Infinite Euler-Bernoulli Beam Research Articles

Analytical solutions for joints in an infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to time-lag loads

This paper investigates the analytical method for the infinite beam with joint parts resting on a foundation under time-lag loads. The ground is simulated as a viscoelastic foundation. The existence of joint parts in beams is explained by introducing virtual forces expressed through the Dirac delta function. Integral transformation simplifies the dynamic governing equations for standard beam sections and joint parts into an algebraic equation. The convolution theorem converts the solutions into the time domain. The analytical solutions for dynamic analysis of standard beam sections and joint parts are obtained conveniently in the frequency domain. The proposed analytical method is verified by comparing it with the traditional methods and also validated by numerical examples with time-lag loads. Parametric discussions study the effects on the dynamic response of joint parts and standard beam sections, such as the amplitude and frequency of time-lag loads and coefficient of bending stiffness reduction of joint parts. The parametric analyses emphasize the dramatic effect of critical parameters on the discontinuous deformation of beams. The damage caused by critical values of some factors is emphasized. The severe effects of discontinuous deformation induced by dynamic loads in the safety of beam structures are revealed in this paper. The differences in dynamic responses between the beam and joint part must be monitored to ensure the safety of whole beam structures.

Parametric Study of the Influence of Nonlinear Elastic Characteristics of Rail Pads on Wheel–Rail Vibrations

The rail pad is the elastic element between the rail and the sleeper that has the role of absorbing the mechanical stresses from the rail and reducing the vibrations and shocks generated by wheel-rail interactions. In this paper, the problem of the influence of the variability of the nonlinear load-deformation characteristic of rail pads (resulting from the manufacturing process) on wheel-rail vibrations is investigated. The limit load-deformation characteristics of a manufactured rail pad and the medium load-deformation characteristic resulting as the arithmetic mean of the two are considered. The nonlinear load-deformation characteristic of the ballast is also considered. All these characteristics are approximated with the help of the bilinear function and are implemented in a track model consisting of an infinite Euler-Bernoulli beam placed on a two-elastic layer continuous foundation with inertial insertion, resulting in a model with an inhomogeneous foundation. The parameters of the inhomogeneous foundation are established from the equilibrium condition under a static load. Wheel-rail vibrations are studied in terms of the contact force and the acceleration of the rail and wheel. The influence of the variability of the elastic characteristics of the rail pad manifests itself in the field of medium frequencies, which amplify or attenuate the vibration levels in certain bands of one-third of an octave.

Settlement of an Existing Tunnel Induced by Crossing Shield Tunneling Involving Residual Jacking Force

An analytical formulation involving residual force is proposed to predict the displacement of an existing structure by simplifying the tunnel as an infinite Euler–Bernoulli beam resting on a two-parameter Pasternak foundation. The feasibility is confirmed by two actual measurements at sites in the published literature. Parametric studies—including consideration of jacking force alone, jacking force and ground loss, and jacking force and equivalent bending stiffness—are carried out to study the influence on the deformation of the existing tunnel. The results show that the residual jacking force can decrease the settlement of the existing tunnel, but the significance of its effect varies under different engineering conditions. With the increase in ground loss, the beneficial effect of jacking force in preventing further aggravation of tunnel settlement gradually becomes obvious, and the jacking force can reduce the deformation more effectively when the tunnel has lower bending stiffness. As a result, it is recommended that some effective measures should be adopted to maintain sufficient residual jacking force in the segments, so as to prolong the life of the tunnel.

Track Model with Nonlinear Elastic Characteristic of the Rubber Rail Pad

This paper presents a new basic nonlinear track model consisting of an infinite Euler-Bernoulli beam (rail) resting on continuous foundation with two elastic layers (rail pad and ballast bed) and intermediate inertial layer (sleepers). The two elastic layers have bilinear elastic characteristic obtained from the load-displacement characteristic of the rail pad and ballast. A time-varying load with two components - time-constant one and harmonic other, representing the wheel/rail contact force is considered as the track model input. Rail deflection due to the time-constant component of the load is obtained solving the nonlinear equations of the balance position. Subsequently, the structure of the nonhomogeneous foundation is determined. Dynamic rail response in terms of receptance due to the harmonic component of the load is calculated using the linearised track model with nonhomogeneous elastic characteristic. Influence of the time-constant component and the reflected waves due to the nonhomogeneous foundation are presented.

A frequency and velocity-dependent impedance method for prediction of rail/foundation dynamics

This paper presents an efficient numerical tool for the prediction of railway dynamic response. A behavior calibration of the infinite Euler-Bernoulli beam resting on continuous viscoelastic foundation is proposed. Constitutive laws of the discrete elements are determined for a rectilinear ballasted track. A three-dimensional model coupled with an adaptive meshing scheme is employed to calibrate the beam model impedances by finding the similarity between the output signals using the genetic algorithm. The model shows an important performance with significant reduction in computational effort. This study emphasizes the major impact of the excitation characteristics on the parameters of the discrete models.

Transition radiation in an infinite one-dimensional structure interacting with a moving oscillator—the Green’s function method

Transition zones in railway tracks are areas with considerable variation of track properties (e.g., foundation stiffness) which may cause strong amplification of the response, leading to rapid degradation of the track geometry. Two possible indicators of degradation in the supporting structure are identified, namely the wheel-rail contact force and the power/energy input by the vehicle. This paper analyses the influence of accounting for the interaction between the vehicle and the supporting structure on the contact force and on the power/energy input. To this end, a one-dimensional model is formulated, consisting of an infinite Euler-Bernoulli beam resting on a locally inhomogeneous Kelvin foundation, interacting with a moving loaded oscillator that has a nonlinear Hertzian spring. The solution is obtained using the Green’s-function method. To obtain the Green’s function of the inhomogeneous and infinite beam-foundation sub-system, the finite difference method is used for the spatial discretization and non-reflective boundary conditions are applied. Accounting for the interaction between the moving oscillator and the supporting structure generally leads to stronger wave radiation, caused by the variation of the vertical momentum of the moving mass. Results show that for relatively high velocity and small transition length, the maximum contact force as well as the energy input exhibit a significant increase compared to the moving constant load case. Furthermore, for relatively high velocities, the maximum contact force also increases significantly with increasing stiffness dissimilarity, findings which supplement the existing literature. Finally, the two degradation indicators can be used in the preliminary stages of design to assess the performance of railway track transition zones.

A numerical study on an infinite linear elastic Bernoulli-Euler beam on a viscoelastic foundation subjected to harmonic line loads

This paper presents a numerical study on the low-amplitude responses of an infinite Bernoulli-Euler beam resting on a viscoelastic foundation subjected to harmonic line loads. To simulate the linear response, a semi-analytical solution procedure that was theoretically proposed by Jang (2016) is utilized and several numerical experiments are conducted to investigate the influence of key model parameters characterizing stiffness and damping. The properties of the viscoelastic foundation are based on theoretical and empirical values for cohesionless sand type foundation. According to the numerical experiments, the obtained responses are compared with those from the closed-form solution and found to have a good agreement with them.

Theoretical method for an elastic infinite beam resting on a deformable foundation with a local subsidence

Local subsidence induced by underneath erosion or excavation is usually encountered by buried pipelines and tunnels, which in turn may cause distress to the existing pipelines and tunnels. A semi-analytical method is proposed for this kind of problem by simplifying the existing pipeline or tunnel as an elastic infinite Euler-Bernoulli beam and the ground as a deformable foundation with a local subsidence. The beam-foundation system is subject to a symmetrical distribution load. Closed-form solutions for the beam responses, including deflection and bending moment, are derived based on the elastic foundation beam theory by expanding the load expression as a Maclaurin series. The proposed method and solutions are validated by a series of 3D FEM numerical simulation and model test results. Sensitivity analyses are then carried out to investigate the influence of different parameters. The results show that the beam response is highly dependent on the length of local subsidence and the beam flexural rigidity. Moreover, comparations between the calculation results based on the Pasternak model and the Winkler model that can be degenerated from the former are also presented to illustrate the rationality of the proposed method in predicting the responses of the foundation beam in this typical problem.

Analytical solutions for steady state responses of an infinite Euler-Bernoulli beam on a nonlinear viscoelastic foundation subjected to a harmonic moving load

The paper deals with the steady state responses of an infinite Euler-Bernoulli beam resting on a nonlinear foundation under a harmonic moving load. Greatly different from previous works, the nonlinear partial differential governing equation of the beam motion is converted to two nonlinear Volterra integral equations by using the Fourier transform, residues Theorem and the convolution theorem. The Volterra integral equations have four different expressions depending on the damping coefficient of the foundation, the linear part of the foundation stiffness and the frequency of the moving load. The modified Adomian decomposition method in conjunction with a simple iterative formula derived from the integral equation theorems are applied to obtain analytical solutions for the steady state responses of the beam. The closed form solutions presented in this paper do not contain complicated infinite integrals, which are better to show the influence of the nonlinear part of foundation stiffness. The parametric study shows that the nonlinear part of foundation stiffness affects not only qualitative but also quantitative analysis results of the infinite Euler-Bernoulli beam under a harmonic moving load.

A wave number based approach for the evaluation of the Green's function of a one-dimensional railway track model

A new wave number based approach is proposed for the evaluation of the time-domain Green's function for the railway track idealized as an infinite Euler-Bernoulli beam overlying a Pasternak-type viscoelastic foundation. In the proposed approach, the rail beam response is first computed in the time wave number domain by solving a generalized single-degree-of-freedom system problem using the classical Duhamel integral technique. The Green's function is subsequently evaluated by inverting the so-obtained time wave number solution to the time-space domain. In contrast with the conventional approach, the computational time for evaluating the Green's function using the proposed approach is much smaller. Further, the proposed solution of the Green's function is shown to be convenient for directly evaluating the time-domain response of a rail beam subjected to various types of time-dependent loads. As an illustration, the Green's function is applied to obtain the time-domain response of the rail beam (overlying a viscoelastic layer) subjected to a uniformly moving oscillator by using a novel and simple two-step iterative scheme. The time-domain deflection analysis reveals that the introduction of damping into the oscillator system brings the rail beam response much closer to that associated with the constant moving load idealization. However, if the moving oscillator system is acted upon by an arbitrarily time-varying external vertical load, significantly higher values of rail beam deflections are observed.

Transition radiation in a piecewise-linear and infinite one-dimensional structure\u2013a Laplace transform method

Transition zones in railway tracks are areas with considerable variation of track properties (i.e., foundation stiffness), encountered near structures such as bridges and tunnels. Due to strong amplification of the response, transition zones require frequent maintenance. To better understand the underlying degradation mechanisms, a one-dimensional model is formulated, consisting of an infinite Euler–Bernoulli beam resting on a locally inhomogeneous and nonlinear Winkler foundation, subjected to a moving load. The nonlinearity is characterized by a piecewise-linear stiffness, and the system thus behaves piecewise linearly. Therefore, the solution can be obtained by sequentially applying the Laplace transform combined with the Finite Difference Method for the spatial discretization and derived non-reflective boundary conditions. Results show that the plastic deformation is a consequence of constructive interference of the excited waves and the response to the load’s deadweight, particularly for the soft-to-stiff transition. The plastic deformation area decreases quasi-monotonically with increasing transition length, and for super-critical velocities, small transition lengths and/or large stiffness dissimilarities, parts of the foundation experience plastic deformation even in the second loading–unloading cycle. Furthermore, the nonlinearity causes the maximum energy associated with the waves radiated forward and the maximum energy input not to occur for the smallest transition length, contrary to findings in corresponding linear systems. Moreover, the energy input drastically increases for the second passage of the moving load, making it a possible indicator of the damage in the supporting structure. The novelty of the current work lies in the computationally efficient solution method for an infinite system which locally exhibits nonlinear behaviour and in the study into the influence of the foundation’s nonlinear behaviour on the generated waves (i.e., transition radiation), and on the resulting plastic deformation. The model presented here can be used for the preliminary design of transition zones in railway tracks.

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.

New approach for the assessment of train/track/foundation dynamics using in-situ measurements of high-speed train induced vibrations

This paper presents an experimentally validated numerical model for the simulation of the complex dynamic response of the ballasted track foundation under the excitation of a moving high-speed train. It consists of an infinite Euler-Bernoulli beam that rests on continuous viscoelastic foundation. This study is carried out within the frame of the project: high-speed line “Bretagne” - “Pays de la Loire” (BPL) in which an instrumented straight track with classical ballasted formation is considered. A huge database of measurements is obtained, that's why a preliminary treatment of the field measurements (rail-foundation displacement histories) is performed in order to eliminate the unwanted high-frequency components (noise) from the raw data. Subsequently, the treated data is used to calibrate the dynamic impedances of the simplified foundation model using an iterative curve fitting procedure. In this work, an additional parameter is introduced to model the wheel load diffusion in the track structure. It is represented by the length of the loaded segment beneath the wheel in which the load distribution is inspired from the wheel-induced displacement field. Appropriate charts are given in the velocity range 162–342 km/h. The numerical experiments prove an important capacity of the model to achieve satisfactory results with significantly limited computational effort in view of the modest number of involved degrees of freedom. The use of relevant input parameters increases the efficiency in minimizing the error between measurements and predictions. The findings are very useful for researchers and practitioners working in railway track design.

Non-stationary localized oscillations of an infinite Bernoulli-Euler beam lying on the Winkler foundation with a point elastic inhomogeneity of time-varying stiffness

We consider non-stationary localized oscillations of an infinite Bernoulli-Euler beam. The beam lies on the Winkler foundation with a point inhomogeneity (a concentrated spring with negative time-varying stiffness). In such a system with constant parameters (the spring stiffness), under certain conditions a trapped mode of oscillation exists and is unique. Therefore, applying a non-stationary external excitation to this system can lead to the emergence of the beam oscillations localized near the inhomogeneity. We provide an analytical description of non-stationary localized oscillations in the system with time-varying properties using the asymptotic procedure based on successive application of two asymptotic methods, namely the method of stationary phase and the method of multiple scales. The obtained analytical results were verified by independent numerical calculations. The applicability of the analytical formulas was demonstrated for various types of an external excitation and laws governing the varying stiffness. In particular, we have shown that in the case when the trapped mode frequency approaches zero, localized low-frequency oscillations with increasing amplitude precede the localized beam buckling. The dependence of the amplitude of such oscillations on its frequency is more complicated in comparison with the case of a one degree of freedom system with time-varying stiffness.

The equivalent stiffness of a saturated poroelastic halfspace interacting with an infinite beam under a moving point load

The equivalent stiffness of a saturated poroelastic halfspace that supports an infinite Euler-Bernoulli beam under a moving point load has been investigated in this paper. The smooth contact condition is assumed for the interface of beam and halfspace, however, with an improved continuity condition, i.e. the continuities of contact normal stress and contact vertical displacements are imposed across the width of the beam. The equivalent stiffness of the halfspace has been evaluated using a contour integration approach such that contributions of the Rayleigh pole and the dispersion branches can be explicitly taken into account. Influences of the continuity condition and the permeability on the equivalent stiffness have been studied. It is found that the imaginary part of the equivalent stiffness is nonzero since the viscous coupling between two phases of the saturated halfspace has been considered. This observation fundamentally differentiates the present paper to the work by Shi and Selvadurai (2016) where an infinite permeability has been assumed for the halfspace, i.e. null viscous coupling.

Closed-form solution of beam on Pasternak foundation under inclined dynamic load

The dynamic response of an infinite Euler–Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution. The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction. Thus, it is more realistic than the Winkler model, which only represents compressive soil resistance. However, the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation; hence, the beam under inclined loads cannot be considered in the model. In this study, a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model, which tends to disregard the tangential interaction between the beam and the foundation. The horizontal spring reaction is assumed to be proportional to the relative tangential displacement. The governing equation can be deduced by theory of elasticity and Newton's laws, combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition. Double Fourier transformation is used to simplify the geometric equation into an algebraic equation, thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam. Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution. The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS. Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam. Results are realistic and can be used as reference for future engineering designs.

Dynamic response of an infinite beam on a transversely isotropic multilayered half-space due to a moving load

This paper investigates the dynamic problem for an infinite Euler–Bernoulli beam supported by a transversely isotropic multilayered half-space due to moving loads. By introducing a moving system and using double Fourier integral transform, the analytical layer-element solution for the transversely isotropic layered half-space is obtained, which can be used to derive the equivalent dynamic stiffness of the transversely isotropic layered medium. By virtue of the theory of Euler–Bernoulli beam, the equivalent dynamic stiffness and the boundary conditions, the deflection of beam in integral transform domain is derived. Solutions in physical domain are further acquired by numerical inverse integral transform. The validation examples demonstrate that the presented approach in this paper is accurate and effective. Other numerical examples are carried out to analyze the effect of flexural rigidity of beam, load speed, transverse isotropy and stratified characters of medium.

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

• Analytical solution for dynamic response of infinite beam on foundation is developed. • The elastic foundation considered the horizontal tangential effect. • Double Fourier transformation is used for deduction and the residue theorem is employed to obtain the solution. • The analytical solution for the beam under inclined dynamic load is presented. The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.

A coupled FE-BIE model for the static analysis of Timoshenko beams bonded to an orthotropic elastic half-plane

Interface displacements, surface tractions and stresses of a flexible beam bonded to an elastic orthotropic half-plane are analysed by means of a Finite Element-Boundary Integral Equation (FE-BIE) method. Numerical results are obtained by using locking-free shear deformable beams and piecewise constant interfacial reactions. Making use of the generalised Green's function for the half-plane, the mechanical behaviour of fully bonded or detached beams subjected to force, couple or thermal load is investigated. The special cases of a beam in bilateral frictionless contact with the half-plane and a beam having a vanishing bending rigidity (thin film) are considered also. In particular, the maximum bending moment of beams subjected to a vertical point force are compared with some closed-form solutions of the contact problem of a rigid indenter and with the solution of an infinite Euler–Bernoulli beam in bilateral frictionless contact with an isotropic substrate.

Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition

Abstract Complex orthogonal decomposition (COD) is applied to an experimental beam to extract the dispersive wave properties from response measurements. The beam is made of steel and is rectangular with a constant cross section. One end of the beam is free and is hung by a soft elastic cord. An impulse is applied to the free-end. The other end is buried in sand to absorb the wave as it travels from the impact site on the free-end; this effectively prevents reflections of the wave off the buried end and emulates a semi-infinite beam. The beam response is measured with an array of accelerometers, whose signals are integrated to obtain an ensemble of displacement signals. Acceleration responses are also compared in the frequency domain to predictions from the Euler–Bernoulli model. COD is applied to the displacement ensemble to obtain complex modal vectors and associated complex modal coordinates (COCs). The spatial whirl rates of nearly harmonic modal vectors are used to extract the modal wave numbers, and the temporal whirl rates of the modal coordinates are used to estimate the modal frequencies. The dispersion relationship between the frequencies and wave numbers compare favorably to those of the theoretical infinite Euler–Bernoulli beam.