Dynamic Response Of Railway Bridges Research Articles

Dynamic Amplification of Railway Bridges under Varying Wagon Pass Frequencies

Train configurations give rise to a primary wagon pass forcing frequency and their multiples. When any one of these frequencies coincides with the natural frequency of vibration of the bridge, a resonant response can occur. This condition can amplify the dynamic response of the bridge, leading to increased levels of displacement, stresses and acceleration. Increased stress levels on critical bridge structural elements increases the rate at which fatigue damage accumulates. Increased bridge acceleration levels can affect passenger comfort, noise levels, and can also compromise train safety. For older bridges the effects of fatigue, and being able to predict the remaining life, has become a primary concern for bridge engineers. Better understanding of the sensitivity of fatigue damage to the characteristics of the passing train will lead to more accurate remaining life predictions and can also help to identify optimal train speeds for a given train–bridge configuration. In this paper, a mathematical model which enables the dynamic response of railway bridges to be assessed for different train configurations is presented. The model is based on the well established closed from solution of the Euler–Bernoulli Beam (EBB) model, for a series of moving loads, using the inverse Laplace–Carson transform. In this work the methodology is adapted to allow different train configurations to be easily implemented into the formulation in a generalised form. A generalised equation, which captures the primary wagon pass frequency for any train configuration, is developed and verified by presenting the results of the bridge response in the frequency domain. The model, and the accuracy of the equation for predicting the primary wagon pass frequency, is verified using independently obtained measured field train–bridge response data. The main emphasis of this work is to enable the practicing engineer, railway operators and bridge asset owners, to easily and efficiently make an initial assessment of dynamic amplification, and the optimal train speeds, for a given bridge and train configuration. This is visually presented in this work using a Campbell diagram, which shows dynamic amplification and compares this with those calculated based on the design code, across a range of train speeds. The diagram is able to identify train speeds at which a resonance response can occur, and the wagon pass frequency, or its multiples, which are causing the increased dynamic amplification. The model is implemented in Matlab and demonstrated by analysing a range of short- to medium-single span simply supported plate girder railway bridges, typically found on the UK railway network, using the standard BS-5400 train configurations. The model does not consider the effects of the train mass and suspension system as this would require a non-closed form numerical solution of the problem which is not practical for the purposes of an initial assessment of the train–bridge interaction problem.

Effect of linear and nonlinear stiffness support on the dynamic response of high-speed railway bridges

In this paper, the effect of linear and nonlinear support vertical stiffness of elastic support on the dynamic response of the railway bridge is investigated. The bridge is modeled by a Euler-Bernoulli beam supported by linear and nonlinear supports stiffness. A constant moving load model HSLM-A is utilized to simulate the moving real train, and based on Hamilton’s principle, the governing partial differential equation of motion of the studied system is derived. A method to deal with the nonlinear boundary conditions is proposed, the fourth-order Runge-Kutta method in combination with the Galerkin method is used to obtain the dynamic response of the bridge. The finite difference method (FDM) is used to validate the proposed solution. The obtained results have shown that the dynamics of the system are governed essentially by a Duffing-like oscillator and proved that the elastic bearings and the cubic nonlinearities of supports have a significant effect on the global behavior of the railway bridge particularly at resonance.

Effect of Elastomeric Bearing Stiffness on the Dynamic Response of Railway Bridges Considering Vehicle–Bridge Interaction

This article presents a numerical study that aims to explore the impacts of the stiffness of elastomeric bearings on the dynamic behavior of railway bridges under train-induced vibrations. For this purpose, a finite element code that considers vehicle–bridge interaction using a coupled approach was developed. The software was validated by comparing the numerical response to the analytical solution. The numerical analysis of single- and multi-span bridges with varying bearing stiffness values under passenger trains showed the interplay between bearing stiffness, its impact on the natural frequency of the bridge and the loading frequency. It is demonstrated that the amplitude of the maximum acceleration on the bridge depends heavily on the stiffness of the bearings. Furthermore, the bearing stiffness significantly impacts the location of the maximum acceleration on the bridge. The results of the extensive numerical analyses improve the understanding of the impact of the bearing stiffness on the dynamic behavior of bridges and highlight the importance of quantifying the boundary conditions correctly for reliable estimation of dynamic response of railway bridges under train-induced vibrations.

Ballast shear effects on the dynamic response of railway bridges

Single-track railway bridges are susceptible of experiencing high levels of vertical acceleration on the deck that may be dangerously accentuated at resonance. This is especially critical for short-to-medium span simply supported bridges. This problem can compromise the safety of the trains and increase the maintenance costs of the track. The main objective of this work is to investigate the influence of the ballasted track on the dynamic behaviour of these structures. The present contribution provides a detailed sensitivity analysis over a wide single-track bridge catalogue covering span lengths from 10 to 25 m and considering two common deck structural typologies: girder-deck and slab-deck bridges. The effect of the vertical stiffness of the neoprene bearings is also evaluated. A 2D Finite-Element track–bridge interaction model is implemented and used to analyse the effect of the track on the modal parameters, harmonic response and vertical acceleration of the bridges under train passages. Additionally, the weak coupling exerted by the track is studied for structures with an increasing number of consecutive spans. The results obtained reveal a notable influence of the mobilised ballast shear transfer mechanism on the dynamic response of the structures, especially for the shortest girder bridges. Finally, a track–bridge interaction model of an existing short girder bridge from a conventional railway line is updated and used to predict the experimental response measured under operating conditions. The adequacy of the numerical tool and influence of the ballast shear parameters on the dynamic response are shown.

Experimental and Analytical Investigation into the Effect of Ballasted Track on the Dynamic Response of Railway Bridges under Moving Loads

Ballasted tracks are among the most widespread railway track typologies. The ballast possesses multiple functions. Among them, it significantly affects the dynamic interaction between a rail bridge and a moving load in terms of damping and load distribution. These effects entail accurate modeling of the track–ballast–bridge interaction. The paper presents a finite-difference formulation of the governing equations of the track and the bridge, modeled as Euler–Bernoulli (EB) beams, and coupled by a distributed layer of springs representing the ballast. The two equations are solved under a moving load excitation using a Runge–Kutta family algorithm and the finite-difference method for the temporal and spatial discretization, respectively. The authors validated the mathematical model against the displacement response of a rail bridge with a ballasted substructure. In a first step, the modal parameters of the bridge, obtained from ambient vibration measurements, are used to estimate the bending stiffness of an equivalent EB beam representative of the tested bridge. In a second step, the authors estimated the coupling effect of the ballast by assessing the model sensitivity to the modeling parameters and optimizing the agreement with the experimental data. Comparing the bridge’s experimental displacement responses highlights the ballast’s significant effect on the load distribution and damping. The considerable difference between the damping estimated from output-only identification and that determined from the displacement response under moving load proves the dominant role of the ballast in adsorbing the vibrations transmitted to the bridge under the train passage and the different damping sources under high-amplitude excitation. The authors discuss the tradeoff between model accuracy and computational effort for a reliable estimation of ballasted tracks response under moving loads.

Parametric analysis of the dynamic response of railway bridges due to vibrations induced by heavy-haul trains

This article presents a numerical study that aims to explore the dynamic behavior of railway bridges under vibrations induced by heavy-haul traffic. For this purpose, a finite element code that can conduct moving load and moving mass analysis of single span bridges was developed. The software was validated by comparing the numerical response to the analytical solution for various speeds. The numerical analysis of the benchmark bridge under the benchmark train showed the interplay between the natural frequency of the bridge, the mass of the train and the loading frequency. A comprehensive parametric study to investigate the impact of different parameters on the dynamic behavior of railway bridges is also provided. The bridge span length, normalized train length, normalized mass of the train, bridge deck stiffness, and train speed are the variables considered in the parametric study. The results of the extensive numerical analyses improve the understanding of railway bridge behavior under heavy-haul trains, and highlight the impact of the inertial effect of the trains on bridges, especially for varying span length and deck stiffness. It is also demonstrated that, when the train-to-bridge mass ratio exceeds 40%, the inertial effects of the train mass needs to be included in the analysis in order to obtain a reliable estimate of the bridge behavior under different train speeds.

Resonance and cancellation phenomena in partially clamped simply supported beam bridges under moving trains

Since the development of new railway lines of high speed, the dynamic response of bridges belonging to them constitutes a subject interest for researchers and engineers, in particular at resonance. In the present work, an analytical approach is developed to analyze the dynamic response of a single ballasted track railway bridges supported by identical rotational springs and subjected to the circulation of moving loads at constant speeds. Although the introduction of rotary springs at the ends of the deck enhances the choice of the appropriate analysis model, where most of the contributions reported in this field do not consider this effect. The central idea of the proposed model is based on analyzing the continuity effect of the ballasted track (rails and ballast) on the dynamic response of railway bridges, with taking into account an axial force that models the effect of prestressing, longitudinal ballast-bridge interface, axial displacement, force braking. The equation of motion of the system is derived by using the Hamilton’s principle, two dynamic case studies of a simply supported and simply supported partially clamped Euler-Bernoulli beam are presented. The results revealed that the compression force presents an additional stiffness which affects the critical velocity and that the continuity of the track modeled by rotational springs at the beam’s end, as it increases the dynamic response decreases. Also, a dynamic analysis for the conditions for maximum response and cancellation in free vibration are derived and interpreted particularly the effect of the geometric scheme of train axles. Equating the condition for resonance of maximum free response and cancellation, cancelled resonance are obtained.

Dynamic response of railway bridges during train passage considering track irregularity uncertainty

Dynamic response of railway bridges during train passage considering track irregularity uncertainty

Dynamic analysis of railway bridges exposed to high-speed trains considering the vehicle\u2013track\u2013bridge\u2013soil interaction

A new semi-analytical approach to analyze the dynamic response of railway bridges subjected to high-speed trains is presented. The bridge is modeled as an Euler–Bernoulli beam on viscoelastic supports that account for the flexibility and damping of the underlying soil. The track is represented by an Euler–Bernoulli beam on viscoelastic bedding. Complex modal expansion of the bridge and track models is performed considering non-classical damping, and coupling of the two subsystems is achieved by component mode synthesis (CMS). The resulting system of equations is coupled with a moving mass–spring–damper (MSD) system of the passing train using a discrete substructuring technique (DST). To validate the presented modeling approach, its results are compared with those of a finite element model. In an application, the influence of the soil–structure interaction, the track subsystem, and geometric imperfections due to track irregularities on the dynamic response of an example bridge is demonstrated.

Fast simulation of railway bridge dynamics accounting for soil–structure interaction

A novel numerical methodology is presented to solve the dynamic response of railway bridges under the passage of running trains, considering soil–structure interaction. It is advantageous compared to alternative approaches because it permits, (i) consideration of complex geometries for the bridge and foundations, (ii) simulation of stratified soils, and, (iii) solving the train-bridge dynamic problem at minimal computational cost. The approach uses sub-structuring to split the problem into two coupled interaction problems: the soil–foundation, and the soil–foundation–bridge systems. In the former, the foundation and surrounding soil are discretized with Finite Elements (FE), and padded with Perfectly Match Layers to avoid boundary reflections. Considering this domain, the equivalent frequency dependent dynamic stiffness and damping characteristics of the soil–foundation system are computed. For the second sub-system, the dynamic response of the structure under railway traffic is computed using a FE model with spring and dashpot elements at the support locations, which have the equivalent properties determined using the first sub-system. This soil–foundation–bridge model is solved using complex modal superposition, considering the equivalent dynamic stiffness and damping of the soil–foundation corresponding to each natural frequency. The proposed approach is then validated using both experimental measurements and an alternative Finite Element–Boundary Element (FE–BE) methodology. A strong match is found and the results discussed.

Real-time hybrid testing for efficiency assessment of magnetorheological dampers to mitigate train-induced vibrations in bridges

ABSTRACT The dynamic response of railway bridges is important to consider, as the high velocity loading from the trains increases the risk of resonance. Increasing train speeds and loads result in a need for adaptable properties for railway bridges. One solution is to install magnetorheological dampers, as the response of the structure originating from moving loads could be controlled by alternating the input current to the damper. However, the complexity of such devices results in difficulties to accurately model their behaviour. From this context, real-time hybrid simulations, consisting of a numerical bridge model and a full-scale physical magnetorheological damper in a test frame, are presented in this paper. The novelty of the present paper is the application of RTHS in railway bridge engineering and the substructure combination with magnetorheological dampers. The results from the real-time hybrid simulations show that the bridge deck responses can be reduced to permissible levels.

On the dynamic characterisation of railway bridges through experimental testing

This work is devoted to the experimental evaluation of the dynamic response of railway bridges with common typologies in short-to-medium span lengths. An extensive experimental campaign is performed in several railway bridges belonging to the High-Speed (HS) and conventional Spanish railway network. The main objectives are to (i) characterise the structures main features affecting their dynamic behaviour, (ii) determine the soil properties at the structures respective locations; and (iii) analyse the bridge dynamic responses under different operating conditions. All the bridges are composed by simply-supported (S-S) spans except the last one which is a continuous beam bridge. The structures present different steel, concrete and steel–concrete composite typologies. Five bridges have been experimentally investigated. The five bridges are nowadays at full operation. The study includes the identification of the bridges modal parameters and dynamic soil properties at the sites, a discussion on the identified structural damping in all the bridges, the measurement of the vibration levels induced by railway traffic under operational conditions in the first four bridges, and under construction operation machinery in the fifth case, and a final analysis on the differences found in the structural behaviour of the different typologies for similar traffic. This paper summarizes a vast experimental campaign. The data is available and may be useful to other researchers in order to estimate the response in similar cases and to validate numerical models.

Dynamic Response of Railway Bridges under Heavy-Haul Freight Trains

In the freight railway bridge, the increase of the train running speed and train axle loads can enlarge dynamic response (DR) of the railway bridges, which leads to excessive vibration of bridges and endangers the structural safety. In this paper, a three-dimensional coupled finite element (FE) model of a heavy-haul freight train-track-bridge (HHFTTB) is established using multibody dynamics theory and FE method, and the DR for the coupled system of HHFTTB are solved by ABAQUS/Explicit dynamic analysis method. The field-measured data for a 32 m simply supported prestressed concrete beam of a heavy-haul railway in China are analyzed, and the validity of the FE model is verified. Finally, the effects of train formation number, train running speed, and train axle loads on DR of the heavy-haul railway bridge structures are studied. The results show that increasing the train formation number only has an influence on DR duration of the bridge structure, rather than the peak value of DR, when the train formation number exceeds a certain number; besides, the train axle loads and train running speed have significant influence on DR of the bridge structure. The results of this study can be used as reference for the design of heavy-haul railway bridges and the reinforcement transformation of existing railway bridges.

The Influence of Deck Flexibility on the Dynamic Response of Railway Bridges

A simplified methodology is proposed for railway bridge vibration analysis under train loading. Dynamic models of both train and bridge were assumed to be initially uncoupled, yet bound by contact forces. These forces were evaluated from the dynamic response of the vehicle subjected to support excitation caused by rail and wheel geometric irregularities. Such forces were statically condensed in the vehicle centre of gravity and applied to a 3D-beam finite-element model of the bridge. The influence of the rigid-deck hypothesis on the dynamic response is assessed via an iterative procedure to redefine updated an equivalent rail irregularities� as the sum of he deck displacements and the actual� rail irregularities. The new interaction forces are re-applied to the bridge model to determine new displacements, repeating the procedure until the results converge.

Dynamic response of railway bridges traversed simultaneously by opposing moving trains

Bridges are vital components of the railroads. High speed of travel, the periodic and oscillatory nature of the loads and the comparable vehicle bridge weight ratio distinguish the railway bridges from the road bridges. The close proximity between estimations by some numerical methods and the measured data for the bridge-vehicle dynamic response under the moving load conditions has boosted the confidence in the numerical analyses. However, there is hardly any report regarding the responses of the railway bridges under the effect of the trains entering from the opposite directions while running at unequal speed and having dissimilar geometries. It is the purpose of this article to present an analytical method for the dynamic analysis of the railway bridges under the influence of two opposing series of moving loads. The bridge structural damping and many modes of vibrations are included. The concept of modal superposition is used to solve for the system motion equations. The method of solution is indeed a computer assisted analytical solution. It solves for the system motion equations and gives output in terms of the bridge deflection. Some case studies are also considered for the validation of the proposed method. Furthermore, the effects of varying some parameters such as the distance between the bogies, and the bogie wheelset distance are studied. Also, the conditions of resonance and cancellation in the dynamic response for a variety of vehicle-bridge specifications are investigated.

Frequency characteristics of railway bridge response to moving trains with consideration of train mass

The dynamic response of railway bridges is known to be influenced by a combination of factors including the bridge natural frequency, train speed, and bridge and carriage lengths. However, the intrinsic relationships among these parameters have seldom been elaborated in common dynamics terms so as to enable more effective implementation in practice. This paper attempts to approach this classic problem from a frequency perspective, by investigating into the frequency characteristics in the bridge response as well as in the moving trainloads. In particular, the significance of the so-called “driving” and “dominant” frequencies arising from the moving load is examined. Based on numerical results and a securitization using a generalised trainload pattern, it is demonstrated that the primary frequency contents in the trainload, and consequently in the dynamic response of the bridge, is largely governed by the bridge-to-carriage length ratio. Namely, for short bridges (with a length ratio below the order of 1.5), well-distributed frequency peaks occur at a number of dominant frequencies, whereas for longer bridges the main frequency peak tends to concentrate towards the lowest dominant frequency. Such a characteristic affects directly the resonance condition and resonance speeds for bridges of different length categories, and this observation echoes well the predictions of the resonance severity using a so-called Z-factor. For the special case of bridge response under a single carriage/vehicle, the influence of the carriage mass is examined in association with the concept of critical speed, and the abnormal acceleration spikes that could occur when the vehicle moves at the critical speed is highlighted.

Critical Speed and Resonance Criteria of Railway Bridge Response to Moving Trains

The dynamic response of railway bridges to moving trains is complicated because of the involvement of the moving loads and the moving masses. Among various response characteristics, bridge resonance is of particular interest in terms of the structural effect and safety of the bridge. As far as the global bridge response is concerned, it is generally understood that when one of the apparent trainload excitation frequencies coincides with the fundamental natural frequency of the bridge, resonance could occur. However, such a general criterion is of little practical use because a typical trainload would involve numerous apparent frequencies (at equal intervals); consequently, for a given bridge (natural frequency), there could be many train speeds that satisfy the preceding resonance condition. Therefore, it is necessary to establish the relative severity of the resonance associated with each resonance scenario. This paper presents the development of a new resonance severity indicator, called Z factor, for the assessment of the resonance effect. It is found that the resonance severity is essentially governed by the ratio between the bridge and carriage lengths. When the carriage mass is significant, the same Z factor will apply; however, the underlying resonance speeds will change because of the altered natural frequency of the bridge-train system. Numerical results demonstrate that the proposed methods are effective for the determination of the resonance effects associated with the potential resonance speeds.

Adoption of the response spectra method for calculating the dynamic response of railway bridges

AbstractThis paper introduces a simplified method for calculating the dynamic response of railway bridges under train service loads. Response spectrum analysis, which has proved itself in structural dynamics, has been adopted for the given problem. If essential parameters such as the natural frequency, mass, span and damping of a bridge are known, the maximum response quantities can be looked up in a diagram. Preparation of the response spectra diagrams is computationally intensive but they do provide an easy and quick way of obtaining accurate results. This method is exemplified for single‐span bridges and the load models defined in the Eurocode. However, it is possible to expand the theory to more complex bridge systems and other load models without difficulty. With the aid of this method, the maximum dynamic response of a bridge under moving loads can be calculated in an efficient and convenient way with the accuracy required by the current design codes.

Effects of speed, load and damping on the dynamic response of railway bridges and vehicles

In this paper, the effects of various parameters influencing the dynamic response of railway bridges are investigated. A versatile and computationally efficient numerical model was developed for this purpose. The model incorporates three-dimensional multi-body train and finite element bridge subsystems. The corresponding equations of motion are integrated numerically by applying the Newmark’s method combined with a modified Newton–Raphson iterative procedure. The model was verified by comparisons with analytical and numerical solutions available in the literature and good agreement was found. A parametric analysis was carried out to establish the key variables influencing the dynamic response of railway bridges. The speed of the train, train-to-bridge frequency, mass and span ratios, as well as bridge damping were identified as significant variables. Vehicle damping was found to have negligible influence on bridge response. Particularly strong dynamic amplification was found for train with shortly and regularly spaced axles travelling at the critical speeds. Other general conclusions were drawn on the basis of the parametric study.

Impact Factors of Concrete Girders Coping with Train Speed-up

The dynamic response of railway bridges is a problem of the resonant response of girders that occurs because bridges are excited at a constant period by running trains. In general, this is referred to as "the speed effect of multiple-axle moving loads." In this study, vehicle running tests were performed to examine 56 bridges of various structural types and span lengths. The impact factor was measured to be about 1.0 on four girders, and the damping ratio as low as 0.02. Based on the results of these running tests and numerical analysis, a revision of the railway design standard was proposed, to cope with the speed-up of trains.