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The design of high-speed simply supported bridges in critical conditions, such as road or canal crossings, is a matter of concern, particularly under two-way crossings with sequential wheel loads. Contrarily to simple classical support, elastic bearings can serve towards vibration mitigation under high speed. In this paper, a non-dimensional framework is developed to understand the impact mechanism of an elastically supported (ES) beam under the action of single as well as opposite traversing loads. A good agreement with the available literature has been confirmed for the one-way crossing using the present methodology. Moreover, a parametric study is conducted for an ES beam, in which the effects of speed parameter (η), speed parameter ratio between two trains (R), interspatial distance between loads to the length of the bridge (ϵ), damping ζ, stiffness parameter (κ), and traversing time difference between two trains (t os ) are investigated. The theoretical results suggest the effective use of the ES beam under higher values of η for one-way and R for two-way traffic. Additionally, various resonances and cancellation speed ratios under particular values of ϵ have been identified under varying values of κ. [ABSTRACT FROM AUTHOR]

In this paper, the internal and external cancellation phenomena for damped beams subjected to multi-moving loads are investigated in detail. To start, the theory for the vibration of a simply supported beam is revisited by including the effect of damping. For the first time, a simple expression is derived for the free vibration of the damped beam under multi-moving loads. Based on the concept of local minimum, two cancellation conditions are identified. One is the internal cancellation, which relates to the inherent property of the beam and is conventionally known. The other is the newly formulated external cancellation that relates to the number and spacing of moving loads. For comparison, both the resonant condition and the optimal criterion for span length of the bridge are also briefed. By comparing with the classical solution, the present simple expression for the free vibration of the beam is firstly validated. Then the factors affecting the cancellation are investigated against various load cases and damping levels. The results show that external cancellation occurs more frequently due to the increase in the number and spacing of the moving loads. The damping of the beam has a leaking effect on cancellation, in that nonzero vibration may occur, but it is also quickly damped out by damping itself. [ABSTRACT FROM AUTHOR]

The advent of railways and especially highspeed railways marks great strides in human transportation history. To guarantee exclusive right-of-way, highspeed railways are often built on equal simply supported beams resting on piers. In this paper, a historical review will be given of the resonance and cancellation phenomena observed for simply supported beams traveled by a set of moving loads, as they are typical of highspeed railways. The phenomenon of resonance was observed by early investigators including Timoshenko, Bolotin, Frýba, Matsuura, etc. However, the phenomenon of cancellation was noted lately in 1997 by Yang et al. By letting the conditions of resonance and cancellation coincident, they proposed the optimal span length for suppressing the resonance of simple beams, which is equal to 1.5 times the car length. This 1.5 times rule has been verified and adopted in the design of some highspeed railways. In this article, the theoretical solution for the problem will be revisited for unveiling the key parameters such as the resonance speed (in temporal domain) and resonance wavelength (in spatial domain). Then a rather in-depth review will be given of existing works on the resonance and cancellation of railway bridges from the waves perspective. Some new developments along these lines will also be identified. [ABSTRACT FROM AUTHOR]

This paper proposes a three-dimensional dynamic model for high-speed railway trains moving over curved bridges considering the transition curves, circular curves, and superelevation. Key features of this study are to consider the nonlinear geometrical relationships and creep relationships between the wheels and rail, for which the interactive iterative numerical algorithms are developed based on the equations of vertical displacement and rolling of wheelset, and the torsional resonance conditions of the vehicle–bridge system are verified. The results show that the torsional vibration will cause amplification on vertical dynamic response of the beam on the outside edge of the curve. The deficient/surplus superelevation plays an important role in the lateral and torsional angular displacements of the bridge, and the peak of the torsional resonance response can be reduced by adjusting the practical superelevation of the curve. The variations of wheel–load reduction rate and derailment coefficient in the curve section are positively correlated to the deficient/surplus superelevation. The curve radius is the key factor affecting the wear and fatigue of wheel–rail, and when the curve radius is greater than 7000 m, the wear and fatigue can be significantly reduced. Running at a deficient superelevation level can also reduce the wear and fatigue. [ABSTRACT FROM AUTHOR]

This paper investigates comprehensively the resonance and cancellation conditions for the free vibration of elastically-supported (ES) beams subjected to successive moving loads. Focus is placed on application of the cancellation condition to minimize bridge vibrations, considering particularly the effect of elastic supports. In terms of the modal amplitude R of free vibration of the ES beam, both resonance and cancellation conditions are identified. This paper is featured by the fact that the cancellations are classified into two types as the external (load-related) and internal (structure-related) ones. Through the (internal) cancellation function, the criterion for selecting the optimal support stiffness ratio (SSR) is derived for the first time for suppressing the resonance of short to medium-span railway bridges. It depends solely on the bridge/vehicle length ratio L / d , and can be utilized to achieve near-perfect cancellation. The theoretical findings are validated by the finite element method (FEM) for various parameters. The results reveal that for beams with lengths in the ranges of (0.5 d , d ] and (1.5 d , 2 d ], an SSR closer to the lower bound of the acceptable range should be selected to achieve the best effect. And for beams with lengths in the range of (d , 1.5 d ], the SSR should be selected as close to the optimal value as possible. Besides, it was found that damping in the beam and supports contributes to further suppression of vibration for bridges designed with optimal SSR. • Derive new resonance and cancellation conditions for elastically supported beams. • Classify cancellations as external (load-related) and internal (structure-related) ones. • Optimal support stiffness ratio (SSR) for near-perfect cancellation of resonance. • Optimal SSR relies solely on the bridge/vehicle length ratio and is easy to use. • Verify the effectiveness of SSR criterion for various factors. [ABSTRACT FROM AUTHOR]

Highlights • A new theory was given for the coupled vertical and rotational motions of the two-axle asymmetric vehicle, which was rarely studied before. • The coupled two-axle asymmetric vehicle can degenerate into the uncoupled one under some conditions, which however cannot exist in practice due to perturbation. • Resonance and cancellation conditions can be deduced from the frequency response function of the asymmetric vehicle. • The cancelation phenomenon distinguishes the two-axle asymmetric vehicle from the two-DOF symmetric vehicles and single-DOF vehicles. • Two-axle asymmetric vehicle and its contact-point response show advantages for application to scanning bridge dynamic properties. Abstract A closed-form solution is presented for the dynamic response of a two-axle asymmetric vehicle moving over bridges, along with potential applications highlighted. First, by modeling the vehicle as two concentrated loads moving over the bridge, closed-form solutions for the responses of the contact points are obtained. Second, by treating the vehicle as a two degree-of-freedom (DOF) system, closed-form solutions for the coupled vertical and rotational responses of the vehicle due to contact-point excitations are obtained. From vehicle's frequency response functions (FRFs), various phenomena are studied, including coupling, uncoupling, beat, resonance and cancellation. Resonance occurs when any excitation frequency coincides with any of the vehicle frequencies. No cancellation is observed when the two-DOF asymmetric vehicle degenerates into the uncoupled mode or a single-DOF vehicle. The results obtained are verified by an independent finite element analysis with virtually no assumptions. The advantage of using the two-axle asymmetric vehicle to scan the frequencies of the sustaining bridge is explored theoretically and numerically. [ABSTRACT FROM AUTHOR]

Railway induced vibrations on short-to-medium span simply-supported (SS) bridges is addressed in this contribution. Such structures may experience high levels of vertical acceleration at the platform, leading to adverse consequences such as a premature degradation of the ballast layer and passenger discomfort. In the present study, the evolution of the bridge dynamic response when soil-structure interaction (SSI) is taken into account is investigated. To this end a coupled three-dimensional (3D) Boundary Element-Finite Element model (BEM-FEM) formulated in the time domain is implemented to reproduce the soil and structural behaviour, respectively. First, a set of soil-bridge systems of interest is defined, covering a wide range of lengths and natural frequencies for the structures, and an interval of expectable elastic properties and damping levels for the soil. Then, different types of analyses are performed on the soil-bridge systems extracting conclusions regarding the effect of including SSI in numerical models for predicting the bridge behaviour under railway traffic. In particular natural frequencies and modal damping levels are identified, and the structure amplification after the passage of a moving load in free vibration is investigated. Conclusions regarding how resonance and cancellation conditions may be affected by soil properties are extracted. Finally, the dynamic response of a real bridge, belonging to the Spanish railway network, is evaluated under the circulation of trains that induce second and third resonances of the bridge fundamental mode. The effect of the soil flexibility, soil material damping and the bridge resonance order are evaluated. Conclusions regarding the appropriateness of the results provided by common models which do not include SSI effects are extracted. [ABSTRACT FROM AUTHOR]

This paper is concerned with the lateral and torsional coupled vibration of monosymmetric I-beams under moving loads. To this end, a train is modeled as two subsystems of eccentric wheel loads of constant intervals to account for the front and rear wheels. By assuming the lateral and torsional displacements to be restrained at the two ends of the beam, both the lateral and torsional displacements are approximated by a series of sine functions. The method of variation of constants is adopted to derive the closed-form solution. For the most severe condition when the last wheel load is acting on the beam, both the conditions of resonance and cancellation are identified. Once the condition of cancellation is enforced, the resonance response can always be suppressed, which represents the optimal design for the beam. Since the condition for suppressing the torsional resonance is exactly the same as that for the vertical resonance, this offers a great advantage in the design of monosymmetric I-beams, as no distinction needs to be made between the suppression of vertical or torsional resonance. [ABSTRACT FROM AUTHOR]

This article addresses the dynamic behaviour of double track simply supported bridges of short to medium span lengths (10 m < L < 25 m) belonging to conventional railway lines. These structures are susceptible to experience inadmissible levels of vertical vibrations when traversed by trains at high speeds, and in certain cases their dynamic performance may require to be re-evaluated in case of an increase of the traffic velocity above 200 km/h. In engineering consultancies, these structures have been traditionally analysed under the passage of trains at different speeds using planar models, neglecting the contribution of transverse vibration modes and also the flexibility of the elastomeric bearings. The study presented herein endeavours to evaluate the influence of these two aspects in the verification of the Serviceability Limit State of vertical accelerations, which is of great interest in order to guarantee a conservative prediction of the dynamic behaviour. In the present study, the dynamic response of representative slab and girder bridges has been evaluated using an orthotropic plate finite element model, leading to practical conclusions regarding the circumstances under which the above mentioned factors should be considered in order to adequately evaluate the transverse vibration levels of the deck. [ABSTRACT FROM AUTHOR]

The objective of this paper is to elucidate the general conditions of resonance and cancellation for Bernoulli-Euler beam-based bridges subject to moving train loads. Firstly, the bridge's vibration shape is approximated based on the modal superposition method and Fourier series expansion. Then, the semi-analytical solution of the bridge vibrations under moving train loads is derived. After that, the general conditions of train-induced bridge resonance and cancellation are further derived, analyzed, and classified. In particular, cancellation is divided into three classes: case 1, case 2-I, and case 2-II. It is found that the resonance condition for simply-supported bridges is applicable to arbitrary Bernoulli-Euler beam-based bridges, while the cancellation conditions are related to the cancellation classes and modal characteristics of bridges. Finally, three examples are used to validate the semi-analytical solution of bridge vibrations as well as the general conditions of resonance and cancellation, including a simply-supported bridge, a continuous-beam bridge, and a long-span extradosed cable-stayed bridge. Besides, these examples show that for simply-supported bridges, both the resonance and cancellation should be considered, while for bridges with a long-span length or low fundamental frequencies, cancellation is more significant than resonance. [ABSTRACT FROM AUTHOR]

The dynamic transverse response of beam type bridges under railway traffic is addressed in this contribution. In particular, how soil-structure interaction may affect the critical or resonant velocities and the associated vibratory amplitudes is evaluated in detail. Resonance in beams, due to the circulation of equidistant loads, is highly influenced by the free vibration response that every single load leaves after traversing the structure. On this basis a numerical investigation is carried out analysing the effects of the wave propagation problem on the free vibration response of simply-supported beams in a wide range of travelling velocities. To this end a coupled three-dimensional boundary element-finite element model formulated in the time domain is used to reproduce the soil and structural behaviour, respectively. A catalogue of bridge deck typologies is defined, covering lengths, associated linear masses and fundamental frequencies that may experience high levels of transverse accelerations under resonant conditions, for nowadays existing trains and design velocities. Lengths ranging from 12.5 to 25 m are evaluated, along with fundamental frequencies covering most common typologies. A homogeneous soil is considered with shear wave velocities in the interval 150 to 365 m/s. From the single load free vibration parametric analysis conclusions are derived regarding the conditions of maximum free vibration and cancellation of the deck response. These conclusions are used afterwards to justify how resonant amplitudes of the bridge under the circulation of railway convoys may be affected by the soil properties, leading to substantially amplified responses or to almost imperceptible ones, and a numerical example is included to show the aforementioned situations. [ABSTRACT FROM AUTHOR]

In this paper, the resonance and cancellation phenomena occurring in a periodic structure-- a pile-supported periodic viaduct subjected to multiple equidistant moving loadings (MEML) are investigated. For this purpose, the representation for the dynamic response of the periodic viaduct to a single moving loading (SML) is introduced first. In order to take account of the pile-soil-structure interaction when evaluating the response of the periodic viaduct to a SML, a wavenumber domain boundary element method (WDBEM) model for the pile-row-soil system is employed to determine the compliances of the piles. By superposing the responses of the viaduct to all SMLs constituting the MEML, the representation for the dynamic response of the periodic viaduct to the MEML is obtained. With the representation for the MEML response of the viaduct, the resonance and cancellation conditions for the viaduct are derived. For the high speed MEML, when the time interval between two neighboring moving loadings is equal to the integer or half-integer multiples of the period of one of the dominant resonance peaks of the corresponding SML response of the viaduct, resonance or cancellation occurs. The predicted resonance and cancellation phenomena are confirmed by the numerical results presented in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a method that can be used during the preliminary design stage to determine the optimal span length of a railway bridge that minimizes its dynamic responses to the passage of high-speed trains. Since the designed speed for passage over a railway bridge is generally higher than the speed at which the resonance phenomenon occurs, the peak dynamic responses of the bridge are likely to occur when a train travels across a bridge. Therefore, this study suggests inertia and elastic spectra that represent the inertial force and elastic force per unit length of the bridge at the resonance condition, respectively. These spectra are defined as functions of the span length of the bridge, for which the type of train and damping ratio are fixed, and based on the relationship between the dynamic responses of two different railway bridges that was used as theoretical basis for the design diagram proposed by the European Rail Research Institute. The optimal span length of the bridge that produces the smallest responses is determined using the suggested spectra by quantitatively comparing the responses at resonance under various train loads as a function of the span length of the bridge. The proposed method is validated using numerical examples that consider the computation of the optimal span length of a simply supported single-span bridge crossed by individual KTX, HEMU and ICE 2 trains and combinations of these trains. [ABSTRACT FROM AUTHOR]

This paper revisits the creation and cancellation of the dynamic resonance phenomenon that occurs in bridge structures on high-speed lines when crossed by wheel loads. The resonance and its cancellation are mathematically formulated for a Bernoulli-type beam with general boundary conditions and subjected to loads moving at a regular spacing. The resonance of the bridge caused by the travelling loads occurs, regardless of the mode shape, when the natural frequency of the structure coincides with the loading frequency produced by the loads moving at a constant speed. In this study, the dependency of the cancellation phenomenon on the mode shape is determined based on the boundary conditions of the structure. In addition, the optimal span length that suppresses the response at resonance is proposed using the cancellation phenomenon for a simple beam with pinned-pinned, clamped-clamped and clamped-pinned boundary conditions; and a simply supported continuous beam. [ABSTRACT FROM AUTHOR]

Dynamic behaviour of simply supported uniform beams subjected to a single moving point load is analysed in this paper. A simple closed-form expression for free vibration response, valid for both lightly and heavily damped beams, is formulated. A detailed investigation of the cancellations of free responses is carried out. New interpretations for the cancellation mechanism, from the perspectives of free vibration amplitude and phase angle, are presented. Expressions for cancellation speed ratios are formulated based on the initial velocity and displacement conditions for free vibration. The effect of damping on the cancellations of free responses is also studied. [ABSTRACT FROM AUTHOR]

Based on the theoretical solution for vibrations of simply supported bridges under moving train loads, resonance and cancellation conditions are derived. The results reveal two types of vibration cancellations for simply supported bridges under moving train loads: the first is related to a single moving load, which is independent of the number and the composition of the train, while the second is related to the interval between a pair of moving loads. When cancellation occurs, the free vibrations induced by the moving loads cancel to null, and the bridge response is determined by the loads still on it. A resonance disappearance effect occurs when train speed meets both the resonance and cancellation conditions, while the cancellation plays a predominant role. Bridge damping has an influence on the cancellation effect: the higher the damping and the longer the interval between the loads, the lower is the cancellation efficiency. In a case study, dynamic responses of a bridge under high-speed train loads are analyzed and the mechanisms of resonance and cancellation are explained to support the theoretical solution. [ABSTRACT FROM AUTHOR]

• Resonance and cancellation phenomena in two-span continuous bridges. • Non?dimensional analytical formulation. • Maximum acceleration assessment in railway bridges. • Most aggressive train according to geometrical considerations. • Application to real bridges from the Swedish railway network. The objective of this study is to evaluate the vibratory response of two-span continuous beams subjected to moving loads and, in particular, to investigate the maximum resonance and cancellation of resonance phenomena. The main practical interest is the evaluation of the maximum acceleration response in railway bridges, which is one of the most demanding Serviceability Limit States for traffic safety according to current regulations. Two-span continuous bridges, in their simplest version (i.e. uniform identical spans), present antisymmetric and symmetric modes with closely spaced natural frequencies, leading to a more involved dynamic behaviour than that of simply-supported bridges. First, the free vibration response of a Bernoulli-Euler two-span beam after the passage of a single load at constant speed is formulated analytically, and non-dimensional speeds leading to cancellation or maximum response in free vibration are obtained for each mode. Then, these conditions are equated to resonant speeds induced by equidistant load series, and span length-to-characteristic distance ratios causing cancelled out resonances, or remarkably prominent ones, are obtained. Based on the previous derivations, a methodology for detecting which could be the most aggressive trains for a particular structure based on pure geometrical considerations is discussed. Finally, the applicability of the theoretical derivations is shown through the numerical analysis of two real bridges belonging to the Swedish railway network. [ABSTRACT FROM AUTHOR]

• Maximum free vibration and cancellation conditions formulated in rectangular plates. • L/d ratios for maximum resonance and cancellation of it proposed. • Simply and elastically-supported boundary conditions envisaged. • Adequacy on 112 skewed bridges investigated covering realistic parameters. • Exemplification through experimental measurements at resonance. The vibrational response of railway bridges is an issue of main concern, especially since the advent of High-Speed traffic. In the case of short-to-medium lengths and simply-supported spans excessive transverse acceleration levels may be induced at the platform, with detrimental consequences for passengers and infrastructures. The orthotropic plate has proven to be an appropriate model for the prediction of the response of certain typologies in the aforementioned cases such as multiple girder decks, solid or voided slabs or filler-beam multiple-track decks. In this contribution, the vibrational response of orthotropic plates, simply and elastically supported, circulated by vertical moving loads is investigated. First, maximum free vibration and cancellation conditions are derived analytically. From these, bridge span length-characteristic distance ratios leading to maximum and minimum resonances under series of equidistant loads are depicted. Second, the applicability of these ratios in oblique decks is analysed for the most common first three mode shapes: first longitudinal bending, first torsion and first transverse bending modes, and the errors in relation to the straight reference case are bounded. To this end, an extensive bridge catalogue of girder bridges is designed in the range of lengths of interest, covering flexural stiffnesses typical from both conventional and High-Speed railway lines. Finally, the applicability of the previous theoretical results is exemplified with experimental measurements performed on a bridge from the Spanish railway network. [ABSTRACT FROM AUTHOR]