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.     

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.     

Numerical analysis on seismic response of Shinkansen bridge-train interaction system under moderate earthquakes

This study is intended to evaluate the influence of dynamic bridge-train interaction (BTI) on the seismic response of the Shinkansen system in Japan under moderate earthquakes. An analytical approach to simulate the seismic response of the BTI system is developed. In this approach, the behavior of the bridge structure is assumed to be within the elastic range under moderate ground motions. A bullet train car model idealized as a sprung-mass system is established. The viaduct is modeled with 3 D finite elements. The BTI analysis algorithm is verified by comparing the analytical and experimental results.The seismic analysis is validated through comparison with a general program. Then, the seismic responses of the BTI system are simulated and evaluated. Some useful conclusions are drawn, indicating the importance of a proper consideration of the dynamic BTI in seismic design.     

Time Domain Flutter and Buffeting Response Analysis of Bridges

A time domain approach for predicting the coupled flutter and buffeting response of long span bridges is presented. The frequency dependent unsteady aerodynamic forces are represented by the convolution integrals involving the aerodynamic impulse function and structural motions or wind fluctuations. The aerodynamic impulse functions are derived from experimentally measured flutter derivatives, aerodynamic admittance functions, and spanwise coherence of aerodynamic forces using rational function approximations. A significant feature of the approach presented here is that the frequency dependent characteristics of unsteady aerodynamic forces and the nonlinearities of both aerodynamic and structural origins can be modeled in the response analysis. The flutter and buffeting response of a long span suspension bridge is analyzed using the proposed time domain approach. The results show good agreement with those from the frequency domain analysis. The example used to demonstrate the proposed scheme focuses on the treatment of frequency dependent self-excited and buffeting force effects. Application to nonlinear effects will be addressed in a future publication.     

Dynamic Stress Analysis of Long Suspension Bridges under Wind, Railway, and Highway Loadings

Computation of the dynamic stress of long suspension bridges under multiloadings is essential for either the strength or fatigue assessment of the bridge. This paper presents a framework for dynamic stress analysis of long suspension bridges under wind, railway, and highway loadings. The bridge, trains, and road vehicles are respectively modeled using the finite-element method (FEM). The connections between the bridge and trains and between the bridge and road vehicles are respectively considered in terms of wheel-rail and tire-road surface contact conditions. The spatial distributions of both buffeting forces and self-excited forces over the bridge deck surface are considered. The Tsing Ma suspension bridge and the field measurement data recorded by a wind and structural health monitoring system (WASHMS) installed in the bridge are utilized as a case study to examine the proposed framework. The information on the concerned loadings measured by the WASHMS is taken as inputs for the computation simulation, and the computed stress responses are compared with the measured ones. The results show that running trains play a predominant role in bridge stress responses compared with running road vehicles and fluctuating wind loading.     

Study of Passive Deck-Flaps Flutter Control System on Full Bridge Model. I: Theory

A passive aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Control flap rotations are governed by prestressed springs and additional cables spanned between the control flaps and an auxiliary transverse beam supported by the main cables of the bridge. The rotational movement of the flaps is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time-domain formulation of self-excited forces for the whole three-dimensional suspension bridge model is obtained through a rational function approximation of the generalized Theodorsen function and implemented in the FEM formulation. This paper lays the theoretical groundwork for the one that follows.     

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.     

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.     

Suppression of Bridge Flutter by Active Deck-Flaps Control System

A theoretical study on an aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Their rotational movement, commanded via feedback control law, is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time domain formulation of self-excited and buffeting forces is obtained through the rational function approximation of the generalized Theodorsen function. The optimal configuration of the deck-flaps system is found with respect to the performance index based on stability robustness of the system. A control system with the rotational center of the flaps that is located on the edges of the deck was found to be the most effective. It is also shown that this control system can provide sufficient aerodynamic damping and satisfactory stability robustness of the system with a relatively small flap size for the considered range of wind speed.     

Nonlinear Rail-Structure Interaction Analysis of an Elevated Skewed Steel Guideway

The use of continuous welded rail (CWR) with direct fixation of track on concrete deck is typical of most modern light-rail aerial structures. The interaction between the CWR and the elevated structure takes place through direct-fixation rail fasteners, which have a nonlinear force-displacement relationship. Factors that have significant influence on this interaction include the following: the bearing arrangement at the substructure units, trackwork terminating on the aerial structure, type of deck construction, and type of rail fasteners. To better understand the interaction mechanism, a nonlinear three-dimensional (3D) finite-element analysis of a straight, skewed, elevated steel guideway was carried out using the commercially available software GT STRUDL. The load cases considered in this study are temperature change, temperature change with rail breaking, and train braking. Results are presented in the form of rail axial stresses along the length of the bridge and normal bearing forces at both abutments and at all pier locations. The study shows that nonlinear 3D modeling can give a comprehensive insight into the rail-structure interaction (RSI) forces.     

Safety Analysis of Moving Road Vehicles on a Long Bridge under Crosswind

This paper presents a safety analysis of high-sided road vehicles running on a long span cable-stayed bridge when the road vehicle enters a sharp-edged crosswind gust while the bridge is oscillating under fluctuating winds. Road vehicle accidents, including overturning, excessive sideslip, and exaggerated rotation, are defined first. The mathematical model and the equation of motion of coupled road vehicle–bridge systems under crosswind are then established, which include road surface roughness, vehicle suspension, and the sideslip of the vehicle tire relative to the bridge deck in the lateral direction. A case study using a real long cable-stayed bridge and a high-sided road vehicle is finally conducted, and an extensive computational work is performed to obtain a series of accident vehicle speed against mean crosswind speed, by which the decision on the threshold of mean wind speed above which the bridge should be closed to the road vehicle can be made. The obtained accident vehicle speeds are also compared with those for the same vehicles running on the ground. It is shown that the oscillation of the cable-stayed bridge will lower the accident vehicle speed when wind speed reaches a certain level.     

Vibration Studies of Tsing Ma Suspension Bridge

A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and towers. Significant coupling between torsional and lateral modes is also observed. The numerical results are in excellent agreement with the measured first 18 natural frequencies and mode shapes. The established dynamic model and computed dynamic characteristics can serve further studies on a long-term monitoring system and aerodynamic analysis of the bridge.     

Noise Evaluation for Pavement Maintenance in Metropolitan Highway Bridges

Dynamic response of a bridge under traffic load induces acoustic energy at the bridge surface. The acoustic energy change generates an additional coupled noise component caused by vibration of a bridge deck associated with the pavement conditions and moving velocity of the vehicle. This paper presents a three-dimensional finite-element method developed for the dynamic response and noise propagation model, and analyzes the coupled effect induced by traffic loading based on different pavement conditions. Even though vibration-induced noise at the bridge is below the audible frequency range of 20–20,000?Hz, it amplifies the traffic noise source to the highly annoyed level of noise in the metropolitan area. Among several factors that contribute to the traffic noise, interaction between pavement and vehicle is considered according to the different surface roughness and vehicle velocity. The result shows that poor pavement condition contributes to the increase of traffic noise at a high traveling speed of the vehicle. In the pavement maintenance stage, the coupled effect as an additional noise source should be considered to mitigate the traffic noise for its added value in conjunction with regulation of engine emission noise and construction of a noise barrier.     

Suspension Bridge Flutter for Girders with Separate Control Flaps

Active vibration control of long span suspension bridge flutter using separate control flaps (SFSC) has shown to increase effectively the critical wind speed of the bridges. In this paper, an SFSC calculation based on modal equations of the vertical and torsional motions of the bridge girder including the flaps is presented. The length of the flaps attached to the girder, the flap configuration, and the flap rotational angles are parameters used to increase the critical wind speed of the bridge. To illustrate the theory a numerical example is shown for a suspension bridge of 1,000 + 2,500 + 1,000 m span based on the Great Belt Bridge streamlined girder.     

Aeroelastic Analysis of Bridges under Multicorrelated Winds: Integrated State-Space Approach

In this paper, an integrated state-space model of a system with a vector-valued white noise input is presented to describe the dynamic response of bridges under the action of multicorrelated winds. Such a unified model has not been developed before due to a number of innate modeling difficulties. The integrated state-space model is realized based on the state-space models of multicorrelated wind fluctuations, unsteady buffeting and self-excited aerodynamic forces, and the bridge dynamics. Both the equations of motion at the full order in the physical coordinates and at the reduced-order in the generalized modal coordinates are presented. This state-space model allows direct evaluation of the covariance matrix of the response using the Lyapunov equation, which presents higher computational efficiency than the conventional spectral analysis approach. This state-space model also adds time domain simulation of multicorrelated wind fluctuations, the associated unsteady frequency dependent aerodynamic forces, and the attendant motions of the structure. The structural and aerodynamic coupling effects among structural modes can be easily included in the analysis. The model also facilitates consideration of various nonlinearities of both structural and aerodynamic origins in the response analysis. An application of this approach to a long-span cable-stayed bridge illustrates the effectiveness of this scheme for a linear problem. An extension of the proposed analysis framework to include structural and aerodynamic nonlinearities is immediate once the nonlinear structural and aerodynamic characteristics of the bridge are established.     

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

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

Hydrodynamic Investigation of Coastal Bridge Collapse during Hurricane Katrina

A number of U.S. coastal bridges have been destroyed by hurricanes, including three highway bridges in Mississippi and Louisiana during Hurricane Katrina (2005). This paper addresses three fundamental questions on the coastal bridge failures: (1) what were the hydrodynamic conditions near the failed bridge during the hurricane; (2) what was the cause of the bridge collapse; and (3) what was the magnitude of the hydrodynamic loading on the bridge under the extreme hurricane conditions. Guided by field observations of winds, waves, and water levels, two numerical models for storm surges and water waves are coupled to hindcast the hydrodynamic conditions. Fairly good agreement between the modeled and measured high watermarks and offshore wave heights is found, allowing an estimate of the surge and wave conditions near the bridges in nested domains with higher resolutions. The output of the coupled wave-surge models is utilized to determine the static buoyant force and wave forces on the bridge superstructure based on empirical equations derived from small-scale hydraulic tests for elevated decks used in the coastal and offshore industry. It is inferred that the bridge failure was caused by the wind waves accompanied by the storm surge, which raised the water level to an elevation where surface waves generated by strong winds over a relatively short fetch were able to strike the bridge superstructure. The storm waves produced both an uplift force and a horizontal force on the bridge decks. The magnitude of wave uplift force from individual waves exceeded the weight of the simple span bridge decks and the horizontal force overcame the resistance provided by the connections of the bridge decks to the pilings. The methodology for determining the hydrodynamic forcing on bridge decks can be used to produce a preliminary assessment of the vulnerability of existing coastal bridges in hurricane-prone areas.     

Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects

The eigenvalues of bridges with aeroelastic effects are commonly portrayed in terms of a family of frequency and damping loci as a function of mean wind velocity. Depending on the structural dynamic and aerodynamic characteristics of the bridge, when two frequencies approach one another over a range of wind velocities, their loci tend to repel, thus avoiding an intersection, whereas the mode shapes associated with these two frequencies are exchanged in a rapid but continuous way as if the curves had intersected. This behavior is referred to as the curve veering phenomenon. In this paper, the curve veering of cable-stayed and suspension bridge frequency loci is studied. A perturbation series solution is utilized to estimate the variations of the complex eigenvalues due to small changes in the system parameters and establish the condition under which frequency loci veer, quantified in terms of the difference between adjacent eigenvalues and the level of mode interaction. Prior to the discussion of bridge frequency loci, the curve veering of a two-degree-of-freedom system comprised of a primary structure and tuned mass damper is discussed, which not only provides new insight into the dynamics of this system, but also helps in understanding the veering of bridge frequency loci. To study this more complicated dynamic system, a closed-form solution of a two-degree-of-freedom coupled flutter is obtained, and the underlying physics associated with the heaving branch flutter is discussed in light of the veering of frequency loci. It is demonstrated that the concept of curve veering in bridge frequency loci provides a correct explanation of multimode coupled flutter analysis results for long span bridges and helps to improve understanding of the underlying physics of their aeroelastic behavior.     

Vehicle Induced Dynamic Behavior of Short-Span Slab Bridges Considering Effect of Approach Slab Condition

In this paper the vehicle induced dynamic bridge responses are calculated by modeling the bridge and vehicle as one coupled system. The dynamic behavior of short slab bridges with different span lengths induced by the AASHTO HS20 truck is investigated. A parametric study is conducted to analyze the effects of different truck speeds and different road surface conditions. Critical truck speeds that result in peaks of dynamic response are found to follow the rule that describes the resonant vibration of bridges due to train loading. The approach slab condition that consists of faulting at the ends and deformation along the span is considered in the analysis. Although the effect of the along-span deformation on the dynamic response of bridges is trivial, the faulting condition of the approach slab is found to cause significantly large dynamic responses in short-span slab bridges. Impact factors obtained from numerical analyses are compared with those values specified in the AASHTO codes.     

Impact Factors for Curved Continuous Composite Multiple-Box Girder Bridges

The results from a parametric study on the impact factors for 180 curved continuous composite multiple-box girder bridges are presented. Expressions for the impact factors for tangential flexural stresses, deflection, shear forces and reactions are deduced for AASHTO truck loading. The finite-element method was utilized to model the bridges as three-dimensional structures. The vehicle axle used in the analysis was simulated as a pair of concentrated forces moving along the concrete deck in a circumferential path with a constant speed. The effects of bridge configurations, loading positions, and vehicle speed on the impact factors were examined. Bridge configurations included span length, span-to-radius of curvature ratio, number of lanes, and number of boxes. The effect of the mass of the vehicle on the dynamic response of the bridges is also investigated. The data generated from the parametric study and the deduced expressions for the impact factors would enable bridge engineers to design curved continuous composite multiple-box girder bridges more reliably and economically.