Determination of Dynamic Track Modulus from Measurement of Track Velocity during Train Passage

The measurement of track stiffness, or track modulus, is an important parameter for assessing the condition of a railway track. This paper describes a method by which the dynamic track modulus can be determined from the dynamic displacements of the track during normal train service, measured using geophones. Two techniques are described for calculating the track modulus—the inferred displacement basin test (DBT) method and a modified beam on an elastic foundation (BOEF) method. Results indicate that the viscoelastic response of the soil will influence the value of track modulus determined using the DBT method. The BOEF method was therefore used to calculate the apparent increase in axle load due to train speed. Hanging or partly supported sleepers were associated with a relatively small increase in dynamic axle loads with train speed.     

Ground Vibration from High-Speed Trains: Prediction and Countermeasure

This paper outlines a test program in southern Sweden for measurement of the vibration induced in the ground and railway embankment by high-speed trains, together with a rigorous numerical model developed for the prediction of embankment/ground response. In this formulation the ground is modeled as a layered viscoelastic half-space, and the railway embankment is modeled as a viscoelastic beam excited by the moving loads of the train. The model uses the Kausel-Ro?sset Green's functions to calculate the soil stiffness matrix at the ground-embankment interface and assembles it with the dynamic stiffness matrix of the embankment. The solution is carried out in the frequency domain, and the time histories of the motions are derived through a Fourier synthesis of the frequency components. Numerous simulations of train-induced vibration are presented for the ground conditions and embankment parameters at the test site and compared with measured records. The simulations agree well with the measurements, both in qualitative and quantitative terms. In particular, the large ground deformations registered for train speeds exceeding 140 km∕h are reproduced by the simulations. With the help of the prediction model, the effectiveness of a remediation measure for the mitigation of ground vibration is explored.     

Effects of Train Characteristics on the Rate of Deterioration of Track Roughness

Track roughness describes in part the up and down waves in the longitudinal geometry of a railway track. A train passing over rough track experiences a degree of bouncing that generates oscillations in the forces exerted by the train’s wheels on the top of the rail, which in turn cause this roughness to worsen. The rate at which the track roughness deteriorates depends on the response of the track to the weight of the train and to the dynamic oscillations in wheel/rail forces, which in turn are affected by the properties of the train vehicles’ components and the speed of the train. The paper develops relationships between the severity of track roughness and the dynamic wheel/rail forces generated by a moving train using field data, and between those forces and the specific vehicle characteristics of speed, total mass, unsprung mass, suspension stiffness, and damping, using NUCARS simulations. These two relationships in turn are combined to show how the speed of the train and the design of the train vehicle’s bogie suspension can worsen or improve the rate of deterioration of track roughness. The relationships also provide a firm basis for the owner of track to set more representative charges levied on the train operator for using the track.     

Transient Dynamics of Stochastically Parametered Beams

The problem of determining the statistics of the transient response of randomly inhomogeneous beams is formulated. This is based on the use of stochastic dynamic stiffness coefficients in conjunction with the fast Fourier transform algorithm. The dynamic stiffness coefficients, in turn, are determined using a stochastic finite-element formulation that employs frequency-dependent shape functions. The approach is illustrated by analyzing the response of a random rod subject to a boxcar type of axial impact and, also, by considering the flexural response of a randomly inhomogeneous beam resting on a randomly varying Winkler's foundation and subjected to the action of a moving force. A discussion on the treatment of system property random fields as being non-Gaussian in nature is presented. Also discussed are the methods for handling nonzero initial conditions within the framework of the frequency domain response analysis employed in the study. Satisfactory comparisons between the analytical results and simulation results are demonstrated.     

Dynamic Response of Suspension Bridge to High Wind and Running Train

A framework is presented for predicting the dynamic response of long suspension bridges to high winds and running trains. A three-dimensional finite-element model is used to represent a suspension bridge. Wind forces acting on the bridge, including both buffeting and self-excited forces, are generated in the time domain using a fast spectral representation method and measured aerodynamic coefficients and flutter derivatives. Each 4-axle vehicle in a train is modeled by a 27-degrees-of-freedom dynamic system. The dynamic interaction between the bridge and train is realized through the contact forces between the wheels and track. By applying a mode superposition technique to the bridge only and taking the measured track irregularities as known quantities, the number of degrees of freedom of the bridge-train system is significantly reduced and the coupled equations of motion are efficiently solved. The proposed formulation is then applied to a real wind-excited long suspension bridge carrying a railway inside the bridge deck of a closed cross section. The results show that the formulation presented in this paper can predict the dynamic response of the coupled bridge-train systems under fluctuating winds. The extent of interaction between the bridge and train depends on wind speed and train speed.     

Dynamic Response of Elevated High-Speed Railway

The study of the dynamic response of the elevated railway for the high-speed train in the Taiwan area at the preliminary design stage is presented. Two types of the elevated reinforced concrete railway, they being the simple-span and the three-equal-span box girders supported on piers; three types of the high-speed train, namely, the French T.G.V., the German I.C.E., and the Japanese S.K.S.; and the maximum operation speed 350 km∕h are under investigation. The general dynamic stiffness matrix of a damped Timoshenko beam is employed for the structural analysis. The influence line of any dynamic response (also called the dynamic influence line) of the elevated railway subjected to the high-speed train, considered as a series of the moving loads, is calculated by the mode-superposition method. A preliminary design of the section of the railway is proposed for this study.     

Ground Motion Induced by Train Passage

Two methods are illustrated to effectively calculate the ground motion induced by constant speed moving loads. On the one hand, the dynamic Betti–Rayleigh reciprocity theorem allows one to take full advantage of the availability of Green’s functions for a homogeneous or a horizontally layered half-space. On the other hand, the spectral element method allows one to deal with complicated configurations, including dynamic soil-structure interaction, with an accuracy that is significantly higher than classical finite element or finite difference methods. Both the Betti–Rayleigh and spectral element methods are considered in three dimensions. After validating both methods, the decay of peak ground motion with distance is analyzed as a function of load speed and frequency. For speeds lower than the Rayleigh wave velocity in the soil, the decay turns out to be much faster than for a stationary point load. This effect is studied in detail by an analytical approach and interpreted in terms of destructive interference. Finally, the previous analytical and numerical results are checked against the records obtained at Ath, Belgium, during a field experiment to study ground motion induced by high-speed trains in soft soil conditions.     

Experimental Study on the Stability of Railroad Silt Subgrade with Increasing Train Speed

The comfort and safety of a moving train is largely determined by the dynamic response of the railway track and its foundation (i.e., subgrade). To study the dynamic stability of a silt subgrade subjected to train traffic loading with increasing speed, cyclic triaxial tests were conducted for compacted silt specimens with varying dry density, water content, dynamic stress, and load frequency. The laboratory test results and field measurements of the subgrade dynamic stress under train loading indicate that with increasing train speed, an increase in dynamic stress and load frequency does not impair the stability of the silt subgrade, provided the subgrade is in sound physical condition (i.e., its natural water content approximates the optimal water content) and the relative compaction is at least 90%. However, if the relative compaction is 85%, the subgrade is stable only at a dynamic stress level that is below 70 kPa, and the subgrade may suffer shear failure at a higher dynamic stress level. The elastic deformation of the subgrade linearly increases with an increase in train speed. However, if the degree of saturation of the silt subgrade increases, the thresholds of both the dynamic stress and resilient modulus decrease markedly, accompanied by sharp increases in elastic deformation and cumulative deformation and can even result in the shear failure of the subgrade. These conditions are unfavorable for the high speeds and stability needed for trains; therefore, train speeds should be limited in wet conditions to reduce subgrade dynamic stress and load frequency.     

Soil Vibrations Caused by Underground Moving Trains

The wave propagation problems caused by the underground moving trains are analyzed by the 2.5-dimensional finite/infinite-element approach. The near field of the half-space, including the tunnel and parts of the soil, is simulated by finite elements, and the far field extending to infinity by infinite elements. The train is simulated as a sequence of wheel loads moving at constant speeds. Using the present approach, a two-dimensional profile with three degrees per node is used to simulate the three-dimensional behavior of the half-space, which is valid for the case when the material and geometry of the system are invariant along the tunnel direction. The factors considered in the analysis of ground-borne vibrations include the damping ratio and stratum depth of the supporting soils, the depth and thickness of the tunnel, and the moving speed and excitation frequency of the trains. It was found that moving train loads with nonzero excitation frequencies can induce significantly higher vibrations than the static moving loads. The effect of stratum depth depends highly on the excitation frequency. For a tunnel constructed in a stiffer soil, the ground surface vibrations can be greatly reduced. Other conclusions useful to practical engineers are contained in the parametric study.     

Model Validation for Bridge-Road-Vehicle Dynamic Interaction System

The dynamic response of highway bridges subjected to moving truckloads has been observed to be dependent on (1) dynamic characteristics of the bridge; (2) truck configuration, speed, and lane position on the bridge; and (3) road surface roughness profile of the bridge and its approach. Historically, truckloads were measured to determine the load spectra for girder bridges. However, truckload measurements are either made for a short period of time [for example, weigh-in-motion (WIM) data] or are statistically biased (for example, weigh stations) and cost prohibitive. The objective of this paper is to present results of a 3D computer-based model for the simulation of multiple trucks on girder bridges. The model is based on the grillage approach and is applied to four steel girder bridges tested under normal truck traffic. Actual truckload data collected using a discrete bridge WIM system are used in the model. The data include axle loads, truck gross weight, axle configuration, and statistical data on multiple presence (side by side or following). The results are presented as a function of the static and dynamic stresses in each girder and compared with code provisions for dynamic load factor. The study provides an alternate method for the development of live-load models for bridge design and evaluation.     

Dynamic Stress Analysis of a Ballasted Railway Track Bed during Train Passage

Scientific design of a railway track formation requires an understanding of the subgrade behavior and the factors affecting it. These include the effective resilient stiffness during train passage, which is likely to depend on the stress history and the stress state of the ground, and the stress path followed during loading. This study investigates the last of these, by means of a two-dimensional dynamic finite-element analysis. The effects of train speed, acceleration/braking, geometric variation in rail head level, and a single unsupported sleeper are considered. Results indicate that dynamic effects start to become apparent when the train speed is greater than 10% of the Rayleigh wave speed, vc, of the subgrade. At a train speed of 0.5vc, the shear stresses will be underestimated by 30% in a static analysis, and at train speeds greater than vc the stresses due to dynamic effects increase dramatically. Train acceleration/braking may increase shear stresses and horizontal displacements in the soil, and hence the requirement for track maintenance at locations where trains routinely brake or accelerate. For heavy haul freight trains, long wavelength variations in rail head level may lead to significantly increased stresses at passing frequencies (defined as the train speed divided by the wavelength of the variation in level) greater than 15, and short wavelength variations at passing frequencies of 60–70. Stress increases adjacent to an unsupported sleeper occur in the ballast and subballast layers, but rapidly become insignificant with increasing depth.     

Wave Barriers for Reduction of Train-Induced Vibrations in Soils

This paper is aimed at studying the effectiveness of different vibration countermeasures in isolating the ground vibrations induced by trains moving at sub- and supercritical speeds, with respect to the Rayleigh wave speed of the supporting soils. The vibration countermeasures considered herein include the installation of open trenches, in-filled trenches, and wave impeding blocks. The 2.5D finite/infinite element approach developed previously by the authors is employed in this study. This approach allows us to consider the load-moving effect of the train in the direction normal to the two-dimensional profile considered, and therefore to obtain three-dimensional results using only two-dimensional elements. The moving train is simulated as a sequence of moving wheel loads that may vibrate at some specific frequencies. The performance of the three types of wave barriers in isolating soil vibrations for trains moving at sub- and supercritical speeds with various excitation frequencies is evaluated with respect to some key parameters, along with suggestions made for enhancing the isolation efficiency.     

Experimental Dynamic Response of a Short-Span Composite Bridge to Military Vehicles

A continued desire for increased mobility in the aftermath of natural disasters or on the battlefield has lead to the need for improved lightweight bridging solutions. Currently, within the U.S. military, there is a need for a lightweight bridging system for crossing short-span gaps up to 4 m (13.1 ft) in length. This paper describes the field testing of a newly developed lightweight fiber-reinforced polymer bridging system to meet the U.S. militaries needs. The study investigates dynamic impact loads of track and wheel vehicles at different crossing speeds to increase understanding of appropriate impact factors used in design. It was found that the impact loads for the bridge treadways were most sensitive to vehicle crossing speed and vehicle type (wheel versus track and axle spacing) with observed impact factors as high as 1.71.     

Train-Induced Vibration Control of High-Speed Railway Bridges Equipped with Multiple Tuned Mass Dampers

This paper deals with the applicability of multiple tuned mass dampers (MTMDs) to suppress train-induced vibration on bridges. A railway bridge is modeled as an Euler-Bernoulli beam and a train is simulated as a series of moving forces, moving masses, or moving suspension masses. According to the train load frequency analysis, resonant effects will occur as the modal frequencies of a bridge are close to the multiple of the impact frequency of the train load to the bridge. An MTMD system is then designed to alter the bridge dynamic characteristics to avoid excessive vibrations. Numerical results from simply supported bridges of the Taiwan High-Speed Railway (THSR) under real trains show that the proposed MTMD is more effective and reliable than a single TMD in reducing dynamic responses during resonant speeds, as the train axle arrangement is regular. It is also found that the inner space of a bridge box-girder of the THSR is wide and deep enough for installation and movement of MTMDs.     

Concentrations of Pressure Between an Elastically Supported Beam and a Moving Timoshenko-Beam

The present paper is concerned with the motion of an elastically supported beam that carries an elastic beam moving at constant speed. This problem provides a limiting case to the assumptions usually considered in the study of trains moving on rail tracks. In the literature, the train is commonly treated as a moving line-load with space-wise constant intensity, or as a system of moving rigid bodies supported by single springs and dampers. In extension, we study an elastically supported infinite beam, which is mounted by an elastic beam moving at a constant speed. Both beams are considered to have distributed stiffness and mass. The moving beam represents the train, while the elastically supported infinite beam models the railway track. The two beams are connected by an interface modeled as an additional continuous elastic foundation. Here, we follow a strategy by Stephen P. Timoshenko, who showed that a beam on discrete elastic supports could be modeled as a beam on a continuous elastic Winkler (one-parameter) foundation without suffering a substantial loss in accuracy. The celebrated Timoshenko theory of shear deformable beams with rotatory inertia is used to formulate the equations of motion of the two beams under consideration. The resulting system of ordinary differential equations and boundary conditions is solved by means of the powerful methods of symbolic computation. We present a nondimensional study on the influence of the train stiffness and the interface stiffness upon the pressure distribution between train and railway track. Considerable pressure concentrations are found to take place at the ends of the moving train.     

Substructural Identification Method without Interface Measurement

Substructural identification provides a novel means by which to reduce a large problem to smaller problems of manageable size, thereby improving numerical convergence and accuracy. Various methods proposed by several researchers thus far require interface response measurements, which are then treated as input to the substructures of concern. In practice, however, it is not always possible to obtain interface measurements, particularly if rotational response is required for beam/frame structures. In this paper, a method for parameter identification of substructures without the need of interface measurements is proposed. On the basis of receptance theory, an inverse problem is formulated in the frequency domain. Interface forces are eliminated by using different sets of measurements in the substructure concerned under the same dynamic excitation. The genetic algorithms approach is employed to determine the unknown parameters, and the fitness function is defined to minimize the difference between the estimates of interface forces obtained using different sets of response measurements. Three numerical examples are presented to illustrate the proposed method, and account for effects of measurement noise.     

Studying Characteristics of Train-Induced Ground Vibrations Adjacent to an Elevated Railway by Field Experiments

This paper investigates the characteristics of the ground vibration induced by moving trains on elevated railways using a number of field measurements at various train speeds. The experimental results indicate that the vibration at the dominant frequencies is quite large in both the vertical and horizontal directions, where the dominant frequencies and their influence factors can be evaluated using two simple theoretical equations deducted from the trainload in the frequency domain. Furthermore, the vibration at the dominant frequencies increases almost proportionally to the train speed, except at the resonance condition, which can be found both in field experiments and using the equation of the influence factor.     

Full Two-Dimensional Model for Rolling Resistance. II: Viscoelastic Cylinders on Rigid Ground

Following the previous paper, “Full two-dimensional model for rolling resistance: Hard cylinder on viscoelastic foundation of finite thickness,” addressing modeling of rigid cylinder rolling against viscoelastic foundation of finite thickness, this paper focuses on the development of a rigorous full two-dimensional semianalytical model for viscoelastic rollers with layered structure rolling against a rigid ground. In this model, the polar coordinate system is used, the solution is expanded into a set of Fourier series corresponding to the angular coordinate, the frequency domain master curves of G′ and G″ characterizing the general viscoelastic properties for a viscoelastic material are used to relate Fourier coefficients, and a special condensed structure model based on the Fourier series is developed to handle viscoelasticity and the rolling contact boundary condition. Examples are given to show the model capabilities to efficiently handle rolling resistance and contact stresses, and capture major characteristics of standing-wave phenomenon, such as sharp rise of rolling resistance, emergence of standing waves and material dynamic softening as the rolling speed approaches a critical value. The methodology may be of interest to industrial roller designers.     

Dynamic Response of Plates on Elastic Foundation to Moving Loads

A procedure incorporating the finite strip method and a spring system has been developed and applied to treat the dynamic response of plate structure resting on an elastic foundation to moving loads. The response to a single moving concentrated load is first investigated and then the effects of velocity, elastic foundation stiffness, moving path, and distance between multiple moving loads are studied. The response under a moving harmonic load with constant velocity is finally treated and the effect of the load frequency is investigated. Results indicate that the foundation stiffness and the velocity and frequency of the moving load have significant effects on the dynamic response of the plate and on resonant velocities. Some of these findings might find use in practical applications.     

Validation of 3DFE Analysis of Rigid Pavement Dynamic Response to Moving Traffic and Nonlinear Temperature Gradient Effects

The response of dowel jointed concrete pavements to the combined effect of nonlinear thermal gradient and moving axle load is examined using three-dimensional finite-element (3DFE) modeling. The 3DFE-computed response to moving axle load was field validated versus measured concrete slab response to a fully loaded moving dump truck. The 3DFE-predicted slab curling due to nonlinear thermal gradient through the slab thickness was validated versus: (1) corner-dowel bar bending as measured using instrumented dowel embedded in an instrumented rigid pavement section in West Virginia; and (2) Westergaard’s closed-form solution. The effects of slab thickness, slab length, axle loading position, and axle type on slab stresses are examined. It is shown that while a negative temperature gradient reduces the intensity of traffic-induced stresses, positive temperature gradient increases it several fold. Formulas are developed for the computation of the peak principal stresses due to the combined effect of tandem axle load and nonlinear thermal gradient.