Vibration Analysis of an Elastic Beam Subjected to a Moving Beam with Flexible Connections

The vibration problem of a simply supported beam subjected to a moving elastic structure is investigated. The model consists of two Euler-Bernoulli beams which are assumed to be connected by flexible springs at two discrete points. The dynamic response of the simply supported beam subjected to the moving elastic beam at a constant speed is studied by the modal superposition method. The elastic stiffness and the inertial effect of the moving beam are included in the analysis. By solving the ordinary differential equations governing the motion of the model, some approximate analytical results are derived and influence factors on the dynamic response of the simply supported beam are discussed in details, including the stiffness ratio, which is defined as the stiffness of the moving beam to that of the simply supported beam, the moving velocity and the connection spring stiffness between the two beams. Results of the study imply that the connection stiffness has an apparent influence on the dynamic behavior of the simply supported beam.     

Dynamic Characteristics of Infinite and Finite Railways to Moving Loads

Some fundamental dynamic characteristics of a railway subjected to a harmonic or constant moving load are established and presented in this paper. The railway is modeled as an infinite or finite Timoshenko beam on viscoelastic foundation. The dynamic-stiffness matrices characterized by the complex wave numbers are employed to deal with this problem. The relationship between the forced frequency and the resonant velocity of the moving load, and the resonant frequency of the railway are especially emphasized and intensively discussed. The fundamental dynamic characteristics of a railway modeled as a Bernoulli-Euler Beam on viscoelastic foundation are also included for comparison.     

Comparative Modeling Study of Reinforced Beam on Elastic Foundation

In this paper, governing ordinary differential equations are derived for a reinforced Timoshenko beam on an elastic foundation. An analytical solution is obtained for a point load on an infinite Timoshenko beam on elastic foundation. Special attention is drawn to the location, tension, and shear stiffness of reinforcement and its influence on settlement/deflection of the beam and reinforcement tension force. A finite element (FE) model is established for the same infinite beam problem. Results from the TB model (Timoshenko beam on elastic foundation) are compared with results from the FE model and from the PB model (the Winkler model, based on the pure bending beam theory). It is found that results from the proposed TB model are, in general, in good agreement with results from the FE model as compared with results from the PB model. The TB model is better than the PB model in considering the shear deformation of the beam. This TB model is particularly useful in modeling a reinforced beam with or without considering the reinforcement shear stiffness. The TB model has practical applications in modeling geosynthetics∕fiber-glass reinforcement of foundation soils or pavement.     

Dynamic Instability of Elastically Supported Beam under Traveling Inertia Load

The dynamic instability of an elastically supported Timoshenko beam under the constant velocity traveling inertia load has been investigated. The regions of dynamic instability are determined for different values of the elastic foundation constant. Floquet theory is utilized to study the parametric regions of instability, which are displayed in graphical form.     

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.     

Second-Order Stiffness Matrix and Loading Vector of a Beam-Column with Semirigid Connections on an Elastic Foundation

The second-order stiffness matrix and corresponding loading vector of a prismatic beam–column subjected to a constant axial load and supported on a uniformly distributed elastic foundation (Winkler type) along its span with its ends connected to elastic supports are derived in a classical manner. The stiffness coefficients are expressed in terms of the ballast coefficient of the elastic foundation, applied axial load, support conditions, bending, and shear deformations. These individual parameters may be dropped when the appropriate effect is not considered; therefore, the proposed model captures all the different models of beams and beam–columns including those based on the theories of Bernoulli–Euler, Timoshenko, Rayleigh, and bending and shear.The expressions developed for the load vector are also general for any type or combinations of transverse loads including concentrated and partially nonuniform distributed loads. In addition, the transfer equations necessary to determine the transverse deflections, rotations, shear, and bending moments along the member are also developed and presented.     

Direct Numerical Procedure for Solution of Moving Oscillator Problems

In this paper, the problem of a 1D elastic distributed system coupled with a moving linear oscillator, often referred to as the “moving oscillator” problem, is studied. The problem is formulated using a “relative displacement” model, which shows that, in the limiting case of infinite oscillator stiffness, the moving mass problem is recovered. The coupled equations of motion are recast into an integral equation that is amenable to solution by a direct numerical procedure. Both the integral equation and the numerical procedure show that the response of the elastic system at the current time depends only on the time history of its response at the positions of the oscillator. Numerical results are presented for the examples of a string and a simply supported beam and are compared to the moving force solutions. It is shown that the oscillator, with its stiffness suitably tuned, can excite the elastic structure into resonance.     

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.     

Substructure Simulation of Inhomogeneous Track and Layered Ground Dynamic Interaction under Train Passage

The vibrations in track and ground induced by train passages are investigated by the substructure method with due consideration to dynamic interaction between an inhomogeneous track system comprising continuous rails and discrete sleepers, and the underlying viscoelastic layered half space ground. Initially, the total system is divided into two separately formulated substructures, i.e., the track and the ground. The rail is described by introducing the Green function for an infinite long Euler beam both for moving axle loads action from a train and for reactions from sleepers. The ground is formulated by the layer transfer matrix approach for wave propagation along the depth. Subsequently, these substructures are integrated to meet the displacement compatibility and force equilibrium via inertia of sleepers and stiffness of railpad springs. The dynamic equations are solved in the frequency–wave-number domain by applying the Fourier transform procedure. Based on the assumption of a constant train speed, the time domain response is evaluated from the inverse Fourier transform computation. The dispersive characteristics of the layered ground and the moving axle loads lead to significantly different response features, depending on the train speed. The response is classified as quasistatic for a low speed, whereas it is dynamic for a high-speed situation. An illustrative case study is presented for Swedish X-2000 train track properties and ground profile.     

Planar Bending of Sandwich Beams with Transverse Loads off the Centroidal Axis

Conventional analysis methods for beams do not distinguish between transverse loads that are applied at the beam centroidal axis and those acting either above or below the centroidal axis. In contrast, this paper formulates a sandwich beam finite element solution which models the effect of load height relative to the centroidal axis. Towards this goal, the governing equilibrium equations and associated boundary conditions are derived based on a Timoshenko beam formulation for the core material. Special shape functions satisfying the homogeneous form of the equilibrium equations are derived and subsequently used to formulate exact stiffness matrices. By omitting the stiffness terms related to the faces, the formulation for a homogeneous Timoshenko beam can be recovered. Also, the Euler–Bernouilli counterpart of the formulation is recovered as a limiting case of the current Timoshenko beam formulation. Effects of load height relative to the centroid are observed to have similarities with those induced by axial forces in beam-columns. For a simply supported beam, downward acting loads located below the centroidal axis are found to induce a stiffening effect while those acting above the centroidal axis are found to induce a softening effect, resulting in higher transverse displacements.     

Buckling of Multiwalled Carbon Nanotubes Using Timoshenko Beam Theory

A Timoshenko beam model is presented in this paper for the buckling of axially loaded multiwalled carbon nanotubes surrounded by an elastic medium. Unlike the Euler beam model, the Timoshenko beam model allows for the effect of transverse shear deformation which becomes significant for carbon nanotubes with small length-to-diameter ratios. These stocky tubes are normally encountered in applications such as nanoprobes or nanotweezers. The proposed model treats each of the nested and concentric nanotubes as individual Timoshenko beams interacting with adjacent nanotubes in the presence of van der Waals forces. In particular, the buckling of double-walled carbon nanotubes modeled as a pair of double Timoshenko beams is studied closely and an explicit expression for the critical axial stress is derived. The study clearly demonstrates a significant reduction in the buckling loads of the tubes with small length-to-diameter ratios when shear deformation is taken into consideration.     

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.     

Buckling of a Weakened Infinite Beam on an Elastic Foundation

An infinite beam attached to an elastic foundation is buckled by an axial force. The beam is weakened by one or more joints or partial cracks. The governing equations are solved analytically and an exact nonlinear characteristic equation gives the buckling criterion. It is found that the buckling force depends on the foundation stiffness and the rotational resistance of the joints. The buckling modes are complex, and may be either antisymmetric or symmetric.     

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.     

Timoshenko Beam with Tuned Mass Dampers to Moving Loads

The structural analysis of a Timoshenko beam system with tuned mass dampers (TMDs) under moving-load excitation is presented. A proposed simplified two-degrees-of freedom system based on the first mode of the Timoshenko beam is employed for the design of TMDs. The dynamic characteristics of a Timoshenko beam, such as the structural-resonant and phase-resonant velocities, and the effectiveness of a TMD for vibrational control are especially emphasized. A practical example of an elevated railway subjected to the Japanese Shinkansen (SKS) high-speed bullet train is included.     

Elastic-Plastic Model of Pinned Beams Subjected to Impulsive Loading

An elastic-plastic dynamic analysis of simply supported beams with end membrane restraints subjected to impulsive loading is developed. The beam is elastically curved. The model takes into account elastic and plastic deformations and their effect on the stretch force and the distribution of inertia forces. Instantaneous plastification of the midspan zone due to bending and axial actions is considered. The effect of the length of the plastic zone is discussed. Equations of motion are derived by virtual work. Using the yield condition, the number of these equations is reduced by relating the variations of the displacement variables. This analysis is compared to the standard form of Lagrange's equation of motion showing that the latter is not energy conservative for this case where the deflection shape is nonlinearly dependent on the generalized displacements. Test results using a spring-powered apparatus are presented. Strain rate sensitivity is accounted for as plastic damping. The model's results are compared to test and rigid-segment model results. The comparison shows that the tests are sensitive to the curvature of the axis.     

Closed-Form Solution for Reinforced Timoshenko Beam on Elastic Foundation

This paper suggests a method for obtaining closed-form solutions for a reinforced Timoshenko beam on an elastic foundation subjected to any pressure loading. A particular solution is obtained for uniform pressure loading at any location of the beam. This solution can be used to calculate settlement, rotation, tension, bending moment, and shear force of the beam. A parametric study is carried out to investigate the effects of geosynthetic shear stiffness and tension modulus and the location of the pressure loading. Results are presented and discussed.     

Vibration of Vehicle on Compressed Rail on Viscoelastic Foundation

This paper presents an analytical method for the instability analysis of an axially compressed rail modeled as an infinite beam resting on a viscoelastic foundation and subjected to a moving vehicle modeled as a single-axle mass-spring-damper system. The vibration equations of the rail and vehicle are first derived. By using Laplace and Fourier transformations, the expressions of the Laplace images of the vibrations of the rail-vehicle interaction are obtained. Then the D-decomposition technique is employed to analyze the instability of the vibration system. It is shown that instability occurs for lower masses as the compression axial force increases. It also is shown that the critical mass in the case of the moving vehicle problem is always less than that in the case of the simple moving mass problem. The steady-state response of the rail is found to depend on the total vertical force acting on the moving vehicle, but it is independent of the masses and other parameters of the moving vehicle. Instability of the rail is possible, although the foundation is overdamped.     

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.     

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.