A model based on the theory of train-track-bridge coupling dynamics is built in the article to investigate how high-speed railway bridge pier differential settlement can affect various railway performance
China has embarked on an extensive program of building high-speed railway lines. The length of its high-speed operating network of approximately 17,000 km through July 2015 accounts for approximately 60% of the world’s total. The majority of these lines have been built on bridges (i.e., 80.5% [
Nevertheless, excessive bridge-pier differential settlement has not been completely avoided, and, for example, such a problem has occurred at the Wangyu River Bridge (see Figure
Excessive bridge-pier differential settlement.
A number of authors have undertaken studies to investigate the vehicle-bridge interaction problem. Yang and Lin [
On the basis of vehicle-bridge coupled models, some researchers have focused on the effect of bridge-pier differential settlement on different performance criterion. For example, Invernizzi et al. [
However, the static response of the track-bridge system due to the pier differential settlement is in fact a process of rebalancing the track structures and bridge beams according to the pier differential settlement under the effect of structure self-weight. Because the bridge beams (especially a simply supported beam bridge) are discontinuous in the longitudinal direction at piers, but the rail is continuous, the rail deformation differs from the bridge beam deformation at certain positions (see Figure
Certain positions where the rail deformation differs from the bridge beam deformation.
To address the issues, which are apparent in the identified existing studies and to simulate the static and dynamic response of the system due to the pier differential settlement in a more proper way, a model based on the theory of vehicle-track coupling dynamics [
The train-track-bridge vertical coupled model includes the dynamic model of vehicle, rail, track slab, concrete base, bridge, and the interaction between these subsystems (see Figure
Dynamic model of coupled train-track-bridge system.
The vehicle subsystem consists of one vehicle body, two bogies, and four wheel sets and has 10 degrees of freedom (DOFs) as shown in Figure
Vehicle subsystem model.
Based on multirigid system dynamics theory, the vehicle dynamics model is written as
The rail is modeled as a simply supported Euler beam with self-weight; the track slab, concrete base, and bridge beam are modeled as a free-free Euler beam with self-weight. The fastener system, emulsified cement asphalt mortar (CA mortar), concrete base-bridge contact, and bridge bearing are regarded as discrete spring-damping systems [
Through mechanical analysis, the differential equation describing the Euler beam oscillation in the vertical direction considering the self-weight can be written as follows:
The partial differential equations of the vertical vibration of the rail, track slab, concrete base, and bridge beam can be obtained by determining the external forces of the structures. The fourth-order partial differential equations can be solved using the Ritz method [
The dynamic coupling relationship of both the wheel and the rail provides the link between the vehicle subsystem and the rail subsystem [
The interactions between the rail, track slab, concrete base, bridge beam, and bridge-pier subsystems are described by fastener force, CA mortar reaction force, bridge deck reaction force, and bridge bearing reaction force. These forces are expressed as
The pier differential settlement is determined from the bridge bearing reaction force:
Solutions of the proposed train-track-bridge dynamics model considering self-weight adopt the explicit integration method suggested in the literature [
The model suggested herein was verified through a comparative analysis of the outputs calculated by the model with those (1) computed by an FEM model for the static responses due to pier differential settlement regardless of wheel-rail contact force and (2) computed by a train-track-bridge dynamics model in the literature [
Parameters of FEM model and the proposed model.
Structure | Parameter (unit) | Value | |
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FEM model | The proposed model | ||
Steel rail | Elastic modulus (N/m2) | 2.1059 × 1011 | |
Inertia moment (m4) | 3.1217 × 10−5 | ||
Linear density (kg/m) | 60.64 | ||
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Fastener | Stiffness (N/m) | 6.0 × 107 | |
Damping (kN s/m) | — | 75 | |
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Track slab | Width (m) | 2.4 | |
Thickness (m) | 0.19 | ||
Elastic modulus (N/m2) | 3.5 × 1010 | ||
Poisson’s ratio | 0.2 | — | |
Density (kg/m3) | 2500 | ||
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CA mortar | Stiffness (N/m) | — | 1.25 × 109 |
Damping (kN s/m) | — | 34.58 | |
Elastic Modulus (N/m2) | 200 × 106 | — | |
Poisson’s ratio | 0.31 | — | |
Density (kg/m3) | 1800 | — | |
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Concrete base | Width (m) | 2.88 | |
Thickness (m) | 0.25 | ||
Elastic modulus (N/m2) | 3.0 × 1010 | ||
Poisson’s ratio | 0.2 | — | |
Density (kg/m3) | 2500 | ||
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Bridge beam | Elastic Modulus (N/m2) | 3.55 × 1010 | |
Poisson’s ratio | 0.2 | — | |
Density (kg/m3) | 2500 | ||
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Bridge bearing | Stiffness (N/m) | 7.5 × 108 | |
Damping (kN s/m) | — | 7.5 × 104 |
Model geometry: (a) three-dimensional view of whole model; (b) elevation view; (c) cross-section view; (d) detail of model.
Figure
Comparison of static responses: (a) rail vertical displacement and (b) fastener force.
Comparison of dynamic responses: (a) vertical acceleration of bridge deck, (b) vertical displacement magnification factor of mid-span, and (c) vertical acceleration of vehicle body.
As shown in Figure
From the comparisons shown in Figures
To obtain a deep insight into the limits of pier differential settlement as a part of the standards for construction and maintenance, a study was carried out to determine the functional performance criteria associated with train safety, stability, and service life of the fastener system as a function of the following: Amplitude of pier differential settlement: 2 mm, 5 mm, 8 mm, and 10 mm Train speed: 200 km/h, 250 km/h, 300 km/h, and 350 km/h Span of the simply supported beam bridge: 24 m and 32 m (note that a span of 24 m or 32 m is most widely used in China)
The measures of functional performance criteria were chosen for the following: Stability: the maximum vertical acceleration of a train, which was in keeping with a number of railway organizations, was used as a measure of stability. In Chinese design standards, an upper limit of the vertical acceleration 0.13 g is specified [ Safety: the likelihood of the derailment of a train is commonly measured by the wheel unloading rate and an upper limit of 0.9 is suggested in the literature [ Service time of the fastener system: the service time of the fastener system is partly related to fastener force.
The train chosen in the calculation is a CRH3 EMU train with an axle load of approximately 16 t, and the bridge is the standard simply supported beam bridge with CRTS I slab track. There are nine piers in the model and only the middle pier of the bridge has a certain settlement.
Figure
Wheel-rail contact force for different pier settlement values: (a) position A and (b) position B.
Figure
Maximum wheel unloading rate at different pier settlement values.
Figure
Vertical acceleration of the vehicle body at different pier settlement values.
Figure
Speed limit for exceeding pier differential settlement.
Pier differential settlement (mm) | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|
Speed limit (km/h) | 300 | 250 | 250 | 250 | 200 |
Maximum vertical acceleration of the vehicle body at different pier settlement values.
As an important component of track structures to link rails and ties, the fastener system must have sufficient strength and durability. For the simply supported beam bridge, the bridge beams are discontinuous in the longitudinal direction at the piers, while the rail is continuous. As a result, the rail deformation differs from the bridge beam deformation when a pier differential settlement occurs, thus leading to initial fastener forces before the train arrives. Figure
Maximum fastener compressive and tensile forces.
Span of the bridge beams | Initial fastener force | Pier differential settlement (mm) | |||
---|---|---|---|---|---|
2 | 5 | 8 | 10 | ||
24 m | Maximum tensile force (kN) | 0.294 | 1.285 | 2.276 | 2.937 |
Maximum compressive force (kN) | 0.712 | 1.238 | 1.733 | 2.065 | |
32 m | Maximum tensile force (kN) | 0.154 | 0.956 | 1.755 | 2.286 |
Maximum compressive force (kN) | 0.647 | 1.048 | 1.449 | 1.717 |
Initial fastener force before train arrives.
As shown in Table
Fastener forces of fasteners subjected to compressive forces alone and fasteners subjected to both compressive and tensile cyclic forces.
This paper describes a numerical model for a coupled dynamics analysis of vertical responses in a train-track-bridge system. The performances of the model are compared with a 3D FEM and numerical model in the literature. Compared to the existing models, the proposed model has two advantages: (i) the model considers the effect of structure self-weight; thus, it can be used to calculate the dynamic response of the system due to the pier differential settlement; (ii) the model is considerably more computationally efficient, and it therefore can be used in environments that do not have access to the computing facilities that are required to run similar FEMs.
The developed model was used to investigate the influence of the pier differential settlement, train speed, and length of the bridge beam on measures of track performance associated with passenger ride quality, railway vehicle safety, and fastener forces. The following findings may be drawn from the analysis. The wheel unloading rate increases with both train speed and pier differential settlement. The wheel unloading rate for a 24 m length of bridge beam is slightly larger than that for a 32 m length of bridge beam, and for all the considered train speeds, its maximum value of 0.695 is well below the allowable limit. The vehicle vertical acceleration increases with the pier differential settlement and train speed. The maximum vertical acceleration, respectively, 0.135 g for a 24 m length of bridge beam and 0.131 g for a 32 m length of bridge beam, exceeds the maximum allowable limit stipulated in the Chinese standards. On this basis, the speed limit for the exceeding pier differential settlement is determined for comfort consideration. The beams of a simply supported beam bridge are discontinuous in the longitudinal direction at piers but the rail is continuous, thus leading to initial fastener forces before the train arrives. The fasteners that have an initial compressive force are always in compression with and without train loading, while fasteners that have an initial tensile force experience both compressive and tensile forces and are likely to have lower service lives than those that are subjected to compressive forces alone.
Acceleration vectors of vehicle system
Velocity vectors of vehicle system
Displacement vectors of vehicle system
Load vectors of vehicle system
Mass matrices of vehicle system
Damping matrices of vehicle system
Stiffness matrices of vehicle system
Vehicle body mass
Moment of inertia of vehicle body
Bogie frame mass
Moment of inertia of bogie frame
Wheel set mass
Secondary suspension stiffness
Distance between bogie centers
Primary suspension stiffness
Bogie wheelbase
Secondary suspension damping
Primary suspension damping
Wheel-rail contact force
Vertical displacement of the Euler beam
Mass of per unit length of the Euler beam
Flexural rigidity of the Euler beam
External force
Number of fasteners
Number of track slab coordinate nodes
Number of concrete base coordinate nodes
Number of bridge beam coordinate nodes
Fastener force
CA mortar reaction force
Bridge deck reaction force
Bridge bearing force
Modal coordinates of the rail
Modal coordinates of the track slab
Modal coordinates of the concrete base
Modal coordinates of the bridge beam
Orthogonal function department of the rail
Orthogonal function department of the track slab
Orthogonal function department of the concrete base
Orthogonal function department of the bridge
Constants of the track slab
Constants of the concrete base
Constants of the bridge beam
Functions of the track slab self-weight
Functions of the concrete base self-weight
Functions of the bridge beam self-weight
Coefficient of the free beam
Displacement of wheel set
Track irregularity at the position of wheel set
Wheel-rail contact constant
Vertical velocity of the rail
Vertical displacement of the rail
Vertical velocity of the track slab
Vertical displacement of the track slab
Vertical velocity of the concrete base
Vertical displacement of the concrete base
Vertical velocity of the bridge beams
Vertical displacement of the bridge beams
Bridge precamber
Damping of the fastener system
Damping of the CA mortar
Damping of the bridge deck
Stiffness of the fastener system
Stiffness of the CA mortar
Stiffness of the bridge deck
Pier settlement value
Damping of the bridge bearing
Stiffness of the bridge bearing
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The work described in this paper was supported by the National Natural Science Foundation of China (51478353).