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Based on a two-and-half-dimensional finite element model (2.5D FEM), the layered ground vibration under combined seismic and high-speed train loads was investigated. On this basis, the effect of site category and earthquake intensity on ground vibration under the combined action of two dynamic loads was analyzed. Numerical examples indicated that ground vibration displacement due to combined loads decreases with the increase of soil stiffness, while the influence of soil stiffness on the ground vibration is small when the hardness of the subsoil is large. The peak ground displacement (PGD) is a reasonable seismic intensity index for predicting the ground vibration displacement at the track center under the combined loads, which has a higher accuracy for hard ground. In view of this, an equivalent shear wave velocity and PGD-based prediction formula was proposed to estimate the ground vibration under combined seismic and high-speed train loads. Reliability of the prediction formula was verified through comparison with results of numerical tests, indicating that the prediction formula has good applicability to different site conditions and seismic events. Compared with the previous study, it demonstrated that the prediction method provided an effective means for estimating ground vibration caused by a high-speed train load during earthquakes.

The increasing frequency of extreme rainfall is leading to a rise in groundwater levels in coastal areas, significantly affecting high-speed railway operations. To address this concern, this study developed a 2.5-dimensional finite element model of a coupled track-embankment-ground system based on Biot's porous media theory to analyze the effect of groundwater level rise on the critical velocity of high-speed railways and vibration responses. The findings reveal a consistent decrease in the critical velocity of high-speed railways with rising groundwater levels. Particularly, the increase in groundwater levels within the embankment significantly influences the critical velocity compared to a similar rise in the foundation's groundwater level. Furthermore, deformations induced by passing trains significantly increase as groundwater levels rise. Specifically, when the groundwater level rises from the foundation bottom to the subgrade surface, subgrade surface deformation increases by approximately 55%. As trains approach the critical velocity, significant vibration phenomena, known as the "Mach effect," occur at the foundation surface. Importantly, as groundwater levels rise, the "Mach effect" intensifies. Analyzing the vibrating frequency spectrum of the displacement response demonstrates a substantial increase in vibration amplitude, particularly in the high-frequency region, as groundwater levels rise. This study highlights that the rise in groundwater level not only amplifies vibrations but also extends the propagation of high-frequency vibrations, underscoring the importance of effective embankment waterproofing in controlling track vibrations. [ABSTRACT FROM AUTHOR]

In this paper, an approach for a two-and-half-dimensional (2.5D) finite element method (FEM)-based analysis, or quasi-three-dimensional (3D) FEM analysis, of an axial flux machine is discussed. By cutting the 3D model laterally and thereby creating cylindrical surface cuts, the 3D model can be split into several cylindrical surfaces. Transforming those cylindrical cuts into planes leads to a layer-based two-dimensional (2D) model with different radii for each layer. By integrating over all lateral surface cuts, the results for the entire axial flux machine can be determined. In comparison to the simulation of a full 3D FEM model, the simulation of the proposed 2.5D model is much faster. To validate the approach, the two main types of axial flux machines are simulated with both 3D-FEM-based model and 2.5D-FEM-based approach, and the results are presented in this paper.

The rise of high-speed railway induces an increased probability of serious derailment accidents of operating high-speed trains during earthquakes. A two-and-half-dimensional finite element model (2.5D FEM) was developed to investigate the ground vibration under combined seismic and high-speed train loads. Numerical examples were demonstrated and the proposed method was turned out to provide an effective means for estimating ground vibration caused by high-speed train load during earthquakes. The dynamic ground displacement caused by combined seismic and high-speed train loads increases with the increase of the train speed, and decreases with the increase of the stiffness of ground soil. Compared with the seismic load alone, the coupling effect of the seismic and high-speed train loads results in the low-frequency amplification of ground vibration. The moving train load dominants the medium–high frequency contents of the ground vibration induced by combined loads. It is observed that the coupling effects are significant as the train speed is higher than a critical speed. The critical train speed increases with the increase of the ground stiffness and the intensity of the input earthquake motion.

The increasing frequency of extreme rainfall is leading to a rise in groundwater levels in coastal areas, significantly affecting high-speed railway operations. To address this concern, this study developed a 2.5-dimensional finite element model of a coupled track-embankment-ground system based on Biot’s porous media theory to analyze the effect of groundwater level rise on the critical velocity of high-speed railways and vibration responses. The findings reveal a consistent decrease in the critical velocity of high-speed railways with rising groundwater levels. Particularly, the increase in groundwater levels within the embankment significantly influences the critical velocity compared to a similar rise in the foundation’s groundwater level. Furthermore, deformations induced by passing trains significantly increase as groundwater levels rise. Specifically, when the groundwater level rises from the foundation bottom to the subgrade surface, subgrade surface deformation increases by approximately 55%. As trains approach the critical velocity, significant vibration phenomena, known as the “Mach effect,” occur at the foundation surface. Importantly, as groundwater levels rise, the “Mach effect” intensifies. Analyzing the vibrating frequency spectrum of the displacement response demonstrates a substantial increase in vibration amplitude, particularly in the high-frequency region, as groundwater levels rise. This study highlights that the rise in groundwater level not only amplifies vibrations but also extends the propagation of high-frequency vibrations, underscoring the importance of effective embankment waterproofing in controlling track vibrations.

Based on Biot's theory of porous media, this paper proposes a coupled vehicle-track-saturated foundation model to study the dynamic responses of saturated soft soil foundation under high-speed train load via two-and-half dimensional finite element method (2.5D FEM). It is found that the ratio of train speed c and soil Darcy permeability kD(c/kD) determines the extent to which maximum excess pore pressures build up in saturated soft soil under the train load when the load speed is lower than the critical speed. For a saturated soil of a particular stiffness, if c/kD is less than or equal to 3×104, the soil can be regarded as highly permeable relating to the load velocity. In this case, almost no excess pore pressure is developed, and the saturated soil can be modeled using a single-phase medium. There is a critical value of c/kD for the development of excess pore pressure, corresponding to which the maximum excess pore pressure increases with increasing c/kD. Above this critical value of c/kD, the maximum excess pore pressure remains independent of c/kD. The amplitude and influencing zone of effective stress are mainly controlled by the train speed and soil permeability for saturated soft soil foundation. The response of displacement is mainly controlled by the train speed. Significant Mach effects have been induced in the saturated soil foundation by the moving train when the train speed reaches or exceeds the critical velocity of the track-foundation system.

Considering the soil as nonlinear material, the ground vibration displacements caused by high-speed train (HST) moving on the elastic-plastic multi-layered ground are investigated using a 2.5-dimensional finite element model (2.5D FEM), in which the improved Mohr-Coulomb model and the post-Eulerian integration algorithm are also introduced. The accuracy of the established model is validated with published data. Results show that, under different train speeds, the ground vibrations at the track center for the multi-layered soil media are predominantly affected by the properties of the topsoil layer. The ground vibration displacements within 2 m from the track center are controlled by resonance conditions. The displacement attenuation curves fluctuate more in the soft layer than the hard one when the train speed reaches or exceeds the Rayleigh wave speed of all soil layers. Additionally, increasing the elastic modulus of topsoil can effectively reduce the ground vibration displacements in the range of 0.0 m–2.5 m from the track center, and the effects of buried depth for the soft soil layer being slight and thus ignorable. • The vibration displacements of elastic-plastic multi-layered soil are analyzed. • The displacements near the track are affected by train speed and all soil layers. • The displacement decay curve fluctuates in the soft layer more than in the hard. • The deeper the soft soil layer embedded, the lower the vibration displacement. [ABSTRACT FROM AUTHOR]

Abstract The rapid development of high-speed railway networks requires advanced methods for analysing vibration and sound radiation characteristics of a fast rotating train wheel subject to a vertical harmonic wheel–rail force. In order to consider the rotation of the wheel and at the same time increase the computational efficiency, a procedure is adapted in this paper taking advantage of the axial symmetry of the wheel. In this procedure, a recently developed 2.5D finite element method, which can consider wheel rotation but only requires a 2D mesh over a cross section containing the wheel axis, is used to calculate the vibration response of the wheel. Then, the vibration response of the wheel is taken as acoustic boundary condition and the 2.5D acoustic boundary element method, which only requires a 1D mesh over the boundary of the above cross section, is utilised to calculate the sound radiation of the wheel. These 2.5D methods and relevant programs are validated by comparing results from this procedure with those from conventional 3D analyses using commercial software. The comparison also demonstrates that these 2.5D methods have a much higher computational efficiency. Using the 2.5D methods, we study the wheel rotation speed influences on the factors including the vertical receptance of the wheel at wheel–rail contact point, sound pressure level at a pre-defined standard measurement point, radiated sound power level, directivity of the radiation, and contribution of each part of the wheel. It can be concluded that the wheel rotation speed splits most peaks of the vertical receptance at the wheel–rail contact point, sound pressure levels at the field, and the sound power level of the wheel into two peaks. The directivity and power contribution of the wheel are also significantly changed by the wheel rotation speed. Therefore, the rotation of a train wheel should be taken into account when calculating its vibration and sound radiation.

Based on a two-and-half-dimensional finite element model (2.5D FEM), the layered ground vibration under combined seismic and high-speed train loads was investigated. On this basis, the effect of site category and earthquake intensity on ground vibration under the combined action of two dynamic loads was analyzed. Numerical examples indicated that ground vibration displacement due to combined loads decreases with the increase of soil stiffness, while the influence of soil stiffness on the ground vibration is small when the hardness of the subsoil is large. The peak ground displacement (PGD) is a reasonable seismic intensity index for predicting the ground vibration displacement at the track center under the combined loads, which has a higher accuracy for hard ground. In view of this, an equivalent shear wave velocity and PGD-based prediction formula was proposed to estimate the ground vibration under combined seismic and high-speed train loads. Reliability of the prediction formula was verified through comparison with results of numerical tests, indicating that the prediction formula has good applicability to different site conditions and seismic events. Compared with the previous study, it demonstrated that the prediction method provided an effective means for estimating ground vibration caused by a high-speed train load during earthquakes. [ABSTRACT FROM AUTHOR]

A two-and-a-half-dimensional finite element method (2.5D FEM) is applied to investigate the dynamic response of an unsaturated ground subjected to moving loads caused by high-speed train. The partial differential equations of unsaturated porous medium in frequency domain are deduced based on the equations of motion and mass conservation of three phases, with consideration of the compressibility of solid grain and pore fluid. Governing equations of unsaturated soil in 2.5D FE form are derived by using the Fourier Transform with respect to the load moving direction. The track structure is simplified as an Euler beam resting on the unsaturated porous half-space and the viscous-elastic artificial boundaries are used to avoid the energy reflection from the boundary. Numerical simulations demonstrate effects of the degree of water saturation and train speed to the ground vibration and the excess pore water pressure. It is concluded that the degree of water saturation has a different influence on the ground displacement and acceleration. The gas phase has varied influence to the ground displacement amplitude at different train speed level at the track center. A very small amount of gas in the saturated ground largely increases the ground acceleration amplitude at a given train speed. Ground displacements attenuate rapidly with almost the same rate for both high and low train speeds near the track center. The maximum amplitude of excess pore water pressure is located at 1.5–2.0 m beneath the ground surface and decreases significantly as the degree of water saturation decreases. • 2.5D FEM was used to study unsaturated ground vibration under moving train loads. • Effects of train speed and degree of saturation to ground vibration were analyzed. • Train speed shows different influences on unsaturated and saturated ground vibration. • Water saturation has different influences on ground displacement and acceleration. • The maximum amplitude of excess pore water pressure is located at 1.5–2.0 m depth. [ABSTRACT FROM AUTHOR]

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The application of finite element simulation to the problem of roll pass design for round bar rolling is considered. Two roll pass sequences were developed by analytical methods and then optimized using 2.5D Finite Element Method (FEM). The first one is a classical oval-round roll pass design. The second one is a combination of flat rolls and round roll passes. Relying on the simulation data obtained by FEM, the roll gaps were adjusted to achieve the required bar shape and the uniform distribution of rolling force between the passes. Advantages and disadvantages of each roll pass design were considered.

The rapid development of high-speed railway networks requires advanced methods for analysing vibration and sound radiation characteristics of a fast rotating train wheel subject to a vertical harmonic wheel-rail force. In order to consider the rotation of the wheel and at the same time increase the computational efficiency, a procedure is adapted in this paper taking advantage of the axial symmetry of the wheel. In this procedure, a recently developed 2.5D finite element method, which can consider wheel rotation but only requires a 2D mesh over a cross section containing the wheel axis, is used to calculate the vibration response of the wheel. Then, the vibration response of the wheel is taken as acoustic boundary condition and the 2.5D acoustic boundary element method, which only requires a 1D mesh over the boundary of the above cross section, is utilised to calculate the sound radiation of the wheel. These 2.5D methods and relevant programs are validated by comparing results from this procedure with those from conventional 3D analyses using commercial software. The comparison also demonstrates that these 2.5D methods have a much higher computational efficiency. Using the 2.5D methods, we study the wheel rotation speed influences on the factors including the vertical receptance of the wheel at wheel-rail contact point, sound pressure level at a pre-defined standard measurement point, radiated sound power level, directivity of the radiation, and contribution of each part of the wheel. It can be concluded that the wheel rotation speed splits most peaks of the vertical receptance at the wheel-rail contact point, sound pressure levels at the field, and the sound power level of the wheel into two peaks. The directivity and power contribution of the wheel are also significantly changed by the wheel rotation speed. Therefore, the rotation of a train wheel should be taken into account when calculating its vibration and sound radiation. [ABSTRACT FROM AUTHOR]

A 2.5D FEM (finite element method) is used to investigate the effects of soil parameters of transversely isotropic (cross anisotropic) saturated soil on ground vibrations and excess pore water pressures induced by moving train loads. The governing equations of transversely isotropic saturated soil are derived from the Boit's theory in frequency domain by applying the Fourier transform with respect to time, and 2.5D FE model is then established using Galerkin method. Correctness of the proposed model is validated with published data. Numerical results illustrate that the decrement of vibration amplitude and excess pore water pressure caused by the increment of vertical elastic modulus is more significant than that of the horizontal direction. Poisson ratios in both directions have little effect on ground vibrations, while an increase in horizontal Poisson ratio results in a significant increment in excess pore water pressure. [ABSTRACT FROM AUTHOR]

Ground vibrations due to a train moving on the pile-supported subgrade are investigated using a 2.5-dimensional finite element model (2.5D FEM). The equivalent pile wall approach is used to model the piles and soils. Two types of subgrades, with or without cement fly-ash gravel (CFG) piles, are analyzed to evaluate the performance of CFG pile-supported subgrade in mitigating ground vibration. The effects of area replacement ratio a R and pile-soil stiffness ratio S ps on vibration mitigation performance are studied. Results show that the CFG pile-supported subgrade exhibits positive performance in vibration mitigation, and this performance gets better away from the track than near it. The improved subgrade could elevate the critical speed, and the resonance-like amplification of ground vibration as the train runs at the critical speed for the unimproved subgrade is completely excluded. Increasing either the a R or S ps can enhance the vibration mitigation performance. However, there are the critical values of a R = 8.70% and S ps = 138.89, beyond which the vibration reduction performance would no longer be improved. • A 2.5D FEM for the coupled track- CFG pile-supported subgrade system is proposed. • Subgrade improved by piles reduces the vibration level and elevates the critical speed. • Vibration decreases with the increasing area replacement ratio or pile-soil stiffness ratio. • There are critical values of area replacement ratio and stiffness ratio in vibration mitigation. [ABSTRACT FROM AUTHOR]

• Ground vibration modelled for high-speed railway track on pile-plank embankment. • Problem simplified to a 2.5D structure coupled with periodically arranged piles. • 2.5D FEM and floquet transform combined to increase computational efficiency. • Pile-plank structure shown to reduce as well as increase ground vibration. Pile-plank structure-enhanced embankments are used in high-speed railways, mainly for controlling embankment settlement and frost heave. The effect of such embankment on train-induced ground vibration has yet received little attention. In this paper, a model is developed to predict ground vibration from such a high-speed railway. In the model, the track/embankment/ground system is idealised as an infinitely long periodic structure. The periodicity is created by the piles which are arranged periodically in the track direction to support the plank and the track. The fictitious pile method is used to divide the periodic track/embankment/ground system into two systems interacting with each other. One is a structure formed by the track, plank, pure soil embankment and ground, all of which are assumed to be uniform in the track direction. Therefore, this structure is uniform in the track direction and termed the 2.5D track/ground structure. The other is an infinite number of fictitious piles which are arranged periodically in the track direction (termed the fictitious piles). The 2.5D finite element method (2.5D FEM) and the Floquet transform are combined to deal with the interaction between the two systems, making the model a 2D FE mesh based and convenient to consider the moving of a train along the track. The focus of the paper is modelling methodology and therefore only preliminary results are produced from the model. The results show that, the pile-plank structure studied in this paper can largely reduce low frequency ground vibrations, for which other measures of ground vibration mitigation are often difficult to apply, but it may also increase ground vibration at other frequencies, depending on the observation location on the ground surface. [ABSTRACT FROM AUTHOR]

The application of finite element simulation to the problem of roll pass design for round bar rolling is considered. Two roll pass sequences were developed by analytical methods and then optimized using 2.5D Finite Element Method (FEM). The first one is a classical oval-round roll pass design. The second one is a combination of flat rolls and round roll passes. Relying on the simulation data obtained by FEM, the roll gaps were adjusted to achieve the required bar shape and the uniform distribution of rolling force between the passes. Advantages and disadvantages of each roll pass design were considered. [ABSTRACT FROM AUTHOR]

An improved two-and-a-half dimensional finite element method (2.5D FEM) model is developed to predict ground-borne vibrations in coastal areas from passing high-speed trains (HSTs). The model treats the ground as nearly saturated layered half-space, and describes the pore water/air mixed fluid by the equivalent fluid approach. Reliability of the proposed model is verified by comparing with the field measurements for X-2000 train and published analytical solutions. On this basis, the ground vibrations and excess pore water pressure under different train speeds and on various soil cases are analyzed in detail. In particular, three kinds of water saturation are discussed to show the sensitivity of vibration responses to the minor change of saturation. Results show that the saturation would cause pronounced influence on vibration displacements and excess pore pressure, and a minor decrease in water saturation would lead to a much higher vibration displacement level and a big difference of excess pore pressure distribution along depth. The influence of saturation is dependent both on soil configurations and on train speeds. It is highlighted that the vibration level on coastal ground would be underestimated by using fully saturated model instead of the nearly saturated model. • A 2.5D FEM for vibration prediction of near-saturated layered ground is proposed. • A minor reduction in saturation results in pronounced increase in vibration displacement. • Decrease in saturation causes difference pore pressure distribution along depth. • The influence of saturation is dependent on soil configurations and train speeds. [ABSTRACT FROM AUTHOR]

Vibration induced by urban rail transit has been a growing public concern worldwide. In this context, medium-low speed maglev trains gradually stand out due to their environmentally-friendly advantages in vibration and noise control. However, few studies have investigated maglev train-induced vibration, and much uncertainty still exists about the environmental vibration caused by underground medium-low speed maglev trains. This paper presents a numerical study of train-induced vibration and proposes a hybrid prediction model based on vehicle-track spatially coupled dynamics in the 2.5D FE framework. This method could consider the nonlinear behaviour of the vehicle-track system in more detail, and the application of Green's function makes this method more efficient when dealing with multiple conditions. Moreover, field measurements are carried out on maglev train-track-embankment system, and measured data show good agreement with the prediction results in the time and frequency domain, verifying the accuracy of the proposed model. Numerical analysis is conducted for the vibration response of the vehicle-track-tunnel-soil system, which reveals the transmission behaviour of the maglev train-induced vibration in layered soils, along with the possible causes of vibration amplification phenomena disclosed. Finally, the vibration levels caused by the maglev and subway trains are compared to quantify their differences. Related results provide insights for unveiling the mechanism of impact brought by daily maglev trains on the neighbouring environment and the quality of residents living alongside. [Display omitted] • Environmental vibration induced by medium-low speed maglev train was conducted. • An efficient prediction model was proposed to calculate underground train-induced vibration. • Field measurements were carried out to verify the reliability of the proposed model. • The vibration differences caused by maglev trains and subway trains were revealed. [ABSTRACT FROM AUTHOR]

During the passage of a train along a railway track, the underlying soil experiences repeated loading. If the soil is saturated, pore pressures will increase as the load passes, and these may or may not start to dissipate before each load is removed. To investigate the dynamic response and excess pore pressures generated in a saturated ground below the track, this study uses a 2.5D finite element model (FEM) of a coupled track-embankment-ground system. The saturated soil is modelled using Biot's theory of elastic wave propagation. The implementation of the method is verified by comparison with semi-analytical solutions for both single-phase elastic and poro-elastic media. It is then used to investigate the influence of load speed c , soil Darcy permeability k D and stiffness on the excess pore water pressures generated. It is found that the ratio c/k D determines the extent to which excess pore pressures build up during passage of the load, at any given depth. For a saturated soil of a particular stiffness, if c/k D is less than 104, the soil can be viewed as highly permeable in relation to the load speed and almost no excess pore pressure is developed. For a single moving load, there is a critical value of c/k D , above which the maximum pore pressure reaches a constant value; this critical value depends on the depth. Below the critical value, the pore pressure accumulated during the passage of a train depends on c/k D but is otherwise independent of the load speed. The pore pressure accumulated during the passage of a bogie pair is greatest for intermediate values of c / k D. For small values of c / k D (high permeability), the pore pressure build-up is small, whereas for large values of c / k D (low permeability) the pore pressure does not dissipate during the loading cycle. The variation in the maximum stress ratio, (τ / σ ′) max , with permeability depends on the depth under consideration. The depth to which pore pressures are generated and the effects of soil stiffness are also discussed in this paper. • Efficient 2.5D model for dynamic response of saturated ground to moving train loads. • Excess pore pressures during load passage determined by load speed and soil permeability. • Above a critical value of c/kD the maximum pore pressure reaches a constant value. • During the train passage pore pressure under each axle increases with each vehicle. [ABSTRACT FROM AUTHOR]