) Theproperty of ground motions and the presence of vertical excitationsaffect drastically the stability of the moving train, especially for near-fault excitations, as represented by the TCU068 record. (4) Variousregions of stability have been established for the train under the fourground motions.

This study represents only parts of a preliminary attempt to dealwith the seismic effects on train–rails–bridge interactions. Furtherstudy should be carried out to consider the effect of more severeearthquakes, say, with a PGA larger than 80 gal, and to includemore representative ground motions, as well as more statistics-basedcriteria for derailment.



Appendix A

Derivation of Response Function P1

in Eq. (2.55)

We are interested in the dynamic response of a simply-supportedbeam of span length L subjected to a sequence of N identical loadsof constant interval d moving at speed v. The most severe case occurswhen the former N − 1 loads have left the beam, and the Nth loadhas entered the beam, or equivalently, when tN < t < tN + L/v,where tN denotes the arriving time of the Nth load on the beam,i.e., tN = (N − 1)d/v. For this particular case, one can obtain theresponse function P1 from Eq. (2.54) as follows:


+

N−1∑j=1

[sinΩ1(t − tj) + sinΩ1

(t − tj − L

v

)]

−S1

N−1∑j=1

sinω1(t − tj) + sin ω1

(t − tj − L

v

)

×H

(t − tN − L

v

), (A.1)

where the term containing H(t − tN ) denotes the response excitedby the Nth moving load at time t and the term containing H(t −tN−1 −∆t), the free vibration caused by the former N − 1 loads. Bydefinition Ω1 = πv/L, it can be shown that

sin Ω1(t − tj) + sin Ω1

(t − tj − L

v

)= 0 . (A.2)

473



Meanwhile,

sin ω1(t − tj) + sin ω1

(t − tj − L

v

)

= 2cosω1L

2vsin ω1

(t − tj − L

2v

). (A.3)

Consequently, Eq. (A.1) reduces to


− 2S1 cosω1L

2v

N−1∑j=1

[sinω1

(t − tj − L

2v

)]

×H

(t − tN − L

v

). (A.4)

Noting that tj = (j − 1)d/v, one may show that

N−1∑j=1

sin ω1

(t − L

2v− tj

)

= sin ω1

(t − L

2v

)+

N−2∑j=1

sin ω1

(t − L

2v− jd

v

). (A.5)

From Mangulis’ (1965) Handbook [Eq. (9) on p. 104],

m∑n=1

sin(nθ + α)

=− sin α + sin(α + θ) − sin[α + (m + 1)θ] + sin(α + mθ)

2(1 − cos θ).

(A.6)


Appendix A. Derivation of Response Function P1 in Eq. (2.55) 475

By substituting n = j, m = N − 2, nθ = −jω1(d/v), and α =ω1(t − L/2v) into the preceding equation, one obtains

N−2∑j=1

sin ω1

(t − L

2v− jd

v

)=

C

2(

1 − cosω1d

v

) , (A.7)

where

C = − sin ω1

(t − L

2v

)+ sinω1

(t − L

2v− d

v

)

− sin ω1

[t − L

2v− (N − 1)

d

v

]

+ sin ω1

[t − L

2v− (N − 2)

d

v

]. (A.8)

By the following formula:

sin a − sin b = 2cosa + b

2sin

a − b

2, (A.9)

one can obtain after some operations the term C as follows:

C = 4 sinω1d

2vsinω1

(t − L

2v− N − 1

2vd

)sin ω1

(N − 2

2vd

). (A.10)

In the meantime, the denominator in Eq. (A.7) can be reducedto 4 sin2(ω1d/2v). With these expressions, one can derive fromEq. (A.5) the following:

N−1∑j=1

sin ω1

(t − L

2v− tj

)

= sinω1

(t − L

2v

)

+1

sin(ω1d/2v)

[sinω1

(t − L

2v− tN

2

)sin ω1

(tN2

− d

2v

)].

(A.11)

By using Eqs. (A.4) and (A.11), Eq. (2.55) can be proved.



Appendix B

Newmark’s β Method

A great number of dynamic problems encountered in engineeringappears in the form of second-order differential equations. For struc-tures that contain only a single degree of freedom (DOF), analyticalsolutions in closed form can be obtained. But for structures withmulti DOFs, finite difference methods are often called for to solvethe second-order differential equations, which have been referred toas the direct integration methods, to distinguish from those basedon modal superposition or others. Newmark’s β method representsa special category of finite difference methods that have frequentlybeen used by engineers and researchers in solving the multi-DOFsecond-order differential equations. The following is a summary ofthe method proposed by Newmark (1959).

In a step-by-step nonlinear analysis, we are interested in the be-havior of a structure within the incremental step from time t tot + ∆t, where ∆t denotes a small time increment. The following arethe equations of motion for the structure at time t + ∆t:

[M ]Ut+∆t + [C]Ut+∆t + [K]Ut+∆t = Pt+∆t , (B.1)

where [M ], [C] and [K] denote the mass, damping and stiffness ma-trices of the structure, assumed to consist of N DOFs, U the nodaldisplacements and P the applied nodal forces. All quantities of thestructure are assumed to be known up to time t.

The method proposed by Newmark is a single-step method, whichrequires only information of the structure at time t for solution.The following are the two basic equations proposed by Newmark

477



for determining the displacements and velocities of the structure attime t + ∆t:

Ut+∆t = Ut + Ut∆t +[(

12− β

)Ut + βUt+∆t

](∆t)2 ,

Ut+∆t = Ut + [(1 − γ)Ut + γUt+∆t]∆t ,

(B.2)

where a dot denotes differentiation with respect to time t. The pa-rameter β denotes the variation of acceleration during the incremen-tal step from t to t+∆t. Different values of β imply different schemesof interpolation for the acceleration over a time step. The value β = 0indicates a scheme equivalent to the central difference method, thevalue β = 1/4 is a constant average acceleration method, and thevalue β = 1/6 is a linear acceleration method. On the other hand,the parameter γ relates to the property of numerical or artificialdamping introduced by discretization in time domain. For the casewith γ < 1/2, there exists some artificial negative damping, while forγ > 1/2, artificial positive damping will occur (Weaver and Johnston,1987). The method has been demonstrated to be unconditionallystable under the conditions when γ ≥ 1/2 and β ≥ 1/4(1/2 + γ)2.Throughout this book, the combination of γ = 1/2 and β = 1/4 willbe selected.

From Eq. (B.2), the accelerations and velocities of the structureat time t + ∆t can be solved as

Ut+∆t = a0(Ut+∆t − Ut) − a2Ut − a3Ut ,

Ut+∆t = Ut + a6Ut + a7Ut+∆t ,(B.3)

where the coefficients a0 ∼ a7 are given as follows:

a0 =1

β∆t2, a1 =

γ

β∆t, a2 =

1β∆t

,

a3 =12β

− 1 , a4 =γ

β− 1 , a5 =

∆t

2

(γ

β− 2)

,

a6 = ∆t(1 − γ) , a7 = γ∆t .

(B.4)


Appendix B. Newmark’s β Method 479

Substituting the preceding expression (B.3) into Eq. (B.1) yields theequivalent stiffness equations

[Keff ]Ut+∆t = Pefft+∆t , (B.5)

where the effective stiffness matrix [Keff ] and the effective load vectorPefft+∆t are defined as follows:

[Keff ] = a0[M ] + a1[C] + [K] ,

Pefft+∆t = Pt+∆t + [M ](a0Ut + a2Ut + a3Ut)

+ [C](a1Ut + a4Ut + a5Ut) .

(B.6)

From Eq. (B.5), the structural displacements U at time t+∆t canbe solved as

Ut+∆t = [Keff ]−1Pefft+∆t . (B.7)

It follows that the velocities and accelerations at time t + ∆t canbe obtained from Eq. (B.3). Since all the structural responses attime t + ∆t have been made available, one can proceed to the nexttime step, treating all the responses solved for the structure at thisstep as the initial conditions, updating the structural and loadingconfigurations, including the mass, damping, stiffness matrices, [M ],[C], [K] and applied loads P when necessary. By repeating theabove procedure for a certain number of time steps, one can computethe time-history response of the structure throughout the durationin which the structural behavior is of interest.



Appendix C

Vertical Frequency of Vibration ofCurved Beam

By letting the determinant in Eq. (5.15) equal to zero, one obtains

(a1 − ω2v1)(b1 − ω2

v1) − a2b2 = 0 , (C.1)

from which the solution can be written as

ωv1 =(a1 + b1) ±

√(a1 + b1)2 − 4(a1b1 − a2b2)

2. (C.2)

Let us discuss whether the term a1b1 − a2b2 is less than zero. FromEq. (5.11), it can be shown that

a1b1 = c1[π2R2L2(E2I2z + G2J2) + EIzGJ(L4 + π4R4)] ,

a2b2 = c1[π2R2L2(E2I2z + G2J2) + EIzGJ(2π2R2L2)] ,

(C.3)

where the constant c1 is

c1 = − 1ρJ

1mR4L4

(π

L

)2< 0 . (C.4)

It follows that

a1b1 − a2b2 = c1EIzGJ(L2 − π2R2)2 . (C.5)

Since c1 is less than zero, it can be ascertained that a1b1−a2b2 is alsoless than zero. On the other hand, because ωv1 is not an imaginarynumber, the positive sign in Eq. (C.2) should be selected. Thus, thefrequency of vibration, ωv1, should be computed as

ωv1 =

√(a1 + b1) +

√(a1 − b1)2 + 4a2b2

2. (C.6)

481



Appendix D

Horizontal Frequency of Vibration ofCurved Beam

Letting the determinant in Eq. (5.35) equal to zero yields

(a1 − ω2h1)(b1 − ω2

h1) − a2b2 = 0 , (D.1)

from which the solution can be written as

ωh1 =(a1 + b1) ±

√(a1 + b1)2 − 4(a1b1 − a2b2)

2. (D.2)

Let us determine whether the term a1b1−a2b2 is less than zero. FromEq. (5.32), it can be shown that

a1b1 = c2

πRIy

[(π

L

)2+

1R2

]2+ π

a1

R2

,

a2b2 = c2

(π

a1

R2− 8a1

πR2

),

(D.3)

where the constant c2 is

c2 =E2a1π

(4π2

− 12

)

m2L2

(8π2

− 56

) > 0 . (D.4)

It follows that

a1b1 − a2b2 = c2

[(π

L

)2+

1R2

]2+

8a1

πR

. (D.5)

483



Since c2 is greater than zero, it is ascertained that a1b1 − a2b2 isalso greater than zero. For the frequency of vibration, ωh1, to be areal number, only the negative sign in Eq. (D.2) should be selected.Thus,

ωh1 =

√√√√(a1 + b1) −√

(a1 − b1)2 + 4a2b2

2. (D.6)


Appendix E

Derivation of Residual Vibration forCurved Beam in Eq. (5.53)

The derivation to be presented below follows basically the procedureof Appendix A. By noting that the time lag for the jth moving load istj = (j − 1)d/v, the residual vibration response of the bridge causedby N − 1 moving loads that passed the beam, as given in Eq. (5.52),can be rewritten as

UN,2

(L

2, t

)= −2PS1 cos

ω1L

2v

×

N−1∑j=1

sin ω1

[(t − L

2v

)− (j − 1)

d

v

]

×H

(t − tN−1 − L

v

), for t − tN−1 ≥ L

v. (E.1)

The series term in Eq. (E.1) can be separated into two terms as

N−1∑j=1

sin ω1

[(t − L

2v

)− (j − 1)

d

v

]

= sin ω1

(t − L

2v

)+

N−2∑j=1

sin ω1

[(t − L

2v

)− j

d

v

]. (E.2)

485



From Manulis’ (1965) handbook (Eq. (9), p. 104), the followingequality is shown to be valid:

m∑n=1

sin(α + nθ)

=− sinα + sin(α + θ) − sin[α + (m + 1)θ] + sin(α + mθ)

2(1 − cos θ). (E.3)

By letting n = j, m = N − 2, α = ω1(t − L/2v), and θ = −ω1d/v inEq. (E.3), one can rearrange the last term in Eq. (E.2) as

N−2∑j=1

sin ω1

[(t − L

2v

)− j

d

v

]=

C

2(1 − cos(ω1d/v)), (E.4)

where

C = − sin ω1

(t − L

2v

)+ sinω1

(t − L

2v− d

v

)

− sin ω1

[t − L

2v− (N − 1)

d

v

]

+ sin ω1

[t − L

2v− (N − 2)

d

v

]. (E.5)

By the relation

sin a − sin b = 2cosa + b

2sin

a − b

2, (E.6)

one can rewrite the expression in Eq. (E.5) as

C = 4 sinω1d

2vsin ω1

(t − L

2v− N − 1

2d

v

)sin ω1

(N − 2

2d

v

). (E.7)

On the other hand, it is known that

1 − cosω1d

v= 2(

sinω1d

2v

)2

. (E.8)


Appendix E. Derivation of Residual Vibration for Curved Beam 487

Using the relations given in Eqs. (E.2), (E.4), (E.7) and (E.8), onecan derive from Eq. (E.1) the following

UN,2

(L

2, t

)=−2PS1 cos

ω1L

2v

[sin ω1

(t − L

2v

)+ sin ω1

(N − 2

2d

v

)

× sin ω1

(t − L

2v− N − 1

2d

v

)sin−1 ω1d

2v

]

×H

(t − tN−1 − L

v

), for t − tN−1 >

L

v. (E.9)



Appendix F

Beam Element and StructuralDamping Matrix

In the first part of this appendix, the mass matrix and stiffness ma-trix of the three-dimensional beam element used throughout the bookwill be summarized. In the second part, the procedure for determin-ing the damping matrix of the structure based on the hypothesis ofRayleigh damping will be presented.

F.1. Equation of Motion for Beam Element

The beam element considered is a prismatic, three-dimensional beamof solid cross sections, with the effect of warping deformations ex-cluded. The longitudinal axis of the beam is denoted by x, and thetwo transverse principal axes of the cross section of the beam by y

and z. There are six degrees of freedom (DOFs), i.e., three trans-lations and three rotations, associated with each of the two ends A

and B of the element, as shown in Fig. F.1. For the case with noexternal loads, the equation of motion for the beam element can bewritten as follows:

[mb]ub + [cb]ub + [kb]ub = 0 , (F.1)

where ub is the displacement vector of the element, which consistsof 12 DOFs, an overdot denotes differentiation with respect to time,[mb], [cb] and [kb] denote the mass, damping and stiffness matrices ofthe element. Both the mass matrix [mb] and stiffness matrix [kb] areavailable in most textbooks on structural and dynamic analyses, e.g.,see Paz (1991) and McGuire et al. (2000). The element displacement

489



(a)

(b)

Fig. F.1. Three-dimensional beam element: (a) nodal DOFs and (b) nodalforces.

vector ub is

ub = 〈uxA uyA uzA θxA θyA θzA

uxB uyB uzB θxB θyB θzB 〉T . (F.2)


Appendix F. Beam Element and Structural Damping Matrix 491

Correspondingly, there are 12 nodal forces for the element, as shownin Fig. F.1(b). The mass matrix [mb] is

[mb] =mL

420

[[m1] [m2]T

[m2] [m3]

], (F.3)

where L is the length of the element and m is the mass per unitlength. By letting A denote the cross area and Ip the polar momentof inertia of the beam, the submatrices in Eq. (F.3) can be given asfollows:

[m1] =

140 Symm.

0 156

0 0 156

0 0 0140Ip

A

0 0 −22L 0 4L2

0 22L 0 0 0 4L2

, (F.4)

[m2] =

70 0 0 0 0 0

0 54 0 0 0 13L

0 0 54 0 −13L 0

0 0 070Ip

A0 0

0 0 13L 0 −3L2 0

0 −13L 0 0 0 −3L2

, (F.5)

[m3] =

140 Symm.

0 156

0 0 156

0 0 0140Ip

A

0 0 22L 0 4L2

0 −22L 0 0 0 4L2

. (F.6)



The stiffness matrix [kb] can be given as follows:

[kb] =[

[k1] [k2]T

[k2] [k3]

]. (F.7)

By letting E and G denote the elastic and shear moduli, respectively,J the torsional constant, and Iy and Iz respectively the moments ofinertia about the y- and z-axes of the element, the submatrices inEq. (F.7) can be given as follows:

[k1] =

EA

LSymm.

012EIz

L3

0 012EIy

L3

0 0 0GJ

L

0 0 −6EIy

L20

4EIy

L

06EIz

L20 0 0

4EIz

L

, (F.8)

[k2] =

−EA

L0 0 0 0 0

0 −12EIz

L30 0 0 −6EIz

L2

0 0 −12EIy

L30

6EIy

L20

0 0 0 −GJ

L0 0

0 0 −6EIy

L20

2EIy

L0

06EIz

L20 0 0

2EIz

L

,

(F.9)



[k3] =

EA

LSymm.

012EIz

L3

0 012EIy

L3

0 0 0GJ

L

0 06EIy

L20

4EIy

L

0 −6EIz

L20 0 0

4EIz

L

. (F.10)

The damping matrix [cb] is usually not computed on the elementlevel, but implicitly implied as a part of the structural damping ma-trix to be shown in the following section.

F.2. Structural Damping Matrix

The damping of structures can appear in various forms. By classicaldamping, it means that the damping matrix of the structure canbe expressed as some linear combinations of the mass and stiffnessmatrices of the structure. With such a property, the natural modesof vibration are orthogonal not only with respect to the mass andstiffness matrices, but also to the damping matrix. One benefit fromthis is that the equations of motion of the structure become decoupledwhen transformed to the generalized coordinates.

Let us consider the equations of motion for a structure that is freeof external loads:

[Mb]Ub + [Cb]Ub + [Kb]Ub = 0 , (F.11)

where Ub denotes the nodal displacements, and [Mb] and [Kb] themass and stiffness matrices of the structure. Based on the assumptionof Rayleigh or classical damping, the damping matrix [Cb] of thestructure can be expressed in a general form as a combination of the



mass matrix [Mb] and stiffness matrix [Kb] (Caughey, 1960), that is,

[Cb] = [Mb]N−1∑i=0

ai([Mb]−1[Kb])i , (F.12)

where N is the total number of DOFs of the structure and ai are thecoefficients to be determined. By the orthogonality properties ofthe vibration modes with respect to the mass and stiffness matrices,the nth modal damping coefficient Cn can be expressed as follows:

Cn = φTn [Cb]φn = φT

n [Mb]N−1∑i=0

ai([Mb]−1[Kb])iφn , (F.13)

where φn denotes the nth vibration mode of the structure with nodamping. The following is the modal equation that must be satisfiedby the nth vibration mode φn:

[Kb]φn = ω2n[Mb]φn , (F.14)

where ωn is the nth vibration frequency of the structure. The fol-lowing is the definition for the nth modal mass:

Mn = φTn [Mb]φn . (F.15)

By using Eqs. (F.14) and (F.15), the nth modal damping coefficientCn in Eq. (F.13) can be written as

Cn =N−1∑i=0

aiω2in Mn . (F.16)

On the other hand, the damping coefficient for the nth modalequation in terms of the generalized coordinates is

Cn = 2ξnωnMn , (F.17)

where ξn is the nth damping ratio. For the case with the modal massnormalized as Mn = 1, the nth damping ratio can be solved fromEqs. (F.16) and (F.17) as

ξn =1

2ωn

N−1∑i=0

aiω2in . (F.18)



For most engineering structures, it is impractical to consider all the N

modes of damping ratios. If only the first two modes are considered,which is the case known as Rayleigh damping, the damping matrix[Cb] in Eq. (F.12) reduces to

[Cb] = a0[Mb] + a1[Kb] . (F.19)

The Rayleigh coefficients a0 and a1 can be determined only if thedamping ratios ξi, ξj and frequencies ωi, ωj are given for any twovibration modes, i.e., from Eq. (F.18),

a0

a1

= 2[

ω−1i ωi

ω−1j ωj

]−1 ξi

ξj

. (F.20)

For the case when the frequencies of vibration of the first two modes,i.e., i = 1 and j = 2, are given and the damping ratios for the twomode are assumed to be the same, i.e., ξ1 = ξ2 = ξ, the precedingequation reduces to

a0

a1

=

2ξω1 + ω2

ω1ω2

1

. (F.21)

The damping matrix [Cb] can then be computed from Eq. (F.19).



Appendix G

Partitioned Matrices and Vector forVehicle, Eq. (9.4)

The partitioned matrices and vectors for the vehicle equation inEq. (9.4) can be given as follows:

[muu] =

Mc 0 0 0 0 0

Ic 0 0 0 0

Mt 0 0 0

It 0 0

Symm. Mt 0

It

, (G.1)

[muw] = [mwu]T = [0] , (G.2)

[mww] =

Mw 0 0 0

0 Mw 0 0

0 0 Mw 00 0 0 Mw

, (G.3)

[lw] =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, (G.4)

497



[cuu] =

2cs 0 −cs 0 −cs 0

2l2ccs −lccs 0 lccs 0

cs + 2cp 0 0 0

2l2t cp 0 0

Symm. cs + 2cp 0

2l2t cp

, (G.5)

[cuw] = [cwu]T =

0 0 0 0

0 0 0 0

−cp −cp 0 0

−ltcp ltcp 0 0

0 0 −cp −cp

0 0 −ltcp ltcp

, (G.6)

[cww] =

cp 0 0 0

0 cp 0 0

0 0 cp 0

0 0 0 cp

, (G.7)

[kuu] =

2ks 0 −ks 0 −ks 0

2l2cks −lcks 0 lcks 0

ks + 2kp 0 0 0

2l2t kp 0 0

Symm. ks + 2kp 0

2l2t kp

, (G.8)

[kuw] = [kwu]T =

0 0 0 0

0 0 0 0

−kp −kp 0 0

−ltkp ltkp 0 0

0 0 −kp −kp

0 0 −ltkp ltkp

, (G.9)


Appendix G. Partitioned Matrices and Vector for Vehicle, Eq. (9.4) 499

[kww] =

kp 0 0 0

0 kp 0 0

0 0 kp 0

0 0 0 kp

, (G.10)

fuet+∆t = 0 , (G.11)

fwet+∆t = 〈−W −W −W −W 〉T . (G.12)

Here, we use Mw, Mt, and Mc to denote the mass of the wheel, bogie,and car body, respectively. It follows that the vehicle weight W inEq. (G.12) can be expressed as

W = (Mw + 0.5Mt + 0.25Mc)g , (G.13)

with g denoting the acceleration of gravity.



Appendix H

Related Matrices and Vectors forCFR Element

The following are the related matrices and vectors for the systemmatrices of the CFR element as given in Eq. (9.10):

[ψ0u] =

l

3l

6l

6l

3

(u1 u2 ) , (H.1)

[ψ1u] =

1l

−1l

−1l

1l

(u1 u2 ) , (H.2)

[ψ0v ] =

l

420

156 22l 54 −13l

4l2 13l −3l2

156 −22l

Symm. 4l2

( v1 θ1 v2 θ2 ) ,

(H.3)

[ψ2v ] =

12/l3 6/l2 −12/l3 6/l2

4/l −6/l2 2/l

12/l3 −6/l2

Symm. 4/l

( v1 θ1 v2 θ2 ) ,

(H.4)

[ψ0u]l =

12λu

(u2) , (H.5)

501



[ψ1u]l =

λu

2(u2) , (H.6)

[ψ0v ]l =

34λv

− 14λ2

v

− 14λ2

v

18λ3

v

( v2 θ2 ) , (H.7)

[ψ2v ]l =

λ3

v −λ2v

−λ2v

3λv

2

( v2 θ2 ) , (H.8)

[ψ0u]r =

12λu

(u1) , (H.9)

[ψ1u]r =

λu

2(u1) , (H.10)

[ψ0v ]r =

34λv

14λ2

v

14λ2

v

18λ3

v

( v1 θ1 ) , (H.11)

[ψ2v ]r =

λ3

v λ2v

λ2v

3λv

2

( v1 θ1 ) , (H.12)

λu =√

kbh

2ErAr, (H.13)

λv = 4

√kbv

8ErIr. (H.14)

The parenthesized symbols in the above expressions indicate the de-grees of freedom of the element with which the column and rowentries in each matrix are associated.


Appendix I

Related Matrices and Vectors for3D Vehicle Model

The following are the partitioned mass matrices involved in the equa-tion of motion for the 3D vehicle model in Eq. (10.4):

[muu] = diag[Mc Mc I∗cx I∗cy I∗cz Mt Mt

I∗tx I∗ty I∗tz Mt Mt I∗tx I∗ty I∗tz ] , (I.1)

[mww] = diag[Mw Mw I∗w Mw Mw I∗w Mw

Mw I∗w Mw Mw I∗w ] , (I.2)

[muw] = [mwu]T = [0] , (I.3)

where diag denotes that the entries following represent the diagonalelements of the matrix and that all the nondiagonal elements arezero. The following are the partitioned damping matrices:

[cuu] =

a 0 0 0 0 −b 0 0 0 0 −b 0 0 0 0

c −d 0 0 0 −e −f 0 0 0 −e −f 0 0

g 0 0 0 h i 0 0 0 h i 0 0

j 0 0 k l 0 0 0 −k −l 0 0

m −n 0 0 0 0 n 0 0 0 0

o 0 0 0 0 0 0 0 0 0

p q 0 0 0 0 0 0 0

r 0 0 0 0 0 0 0

s 0 0 0 0 0 0

t 0 0 0 0 0

Symm. o 0 0 0 0

p q 0 0

r 0 0

s 0

t

, (I.4)

503

Decem

ber

2,2005

9:2

0Veh

icle–B

ridge

Intera

ction

Dynam

icsbk04-0

01

504

Veh

icle–Brid

geIn

teractio

nD

ynam

icswhere

a = 4csy , b = 2csy , c = 4csz , d = 4hcscsz , e = 2csz , f = 2htscsz , g = 4d2scsy + 4h2

cscsz ,

h = 2hcscsz , i = −2d2scsy + 2hcshtscsz , j = 4l2ccsz , k = 2lccsz , l = 2htslccsz , m = 4l2ccsy ,

n = 2lccsy , o = 2csy + 4cpy , p = 2csz + 4cpz , q = 2htscsz − 4htpcpz ,

r = 2d2scsy + 2h2

tscsz + 4d2pcpy + 4h2

tpcpz , s = 4l2t cpz , t = 4l2t cpy ,

(I.5a–t)

[cuw] = [cwu]T

=

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

−2cpy 0 0 −2cpy 0 0 0 0 0 0 0 0

0 −2cpz 0 0 −2cpz 0 0 0 0 0 0 0

0 2htpcpz −2d2pcpy 0 2htpcpz −2d2

pcpy 0 0 0 0 0 0

0 2ltcpz 0 0 −2ltcpz 0 0 0 0 0 0 0

−2ltcpy 0 0 2ltcpy 0 0 0 0 0 0 0 0

0 0 0 0 0 0 −2cpy 0 0 −2cpy 0 0

0 0 0 0 0 0 0 −2cpz 0 0 −2cpz 0

0 0 0 0 0 0 0 2htpcpz −2d2pcpy 0 2htpcpz −2d2

pcpy

0 0 0 0 0 0 0 2ltcpz 0 0 −2ltcpz 0

0 0 0 0 0 0 −2ltcpy 0 0 2ltcpy 0 0

,

(I.6)


Appendix I. Related Matrices and Vectors for 3D Vehicle Model 505

[cww] = diag[ 2cpy 2cpz 2d2pcpy 2cpy 2cpz 2d2

pcpy

2cpy 2cpz 2d2pcpy 2cpy 2cpz 2d2

pcpy ] . (I.7)

The matrix [kuw] = [kwu]T is identical in form to [cuw] given inEq. (I.6), except that the terms cpy, cpz should be replaced by kpy,kpz, respectively. The matrix [kuu] is identical in form to the matrix[cuu] given in Eq. (I.4) except that the nonzero entries are given asfollows:

a = 4ksy , b = 2ksy , c = 4ksz , d = 4hcsksz ,

e = 2ksz , f = 2htsksz , g = 4d2sksy + 4h2

csksz ,

h = 2hcsksz , i = −2d2sksy + 2hcshtsksz , j = 4l2cksz ,

k = 2lcksz , l = 2htslcksz , m = 4l2cksy , n = 2lcksy ,

o = 2ksy + 4kpy , p = 2ksz + 4kpz , q = 2htsksz − 4htpkpz ,

r = 2d2sksy + 2h2

tsksz + 4d2pkpy + 4h2

tpkpz ,

s = 4l2t kpz , t = 4l2t kpy .

(I.8)

The following are the related force vectors:

fuet+∆t = 0 ,

fwet+∆t = 〈−W −W −W −W 〉T ,(I.9)

where W denotes the weight lumped from the vehicle, W = (Mc +2Mt + 4Mw)g/4.



Appendix J

Mass and Stiffness Matrices for Railand Bridge Elements

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

[mA0] = [mB0] =mtl

420

×

140 0 0 0 0 0 70 0 0 0 0 0

156 0 0 0 22l 0 54 0 0 0 −13l

156 0 −22l 0 0 0 54 0 13l 0

140I∗tmt

0 0 0 0 070I∗tmt

0 0

4l2 0 0 0 −13l 0 −3l2 0

4l2 0 13l 0 0 0 −3l2

140 0 0 0 0 0

156 0 0 0 −22l

Symm. 156 0 22l 0

140I∗tmt

0 0

4l2 0

4l2

,

(J.1)

507



[kA0] = [kB0]

=

a 0 0 0 0 0 −a 0 0 0 0 0

b 0 0 0 g 0 −b 0 0 0 g

c 0 −e 0 0 0 −c 0 −e 0

d 0 0 0 0 0 0 0 0

f 0 0 0 e 0 i 0

h 0 −g 0 0 0 j

a 0 0 0 0 0

b 0 0 0 −g

Symm. c 0 e 0

d 0 0

f 0

h

,

(J.2)

where

a =EtAt

l, b =

12EtItz

l3, c =

12EtIty

l3, d = 0 ,

e =6EtIty

l2, f =

4EtIty

l, g =

6EtItz

l2, h =

4EtItz

l,

i =2EtIty

l, j =

2EtItz

l.

(J.3)

J.2. Mass and Stiffness Matrices of the BridgeElement

The mass matrix [mb0] of the bridge element is identical in formto that given in Eq. (J.1) except that the two parameters mt andI∗t should be replaced by mb and I∗b , respectively. In addition, thestiffness matrix [kb0] of the bridge element is identical in form to that


Appendix J. Mass and Stiffness Matrices for Rail and Bridge Elements 509

of Eq. (J.2) but with the following substitutions:

a =EtAt

l, b =

12EbIbz

l3, c =

12EbIby

l3, e =

6EbIby

l2,

d =EbIbx

2(1 + νb)l, f =

4EbIby

l, g =

6EbIbz

l2,

h =4EbIbz

l, i =

2EbIby

l, j =

2EbIbz

l.

(J.4)

J.3. Mass and Stiffness Matrices for the LSR Element

[ml0] =

mt

2λu0 0 0 0 0

3mt

4λv0 0 0 −sgn

mt

4λ2v

3mt

4λwv0 sgn

mt

4λ2w

0

I∗t2λθ

0 0

Symm.mt

8λ3w

0

mt

8λ3v

, (J.5)

[kl0] =

EtAtλu

20 0 0 0 0

EtItzλ3v 0 0 0 −sgnEtItzλ

2v

EtItyλ3w 0 sgnEtItyλ2

w 0

0 0 0

Symm.32EtItyλw 0

32EtItzλv

,

(J.6)

where the sign “sgn” is taken as positive.



J.4. Mass and Stiffness Matrices of the RSR Element

The mass and stiffness matrices of the RSR element, i.e., [mr0] and[kr0], are identical to those of the LSR element except that the sign“sgn” is taken as negative.

J.5. Related Matrices and Vectors for the RailElements

[ψu] =

l

3l

6l

6l

3

(u1, u2) , (J.7)

[ψv] =

13l35

11l2

2109l70

−13l2

420

0l3

10513l2

420−l3

140

0 013l35

−11l2

210

0 0 0l3

105

(v1, ψ1, v2, ψ2) , (J.8)

[ψθ] =

l

3l

6l

6l

3

(θ1, θ2) , (J.9)

[ψw] =

13l35

−11l2

2109l70

13l2

420

0l3

105−13l2

420−l3

140

0 013l35

11l2

210

0 0 0l3

105

(w1, ϕ1, w2, ϕ2) , (J.10)


Appendix J. Mass and Stiffness Matrices for Rail and Bridge Elements 511

[ψvθ] =

7l20

3l20

l2

20l2

303l20

7l20

−l2

30−l2

20

(v1, ψ1, v2, ψ2) and (θ1, θ2) , (J.11)

[ψwθ] =

7l20

3l20

−l2

20−l2

303l20

7l20

l2

30l2

20

(w1, ϕ1, w2, ϕ) and (θ1, θ2) , (J.12)

[ψu]l =[

12λu

](u2) , (J.13)

[ψθ]l =[

12λθ

](θ2) , (J.14)

[ψv ]l =

34λv

− 14λ2

v

− 14λ2

v

18λ3

v

(v2, ψ2) , (J.15)

[ψw]l =

34λw

14λ2

w

14λ2

w

18λ3

w

(w2, ϕ2) , (J.16)

[ψu]r =[

12λu

](u1) , (J.17)



[ψθ]r =[

12λθ

](θ1) , (J.18)

[ψv ]r =

34λv

14λ2

v

14λ2

v

18λ3

v

(v1, ψ1) , (J.19)

[ψw]r =

34λw

− 14λ2

w

− 14λ2

w

18λ3

w

(w1, ϕ1) . (J.20)

The parenthesized symbols in the above expressions indicate theDOFs associated with the row and column entries of the matrix.


References

Aida, T., Green, R., and Hosogi, Y. (1990). “Dynamic behavior of railwaybridges under unsprung masses of a multi-vehicle train,” J. Sound &Vibr., 142(2), 245–260.

Akin, J. E. and Mofid, M. (1989). “Numerical solution for response of beamswith moving mass,” J. Struct. Eng., ASCE, 115(1), 120–131.

Au, F. T. K., Wang, J. J., and Cheung, Y. K. (2001a). “Impact studyof cable-stayed bridge under railway traffic using various models,” J.Sound & Vibr., 240(3), 447–465.

Au, F. T. K., Cheng, Y. S., and Cheung, Y. K. (2001b). “Effects of ran-dom road surface roughness and long-term deflection of prestressedconcrete girder and cable-stayed bridges on impact due to movingvehicles,” Comp. & Struct., 79, 853–872.

Ayre, R. S., Ford, G., and Jacobsen, L. S. (1950). “Transverse vibration ofa two-span beam under action of a moving constant force,” J. Appl.Mech., 17(1), 1–12.

Ayre, R. S. and Jacobsen, L. S. (1950). “Transverse vibration of a two-spanbeam under the action of a moving alternating force,” J. Appl. Mech.,17(3), 283–290.

Bhatti, M. H., Garg, V. K., and Chu, K. H. (1985). “Dynamic interactionbetween freight train and steel bridge,” J. Dyn. Syst., Measurement& Control, 107(1), 60–66.

Biggs, J. M. (1964). Introduction to Structural Dynamics, McGraw-Hill,New York, N.Y.

Blejwas, T. E., Feng, C. C., and Ayre, R. S. (1979). “Dynamic interac-tion of moving vehicles and structures,” J. Sound & Vibr., 67(4),513–521.

513



Bolotin, V. V. (1964). The Dynamic Stability of Elastic Systems, Holden-Day, Inc., San Francisco, California, U.S.A.

Cai, Y., Chen, S. S., Rote, D. M., and Coffey, H. T. (1994). “Vehi-cle/guideway interaction for high speed vehicles on a flexible guide-way,” J. Sound & Vibr., 175(5), 625–646.

Caughey, T. K. (1960). “Classical normal modes in damped linear dynamicsystems,” J. Appl. Mech., ASME, 27, 269–271.

Chang, C. H. (1997). Rigid vehicle–bridge interaction response analysisand impact responses of horizontally curved beams, Master’s thesis,National Taiwan University, Taipei, Taiwan, R.O.C. (in Chinese).

Chang, D. and Lee, H. (1994). “Impact factors for simple-span highwaygirder bridges,” J. Struct. Eng., ASCE, 120(3), 704–715.

Chatterjee, P. K. and Datta, T. K. (1995). “Dynamic analysis of archbridges under travelling loads,” Int. J. Solids Struct., 32(11),1585–1594.

Chatterjee, P. K., Datta, T. K., and Surana, C. S. (1994). “Vibration of sus-pension bridges under vehicular movement,” J. Struct. Eng., ASCE,120(3), 681–703.

Chen, G. and Zhai, W. M. (1999). “Numerical simulation of the stochasticprocess of railway track irregularities,” J. Southwest Jiaotong Univ.,34(3), 138–142 (in Chinese).

Chen, Y. H. (1978). “Dynamic analysis of continuous beams subjectedto moving loads,” Chinese J. Civil & Hydraulic Eng., 5(2), 1–7 (inChinese).

Chen, Y. H. and Li, C. Y. (2000). “Dynamic response of elevated high-speedrailway,” J. Bridge Eng., ASCE, 5(2), 124–130.

Cherchas, D. B. (1979). “A dynamic simulation for a high speed magneti-cally levitated guided ground vehicle,” J. Dyn. Syst. Measurement &Control, 101(3), 223–229.

Cheung, Y. K., Au, F. T. K., Zheng, D. Y., and Cheng, Y. S. (1999).“Vibration of multi-span non-uniform bridges under moving vehiclesand trains by using modified beam vibration functions,” J. Sound &Vibr., 228(3), 611–628.

Chu, K. H., Garg, V. K., and Dhar, C. L. (1979). “Railway–bridge impact:Simplified train and bridge model,” J. Struct. Div., ASCE, 105(9),1823–1844.

Chu, K. H., Garg, V. K., and Wang, T. L. (1986). “Impact in railway pre-stressed concrete bridges,” J. Struct. Eng., ASCE, 112(5), 1036–1051.


References 515

Claus, H. and Schiehlen, W. (1998). “Modelling and simulation of rail-way bogie structural vibrations,” Vehicle Syst. Dyn. Supplement, 28,538–552.

Coussy, O., Said, M., and van Hoove, J. P. (1989). “The influ-ence of random surface irregularities on the dynamic response ofbridges under suspended moving loads,” J. Sound & Vibr., 130(2),313–320.

Delfosse, P. (1991). “Very-high-speed tests on the French railway track,”Vehicle Syst. Dyn., 20, 195–206.

Delgado, R. M. and dos Santos, S. M. (1997). “Modelling of railway bridge–vehicle interaction on high speed tracks,” Comp. & Struct., 63(3),511–523.

Deutsche, B. (1993). DS804 — Vorschrift fur Eisenbahnbrucken und son-stige Ingenieurbauwerke (VEI).

Diana, G. and Cheli, F. (1989). “Dynamic interaction of railway systemswith large bridges,” Vehicle Syst. Dyn., 18, 71–106.

Diana, G., Cheli, F., and Bruni, S. (2000). “Railway runnability and train–track interaction in long span cable supported bridges,” Advances inStruct. Dyn., Vol. 1, eds. Ko, J. M. and Xu, Y. L., Elsevier, Amster-dam, 43–54.

Dodds, C. J. and Robson, J. D. (1973). “The description of road surfaceroughness,” J. Sound & Vibr., 31(2), 175–183.

Dukkipati, R. V. and Dong, R. (1999). “Idealized steady state interactionbetween railway vehicle and track,” Proc. Inst. Mech. Engrs. Part F,J. Rail & Rapid Transit, 213(2), 15–29.

Dugush, Y. A. and Eisenberger, M. (2002). “Vibrations of non-uniformcontinuous beams under moving loads,” J. Sound & Vibr., 254(5),911–926.

Elkins, J. A. and Carter, A. (1993). “Testing and analysis techniques forsafety assessment of rail vehicles: The sate-of-the-art,” Vehicle Syst.Dyn., 22, 185–208.

Esveld, G. (1989). Modern Railway Track, MRT Productions, Duisburg,Germany.

European Committee for Standardization (1995). EUROCODE 1: Basisof design and actions on structures, Part 3: Traffic loads on bridges,ENV 1991-3.

Fertis, D. G. (1973). Dynamics and Vibration of Structures, John Wiley &Sons, New York, N.Y.



Fries, R. H. and Coffey, B. M. (1990). “A state-space approach to thesynthesis of random vertical and crosslevel rail irregularities,” J. Dyn.Syst., Measurement, and Control, 112(1), 83–87.

Fryba, L. (1972). Vibration of Solids and Structures under Moving Loads,Noordhoff International Publishing, Groningen, The Netherlands.

Fryba, L. (1996). Dynamics of Railway Bridges, Thomas Telford, London,England.

Fryba, L. (2001). “A rough assessment of railway bridges for high speedtrains,” Eng. Struct., 23(5), 548–556.

Galdos, N. H., Schelling, D. R., and Sahin, M. A. (1993). “Methodologyfor impact factor of horizontally curved box bridges,” J. Struct. Eng.,ASCE, 119(6), 1917–1934.

Garg, V. K. and Dukkipati, R. V. (1984). Dynamics of Railway VehicleSystems, Academic Press, New York, N.Y.

Gbadeyan, J. A. and Oni, S. T. (1995). “Dynamic behaviour of beams andrectangular plates under moving loads,” J. Sound & Vibr., 182(5),677–695.

Genin, J. and Chung, Y. I. (1979). “Response of a continuous guideway onequally spaced supports traversed by a moving vehicle,” J. Sound &Vibr., 67, 245–251.

Genin, J., Ginsberg, J. H., and Ting, E. C. (1975). “A complete formulationof inertial effects in the guideway–vehicle interaction problem,” J.Sound & Vibr., 38(1), 15–26.

Genin, J. and Ting, E. C. (1979). “Vehicle–guideway interaction problems,”The Shock & Vibr. Dig., 11(12), 3–9.

Genin, J., Ting, E. C., and Vafa, Z. (1982). “Curved beam response due toa moving vehicle,” J. Sound & Vibr., 65, 569–575.

Gere, J. M. and Timoshenko, S. P. (1990). Mechanics of Materials, PWS-KENT, Boston.

Gorman, D. J. (1975). Free Vibration Analysis of Beams and Shafts, JohnWiley & Sons, N.Y.

Grandil, J. and Ramondenc, P. (1990). The Dynamic Behavior of Railwayson High Speed Lines, SNCF, France.

Green, M. F. and Cebon, D. (1994). “Dynamic response of highway bridgesto heavy vehicle loads: Theory and experimental validation,” J. Sound& Vibr., 170(1), 51–78.

Guide Specifications for Horizontally Curved Highway Bridges (1980).American Association for State Highway and Transportation Officials(AASHTO), Washington, D.C.


References 517

Guo, W. H. and Xu, Y. L. (2001). “Fully computerized approach to studycable-stayed bridge–vehicle interaction,” J. Sound & Vibr., 248(4),745–761.

Gupta, R. K. (1980). “Dynamic loading of highway bridges,” J. Eng. Mech.Div., ASCE, 106(2), 337–393.

Guyan, R. J. (1965). “Reduction of stiffness and mass matrices,” AIAA J.,3(2), 380.

Henchi, K., Fafard, M., Talbot, M., and Dhatt, G. (1998). “An efficientalgorithm for dynamic analysis of bridges under moving vehicles usinga coupled modal and physical components approach,” J. Sound &Vibr., 212(4), 663–683.

Hetenyi, M. (1979). Beams on Elastic Foundation, The University of Michi-gan Press, Ann Arbor.

Hsu, L. C. (1996). Impact responses of bridges traveled by high speed trains,Master’s thesis, National Taiwan University, Taipei, Taiwan, R.O.C.(in Chinese).

Huang, D., Wang, T. L., and Shahawy, M. (1993). “Impact studies on multi-girder concrete bridges,” J. Struct. Eng., ASCE, 119(8), 2387–2402.

Humar, J. L. and Kashif, A. M. (1993). “Dynamic response of bridges undertravelling loads,” Can. J. Civ. Eng., 20, 287–298.

Hwang, E. S. and Nowak, A. S. (1991). “Simulation of dynamic load forbridges.” J. Struct. Eng., ASCE, 117(5), 1413–1434.

Ichikawa, M., Miyakawa, Y., and Matsuda, A. (2000). “Vibration analysisof the continuous beam subjected to a moving mass,” J. Sound &Vibr., 230, 492–506.

Inbanathan, M. J. and Wieland, M. (1987). “Bridge vibrations due to ve-hicle moving over rough surface,” J. Struct. Eng., ASCE, 113(9),1994–2008.

Inglis, C. E. (1934). A Mathematical Treatise on Vibration in RailwayBridges, The University Press, Cambridge, England.

Iwnick, S. (1998). “Manchester benchmarks for rail vehicle simulation,”Vehicle Syst. Dyn., 30, 295–313.

Jeffcott, H. H. (1929). “On the vibrations of beams under the action ofmoving loads,” Phil. Magazine, Ser. 7, 8(48), 66–97.

Ju, S. H. and Lin, H. T. (2003). “Numerical investigation of a steel archbridge and interaction with high speed trains,” Eng. Struct., 25,241–250.

Kawatani, M. and Kim, C. W. (2001). “Computer simulation for dynamicwheel loads of heavy vehicles,” Struct. Eng. & Mech., 12(4), 409–428.



Kim, S. and Nowak, A. S. (1997). “Load distribution and impact factorsfor I-girders bridges,” J. Bridge Eng., ASCE, 2(3), 97–104.

Kou, J. W. and DeWolf, J. T. (1997). “Vibrational behavior of continuousspan highway bridge — Influencing variables,” J. Struct. Eng., ASCE,123(3), 333–344.

Kurihara, M. and Shimogo, T. (1978a). “Stability of a simply-supportedbeam subjected to random spaced moving loads,” J. Mech. Design,ASME, 100(7), 507–513.

Kurihara, M. and Shimogo, T. (1978b). “Vibration of an elastic beam sub-jected to discrete moving loads,” J. Mech. Design, ASME, 100(7),514–519.

Lin, C. L. (2001). Reduction in vibration of isolated railway bridges, Mas-ter’s thesis, National Taiwan University, Taipei, Taiwan, R.O.C. (inChinese).

Li, J. Z. and Su, M. B. (1999). “The resonant vibration for a simply sup-ported girder bridge under high-speed trains,” J. Sound & Vibr.,224(5), 897–915.

Lin, B. H., Yau, J. D., and Yang, Y. B. (2000). “Impact factors for sim-ple beams subjected to moving loads,” Struct. Eng., 15(2), 3–20 (inChinese).

Lowan, A. N. (1935). “On transverse oscillations of beams under the actionof moving variable loads,” Phil. Magazine, Ser. 7, 19(127), 708–715.

Ma, K. Q. and Zhu, J. L. (1998). “Seismic responses analysis for a systemof high-speed train and continuous rigid-frame bridge,” J. ShanghaiTiedao Univ., 19(10), 29–36 (in Chinese).

Mangulis, V. (1965). Handbook of Series for Scientists and Engineers, Aca-demic Press, N.Y.

Marcondes, J., Burgess, G. J., Harichandran, R., and Snyder, M. B. (1991).“Spectral analysis of highway pavement roughness,” J. Transp. Eng.,ASCE, 117(5), 540–549.

Marchesiello, S., Fasana, A., Garibaldi, L., and Piombo, B. A. D. (1999).“Dynamics of multi-span continuous straight bridges subject to multi-degrees of freedom moving vehicle excitation,” J. Sound & Vibr.,224(3), 541–561.

Matsuura, A. (1976). “A study of dynamic behavior of bridge girder forhigh speed railway,” Proc. JSCE, 256, 35–47 (in Japanese).

McGuire, W., Gallagher, R. H., and Ziemian, R. D. (2000). Matrix Struc-tural Analysis, 2nd edn., John Wiley & Sons, New York, N.Y.


References 519

Meisenholder, S. G. and Weidlinger, P. (1974). “Dynamic interaction as-pects of cable-stayed guideways for high speed ground transporta-tion,” J. Dyn. Syst., Measurement, & Control, ASME, 74-Aut-R,180–192.

Miura, S. (1996). “Deformation of track and safety of train in earthquake,”Quarterly of RTRI, 37(3), 139–147.

Miyamoto, T., Ishida, H., and Matsuo, M. (1997). “Running safety ofrailway vehicle as earthquake occurs,” Quarterly of RTRI, 38(3),117–122.

Museros, P., Romero, M. L., Poy, A., and Alarcon, E. (2002). “Advances inthe analysis of short span railway bridges for high-speed lines,” Comp.& Struct., 80, 2121–2132.

Nassif, H. H. and Nowak, A. S. (1995). “Dynamic load spectra for girderbridges,” Transp. Res. Rec. 1476, Transp. Res. Board, Nat. Res.Council, Washington, D.C., 69–83.

Newmark, N. M. (1959). “A method of computation for structural dynam-ics,” J. Eng. Mech. Div., ASCE, 85(3), 67–94.

Nielsen, J. C. O. and Abrahamsson, T. J. S. (1992). “Coupling of physicaland modal components for analysis of moving non-linear dynamicsystems on general beam structures,” Int. J. Num. Meth. Eng., 33,1843–1859.

O’Connor, C. and Pritchard, R. W. (1984). “Impact studies on small com-posite girder bridge,” J. Struct. Eng., ASCE, 111(3), 641–653.

Ontario Highway Bridge Design Code. (1983). Ministry of Transportationand Communication, Downsview, Ontario, Canada.

Pan, T. C. and Li, J. (2002). “Dynamic vehicle element method for tran-sient response of coupled vehicle–structure systems,” J. Struct. Eng.,ASCE, 128(2), 214–223.

Paz, M. (1991). Structural Dynamics — Theory and Computation, VanNostrand Reinhold, New York, N.Y.

Paultre, P., Chaallal, O., and Proulx, J. (1992). “Bridge dynamics anddynamic amplification factors — A review of analytical experimentalfindings,” Can. J. Civ. Eng., 19, 260–278.

Pesterev, A. V., Yang, B., Bergman, L. A., and Tan, C. A. (2001). “Re-sponse of elastic continuum carrying multiple moving oscillators,” J.Eng. Mech., ASCE, 127(3), 260–265.

Pesterev, A. V., Bergman, L. A., Tan, C. A., Tsao, T. C., and Yang,B. (2003). “On asymptotics of the solution of the moving oscillatorproblem,” J. Sound & Vibr., 260, 519–536.



Rao, G. V. (2000). “Linear dynamics of an elastic beam under movingloads,” J. Vibr. & Acous., ASME, 122(7), 281–289.

Richardson, H. H. and Wormley, D. N. (1974). “Transportationvehicle/beam-elevated guideway dynamic interactions: A state-of-the-art review,” J. Dyn. Syst. Measurement & Control, 96(2), 169–179.

Sadiku, S. and Leipholz, H. H. E. (1987). “On the dynamics of elasticsystems with moving concentrated masses,” Ingenieur-Archiv, 57,223–242.

Savin, E. (2001). “Dynamic amplification factor and response spectrum forthe evaluation of vibrations of beams under successive moving loads,”J. Sound & Vibr., 248, 267–288.

Schupp, G. and Jaschinski, A. (1996). “Investigation of running behaviourand derailment criteria for freight cars by dynamic simulation,” COM-PRAIL 96, Computers in Railways V, 2, Computational MechanicsPublications, Southampton, UK, 13–22.

Shen, R. (1996). “Study of analytic methods for vertical vibration ofhigh-speed railway girder bridge,” Bridge Construction, 2, 46–49 (inChinese).

Senthilvasan, J., Thambiratnam, D. P., and Brameld, G. H. (2002). “Dy-namic response of a curved bridge under moving truck load,” Eng.Struct., 24, 1283–1293.

Smith, C. C., Gilchrist, A. J., and Wormley, D. N. (1975). “Multiple andcontinuous span elevated guideway–vehicle dynamic performance,” J.Dyn. Syst., Measurement & Control, 97(1), 30–39.

Sridharan, N. and Mallik, A. K. (1979). “Numerical analysis of vibra-tion of beams subjected to moving loads,” J. Sound & Vibr., 65,147–150.

Standard Specifications for Highway Bridges. (1989). 14th edn., Amer-ican Association of State Highway and Transportation Officials(AASHTO), Washington, D.C.

Stanisic, M. M. (1985). “On a new theory of the dynamic behavior of thestructures carrying moving masses,” Ingenieur-Archiv, 55, 176–185.

Stanisic, M. M. and Hardin, J. C. (1969). “On the response of beams to anarbitrary number of concentrated moving masses,” J. Franklin Inst.,287, 115–123.

Stokes, G. G. (1849). “Discussion of a differential equation relating tothe breaking of railway bridges,” Trans. Cambridge Phil. Soc., 8(5),707–735.


References 521

Taheri, M. R., Ting, E. C., and Kukreti, A. R. (1990). “Vehicle–guidewayinteractions: A literature review,” The Shock & Vibr. Dig., 22(6),3–9.

Tan, C. P. and Shore, S. (1968a). “Dynamic response of a horizontallycurved bridge,” J. Struct. Div., ASCE, 94(3), 761–781.

Tan, C. P. and Shore, S. (1968b). “Response of horizontally curved bridgesto moving load.” J. Struct. Div., ASCE, 94(9), 2135–2151.

Tan, G. H., Brameld, G. H., and Thambiratnam, D. P. (1998). “Devel-opment of analytical model for treating bridge–vehicle interaction,”Eng. Struct., 20(1), 54–61.

Timoshenko, S. P. (1922). “On the forced vibrations of bridges,” Phil. Mag-azine, Ser. 6, 43, 1018–1019.

Timoshenko, S. P. and Young, D. H. (1955). Vibration Problems in Engi-neering, 3rd edn., D. Van Nostrand, New York, N.Y.

Ting, E. C. and Genin, J. (1980). “Dynamics of bridge structures,” SMArchives, 5(3), 217–252.

Ting, E. C., Genin, J., and Ginsberg, J. H. (1974). “A general algorithmfor the moving mass problem,” J. Sound & Vibr., 33(1), 49–58.

Ting, E. C., Genin, J., and Ginsberg, J. H. (1975). “Dynamic interactionof bridge structures and vehicles,” The Shock & Vibr. Dig., 7(11),21–29.

Ting, E. C. and Yener, M. (1983). “Vehicle–structure interactions in bridgedynamics,” The Shock & Vibr. Dig., 15(12), 3–9.

T. Y. Lin Taiwan (1993). “Vibrations of elevated bridge structures causedby train loadings of the west Taiwan High Speed Rail Project — Sup-plementary final report,” Provisional Eng. Office of High Speed Rail,Ministry of Transportation and Communications, Taiwan, R.O.C.

Veletsos, A. S. and Huang, T. (1970). “Analysis of dynamic response ofhighway bridges,” J. Eng. Mech. Div., ASCE, 96(5), 593–620.

Vellozzi, J. (1967). “Vibration of suspension bridges under moving loads,”J. Struct. Div., ASCE, 93(4), 123–138.

Wakui, H., Matsumoto, N., Matsuura, A., and Tanabe, M. (1995). “Dy-namic interaction analysis for railway vehicles and structures,” J.Struct. Mech. & Earthquake Eng., JSCE, 513, 129–138 (in Japanese).

Wang, R. T. (1997). “Vibration of multi-span Timoshenko beams to a mov-ing force,” J. Sound & Vibr., 207(5), 731–742.

Wang, T. L., Garg, V. K., and Chu, K. H. (1991). “Railway bridge/vehicleinteraction studies with new vehicle model,” J. Struct. Eng., ASCE,117(7), 2099–2116.



Wang, T. L. and Huang, D. (1992). “Cable-stayed bridge vibrationdue to road surface roughness,” J. Struct. Eng., ASCE, 118(5),1354–1374.

Wang, T. L., Huang, D., and Shahawy, M. (1992). “Dynamic response ofmultigirder bridges,” J. Struct. Eng., ASCE, 118(8), 2222–2238.

Warburton, G. B. (1976). The Dynamical Behaviour of Structures, Perga-mon, Oxford, England.

Weaver, W. and Johnston, P. R. (1987). Structural Dynamics by FiniteElements, Prentice-Hall, Englewood Cliffs, N.J.

Weaver, W., Timoshenko, S. P., and Young, D. H. (1990). Vibration Prob-lems in Engineering, 5th edn., John Wiley & Sons, New York, N.Y.

Wen, R. K. (1960). “Dynamic response of beams traversed by two-axleloads,” J. Eng. Mech. Div., ASCE, 86(EM5), 91–111.

White, R. C., Limbert, D. A., Hedrick, J. K., and Cooperrider, N. K.(1978) “Guideway-suspension tradeoffs in rail vehicle systems,” Ari-zona State University Engineering Research Center Report ERC-R-78035.

Willis, R. (1849). “Appendix to the report of the commissioners appointedto inquire into the application of iron to railway structures,” H. M.Stationary Office, London, England.

Wiriyachai, A., Chu, K. H., and Garg, V. K. (1982). “Bridge impact dueto wheel and track irregularities,” J. Eng. Mech., ASCE, 108(4),648–666.

Wu, C. M. (1998). Dynamic response of curved bridges due to high speedtrains and vibration of continuous bridges, Master’s thesis, NationalTaiwan University, Taipei, Taiwan, R.O.C.

Wu, J. S. and Dai, C. W. (1987). “Dynamic responses of multispan nonuni-form beam due to moving loads,” J. Struct. Eng., ASCE, 113(3),458–474.

Wu, J. S., Lee, M. L., and Lai, T. S. (1987). “The dynamic analysis of aflat plate under a moving load by the finite element method,” Int. J.Num. Meth. Eng., 24, 743–762.

Wu, Y. S. (2000). Dynamic interaction of train–rail–bridge system undernormal and seismic conditions, Ph.D. thesis, National Taiwan Uni-versity, Taipei, Taiwan, R.O.C.

Wu, Y. S. and Yang, Y. B. (2003). “Steady-state response and riding com-fort of trains moving over a series of simply supported bridges,” Eng.Struct., 25(2), 251–265.


References 523

Wu, Y. S., Yang, Y. B., and Yau, J. D. (2001). “Three-dimensional analysisof train–rail–bridge interaction problems,” Vehicle Syst. Dyn., 36(1),1–35.

Xia, H., Xu, Y. L., and Chan, T. H. T. (2000). “Dynamic interaction of longsuspension bridges with running trains,” J. Sound & Vibr., 237(2),263–280.

Xu, Y. L., Xia, H., and Yan, Q. S. (2003). “Dynamic response of suspensionbridge to high wind and running train,” J. Bridge Eng., ASCE, 8(1),46–55.

Yang, F. and Fonder, G. A. (1996). “An iterative solution method for dy-namic response of bridge–vehicles systems,” Earthquake Eng. Struct.Dyn., 25, 195–215.

Yang, F. and Fonder, G. A. (1998). “Dynamic response of cable-stayed bridges under moving loads,” J. Eng. Mech., ASCE, 214(7),741–747.

Yang, Y. B., Chang, C. H., and Yau, J. D. (1999). “An element for analysingvehicle–bridge systems considering vehicle’s pitching effect,” Int. J.Num. Meth. Eng., 46, 1031–1047.

Yang, Y. B. and Kuo, S. R. (1987). “Effect of curvature on stability ofcurved beams,” J. Struct. Eng., ASCE, 113(6), 1185–1202.

Yang, Y. B. and Kuo, S. R. (1994). Theory and Analysis of NonlinearFramed Structures, Prentice-Hall, Englewood Cliffs, New Jersey.

Yang, Y. B., Liao, S. S., and Lin B. H. (1995). “Impact formulas for vehiclesmoving over simple and continuous beams,” J. Struct. Eng., ASCE,121(11), 1644–1650.

Yang, Y. B., Liao, S. S., and Lin, B. H. (1997a). Closure to “impact for-mulas for vehicles moving over simple and continuous beams,” Proc.No. 08917, J. Struct. Eng., ASCE, 123(4), 533–534.

Yang, Y. B. and Lin, B. H. (1995). “Vehicle–bridge interaction analysisby dynamic condensation method,” J. Struct. Eng., ASCE, 121(11),1636–1643.

Yang, Y. B., Lin, C. L., Yau, J. D., and Chang, D. W. (2004). “Mechanismof resonance and cancellation for train-induced vibrations on bridgeswith elastic bearings,” J. Sound & Vibr., 269(1–2), 345–360.

Yang, Y. B., Lin, C. W., and Hsu, T. P. (1996). “Impact responses forbridges subject to random traffic flows,” Proc. 1996 Asia-Pacific Re-gional Meeting, eds. Chou, C. P. and Chang, S. K., Taipei, R.O.C.,4, 43–49.



Yang, Y. B., Wu, C. M., and Yau, J. D. (2001). “Dynamic response of ahorizontally curved beam subjected to vertical and horizontal movingloads,” J. Sound & Vibr., 242(3), 519–537.

Yang, Y. B. and Wu, Y. S. (2001). “A versatile element for analysingvehicle–bridge interaction response,” Eng. Struct., 23, 452–469.

Yang, Y. B. and Wu, Y. S. (2002). “Dynamic stability of trains moving overbridges shaken by earthquakes,” J. Sound & Vibr., 258(1), 65–94.

Yang, Y. B. and Yau, J. D. (1997). “Vehicle–bridge interaction elementfor dynamic analysis,” J. Struct. Eng., ASCE, 123(11), 1512–1518(Errata: 124(4), 479).

Yang, Y. B. and Yau, J. D. (1998). “Vehicle–bridge interactions and appli-cations to high speed rail bridges,” Proc. Natl. Science Coun. R.O.C.(A), 22(2), 214–224.

Yang, Y. B., Yau, J. D., and Hsu, L. C. (1997b). “Vibration of simple beamsdue to trains moving at high speeds,” Eng. Struct., 19(11), 936–944.

Yau, J. D. (1996). Dynamic response of bridges travelled by trains — Ana-lytical and numerical approaches, Ph.D. thesis, National Taiwan Uni-versity, Taipei, Taiwan, R.O.C.

Yau, J. D., Wu, Y. S., and Yang, Y. B. (2001). “Impact response of bridgeswith elastic bearings to moving loads,” J. Sound & Vibr., 248(1),9–30.

Yau, J. D. and Yang, Y. B. (1998). “Effect of axle interactions on thedynamic response of the simple beam and vehicles moving overit,” Proc. 6th East Asia-Pacific Conf. Struct. Eng. & Construction,2, National Taiwan University, Taipei, Taiwan, R.O.C., January,1437–1442.

Yau, J. D. and Yang, Y. B. (2004). “Vibration reduction for cable-stayedbridges traveled by high speed trains,” Fin. Elem. Analysis & Design,40, 341–359.

Yau, J. D., Yang, Y. B., and Kuo, S. R. (1999). “Impact response of highspeed rail bridges and riding comfort of rail cars,” Eng. Struct., 21(9),836–844.

Zhang, Q. L., Vrouwenvelder, A., and Wardenier, J. (2001a). “Numericalsimulation of train-bridge interactive dynamics,” Comp. & Struct.,79, 1059–1075.

Zhang, Q. L., Vrouwenvelder, A., and Wardenier, J. (2001b). “Dynamic am-plification factors and EUDL of bridges under random traffic flows,”Eng. Struct., 23, 663–672.


References 525

Zheng, D. Y., Cheung, Y. K., Au, F. T. K., and Cheng, Y. S. (1998).“Vibration of multi-span non-uniform bridges under moving loads byusing modified beam vibration functions,” J. Sound & Vibr., 212,455–467.



Subject Index

1940 El Centro Earthquake, 4261994 Northridge Earthquake, 4261999 Chi–Chi Earthquake, 409, 426

AASHTO, 20added mass, 50, 63alignment irregularity, 356, 375anti-symmetric mode, 35asymmetric crossing movement, 391axle load decrement ratio (PD), 399

ballast, 14, 158, 160, 188bandedness, 18, 155, 167base-line correction, 433beam element, 489Bernoulli–Euler beam, 30, 313, 413bi-directional excitation, 463bogie, 46, 273bogie-side lateral to vertical force ratio

(BYQ), 401braking, 245, 265bridge codes, 20bridge element, 282, 334, 419, 507bridge model, 9

cancellation, 15, 27, 29, 56, 77, 81,102, 108, 111, 112, 116, 121, 143,146

central finite rail (CFR) element, 279,311, 327, 332, 414, 418

central track segment, 279Central Weather Bureau, 463

centrifugal force, 48, 50, 126, 242characteristic length, 21, 182characteristic number, 284, 339, 422Chaster Rail Bridge, 4classical damping, 493comfort index, 306condensed equation of motion, 165,

243, 287, 346condensed stiffness matrix, 166consistent nodal loads, 19, 167, 206,

212, 236, 242constraint condition, 237, 278, 323contact force, 2, 15, 19, 156, 160, 171,

233, 236, 240, 277, 291, 314, 321,324, 443

continuous beam, 182Coriolis force, 48, 50, 242corrugation, 188critical car length, 56critical speed, 186cross-level irregularity, 380crossing movement, 391crossing of two vehicles, 14, 390curved beam, 125, 481, 483

damped frequency of vibration, 33damping, 30, 38, 43, 62, 68, 69, 82, 85,

93, 245, 289, 349, 489deceleration, 244, 262derailment, 14, 313, 399, 447derailment index, 411Dirac delta function, 45, 72, 106

527



direct integration method, 16, 477

driving frequency, 23, 75, 106Duhamel’s integral, 76

dynamic amplification factor, 20

dynamic condensation, 16, 70, 157

dynamic increment factor, 20

earthquake, 409

effective load vector, 216, 479

effective resistant force vector, 169

effective stiffness matrix, 169, 216, 479elastic bearing, 14, 69, 102, 122

elastically-supported beam, 69, 73

elevation irregularity, 356

envelope formula, 87

equations of motion for the structure,48, 72, 105, 106, 168, 213, 493

equivalent stiffness equation, 163, 169,209, 479

Eurocode, 305

excitation frequency, 4, 21, 33

external damping coefficient, 30, 72

far field excitation, 426

fast Fourier transform (FFT), 271

Federal Railroad Administration(FRA), 303, 357

flexural sine mode, 73

forced vibration, 31, 81, 108, 141France-SNCF, 305

free vibration, 32, 34, 108, 139

frequency equation, 74

frictional coefficient, 244, 288, 348

frictional force, 238, 244

fundamental frequency, 105

Galerkin’s method, 131

general contact force, 244, 287, 347general solution, 132

generalized coordinate, 31, 49, 73, 103

generalized forcing function, 75

governing equations for curved beam,128

gravitational force, 126

Guyan reduction technique, 235

Hermitian function, 205, 242

high-frequency excitation, 367

high modes, 41, 61, 176, 255

homogenous solution, 132, 136

horizontal contact force, 244

horizontal frequency, 483

horizontal moving load, 125

horizontal reaction force, 265

I − S plot, 96, 150

idle train, 435, 448

impact factor, 9, 19, 29, 37, 90, 150,299, 384

impact factor for end shear force, 43

impact factor for midpoint bendingmoment, 40

impact factor for midpointdisplacement, 36

impact formula, 27

in-plane vibration, 126

incremental-iterative analysis, 173

inertia effect, 2, 30, 47

infinite beam, 273

initial pulse phenomenon, 440

instability region, 470

interaction element, 18, 158

interaction force, 171, 234

interlocking action, 243, 262

internal damping coefficient, 30, 72

internal resistant force, 162

L’Hospital’s rule, 55

Lagrange multiplier, 235

Lagrange’s equation, 16

lateral track force (Y ), 402

left semi-infinite rail (LSR) element,279, 283, 311, 326, 337, 342, 420,509

left track segment, 279

light damping, 76

linking action, 291, 346

low-frequency excitation, 367

master–slave relation, 157, 165, 175,235


Subject Index 529

maximum allowable lateral axle forceYlim, 402

maximum allowable speed, 470

maximum static deflection, 37, 76

minimal bridge segment, 273, 277

modal damping coefficient, 494

modal damping ratio, 75

modal equation, 494

modal mass, 494

modal superposition method, 16, 103

modified Newton–Raphson method,172

moving load, 2, 6, 28, 30, 47, 105, 129,167, 184, 249

moving mass, 6, 167, 252

natural deformation, 417

near fault excitation, 426, 427

Newmark’s β method, 16, 63, 163,169, 208, 239, 247, 292, 425, 477

Ontario Code, 20

optimal condition, 64

optimal design criterion, 15, 29, 57

Ormsby filtering, 433

orthogonality property, 494

out-of-plane vibration, 126

particular solution, 132, 137

pavement roughness, 7, 11

pitching, 199, 399

possible derailment zone, 447

power spectral density (PSD), 271,356

procedure of iterations, 171

profile irregularity, 375

Provision DS-804, 434

rail element, 507

rail irregularity, 7, 11, 158, 183, 186

railway bridge, 3, 12, 118, 435, 450

Rayleigh damping, 49, 161, 183, 205,247, 292, 353, 489, 495

Rayleigh’s method, 74

renumbering, 292

residual response, 56, 60, 81, 108, 139,140, 142, 485

resonance, 4, 15, 27, 29, 54, 60, 77, 81,91, 102, 108, 110, 112, 116, 119,143, 149, 372, 386, 390

resonant speed, 56, 81riding comfort, 2, 19, 170, 177, 182,

191, 222, 234, 257, 303, 305right semi-infinite rail (RSR) element,

279, 285, 311, 326, 340, 343, 423,510

right track segment, 279rigid beam, 104rigid car body, 204rigid displacement, 73, 107, 417rigid vehicle–bridge interaction

element, 207, 211rocking motion, 419rolling, 399, 443

seismic force, 102series of moving loads, 45, 106, 140serviceability, 222single wheel lateral to vertical force

ratio (SYQ), 400, 446spectral representation method, 357speed parameter, 21, 33, 106, 110,

133, 384Sperling’s ride index, 271, 306sprung mass, 7, 159, 176, 184, 234, 254stability limit, 460steady-state response, 29, 275, 296stiffness ratio, 73, 104superelevation irregularity, 356support displacement, 417surface roughness, 10, 11, 30suspension damping, 194, 226suspension stiffness, 191, 223symmetric crossing movement, 391symmetric mode, 37symmetry, 18, 155, 167, 213, 292

Taiwan-HSR, 70, 305TAP003 motion, 426TCU068 motion, 426torsional vibration, 14



track classification, 356

track irregularity, 11, 202, 223, 229,303, 323, 354, 372

train–bridge interaction, 413

train–rails–bridge interaction, 8, 411

train–rails–bridge resonance, 298

tuned mass, 186

two-axle system, 217

unbalanced force, 161, 171

uni-directional excitation, 460

unit step function, 45, 106

upper-bound envelope, 38

vehicle equation, 160, 162, 204, 236,277, 314, 322

vehicle–bridge interaction (VBI), 2,28, 155, 157

vehicle–bridge interaction (VBI)element, 165, 343

vehicle–rails interaction (VRI), 286vehicle–rails interaction (VRI)

elements, 286, 311, 425vehicle–rails–bridge interaction, 271,

311, 361, 366, 413vertical contact force, 242, 286vertical frequency, 481vertical moving load, 129

wheel assembly, 46wheel climb, 447wheel load function, 47wheelset, 46wheelset lateral to vertical force (YQ)

ratio, 400, 446Winkler foundation, 284, 338, 422

yawing, 399

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Vehicle-Bridge Interaction Dynamics