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Reducing ground borne vibrations in urban areas is a very challenging task in railway transportation. Many mitigation measures can be considered and applied; among these open trenches are very effective. This paper deals with the study of the effect, in terms of reduction of vertical and horizontal displacements and velocities, of the open trenches. 2D FEM simulations have been performed and several open trench configurations have been analysed varying the main geometric features such as width and depth, distance from the rail, thickness of the soil layer over the rigid bedrock, type of the ground, ratio between the depth of the trench, and the thickness of the soil layer. For quantifying the influence of the above specified parameters in reducing ground-borne vibrations an analysis using artificial neural networks (ANNs) has been carried out. Results show that among the geometric parameters the role of the depth of the trench is very significant; however the influence of the depth must be also evaluated in relation to the thickness of the soil layer.

The problem of the ground-borne vibrations induced by railway traffic has received an increasing interest in the recent years becoming a relevant scientific and technical research area.

Due to the frequent construction of high-speed railways and mass rapid transit systems worldwide, most highly developed cities or metropolitan areas have encountered the problem that the rail tracks inevitably come cross or close to vibration-sensitive sites, as discussed by Hung and Yang [

The economical and environmental aspects of the issue require a careful assessment of the problem before the construction of new rail tracks or the upgrading of the existing ones for heavier and faster traffic.

The main source of excitation of the track is represented by the vertical force determined by the wheel-rail interaction during the passage of the train. Vibrations can be amplified by the passage of trains due to the surface irregularities of wheels and rails, by the rise and fall of axles over sleepers, and by the propagation of deformation patterns in the track and ground. Such vibrations are transmitted through the track structure, including rails, sleepers, ballast, and sublayers and propagate as waves through the soil medium [

The study of the ground-borne vibrations requires the consideration of four main components: the “source,” that generate the vibration, that is, the excitation caused by the motion of the trains over rails with irregular surfaces, the “propagation path” through the soil medium, the “receiver,” that is, the nearby buildings, and finally the “interceptor,” that is, wave barriers, such as piles, in-filled and open trenches, and isolation pads.

At each component is related one specific phase of the process of the transmission of the vibrations and in particular the generation, the transmission, the reception, and the interception.

In regard to this last phase, three groups of mitigation measures can be considered:

mitigations in the source, that include all types of interventions in the railway structures (active isolation);

mitigations in the path, such as barriers to the waves propagation from the source to the receiver (in-filled and open trenches, lime cements columns); these barriers provide an active isolation when they are close to the source, and a passive one when they are far away;

mitigations to the receiver, that include all the measures aimed at reducing the effects on the buildings (passive isolation) or on the other vibration-sensitive sites.

Among the mitigation measures in the path, the open trenches have exhibited a good performance on the screening of the vibrations but, to achieve this effect, it is very important to assign it the proper dimensions.

The first experimental surveys on the effectiveness of open and in-filled trenches were carried out by Barkam (1962) [

In regard to the continuous barriers also Woods [

Several experimental surveys stated that the best screening performance takes place when the depth of the trench is equal to the wavelength while the width of the trench is small [

Numerous researches on the effectiveness of the barriers have been carried out using FEM and BEM modelling [

In particular, Beskos et al. [

Adam and Estorff (2005) [

Yang and Hung (1997) [

Few works focus on the effectiveness of barriers in full scale.

The most important application is based on Gas Cushion Method that consists in a vertical panels filled of gas and flexible cushion with very low impedance installed in a trench having a great depth [

In this work the screening performance of open trenches, excavated in a soil layer over a bedrock, has been studied. By applying a 2D FEM model, an extensive analysis has been carried out to the aim of determining the contribution of the main geometric parameters of the trenches in the interception of the vibrations. The geometric characteristics assumed to be variable parameters are width (

Schematic of the position of the open trench.

The study has been carried out considering different scenarios obtained by varying the above specified geometric parameters of the trenches and the mechanical and geometric characteristics of the soil medium. The values attributed to the geometric features are summarised in Table

Geometric parameters of the trenches.

Width | Distance from the rail | Distance from the site |
---|---|---|

0.5 | 8 | 19 |

1.0 | 12 | 15 |

1.5 | 16 | 11 |

2.5 | 20 | 7 |

— | 24 | 3 |

Depth of the trench and ratio depth/thickness of layer.

Parameters | Assumed values |
---|---|

2, 5, 8, 11,14 | |

1/5, 1/2, 2/3 |

Two types of unsaturated soil, labelled, according to the classification of Eurocode 8, respectively, ground type C (medium dense sand) and ground type D (firm clay), have been considered. These soils have been characterised by the shear modulus (

Mechanical characteristics of the soil medium.

Type of soil | ^{2}) | ^{4} sec^{2}) | ||||
---|---|---|---|---|---|---|

Clay | 20.7 | 1583.8 | 0.3 | 0.06 | 114.3 | 108.1 |

Sand | 138 | 1682.8 | 0.3 | 0.04 | 286.3 | 270.9 |

The value of the damping ratio for clay and sand has been selected according to the results of an Italian comprehensive review [

By combining all the parameters, 540 scenarios have been obtained and investigated. The propagation and interception of ground-borne vibrations for each scenario have been modelled by means of the finite elements method. The scheme adopted is shown in Figure

Modelling scheme.

It has been also established that the railway line is sited over an embankment 1.5 m high and with escarpment’s slope of 2/3; the geotechnical parameters of the embankment are ^{4} s^{2}, and

For each scenario the following responses have been obtained: vertical (dir.

in a “vibration-sensitive site,” large 10 m and far away 27 m from the symmetry axis of the railway;

in a zone, in the following called “site close to the trench,” 5 m wide and sited far away 0.5 m from the trench.

The dimensions assigned to the two sites are well suited to be comparable with the dimensions of receptors such us buildings.

The study of the response in this last site allowed to verify the effectiveness of the open trench as passive isolation.

The propagation of vibrations is a typical three-dimensional issue, specially in railway field where the train acts as a series of incoherent point source rather than a fully coherent line source. So three-dimensional models are certainly more suitable to predict the absolute vibration but require computation times not always consistent with extensive analyses.

Andersen and Jones [

In the light of this, the 2D model has been considered appropriate to output results consistent with the aim of the present study.

Linear viscoelastic constitutive model for the embankment and the soil medium materials has been adopted to carry out the investigation, according to some previous studies [

The presence in the analysed scheme of a rigid base has been taking into account by introducing in the model fixities at bedrock.

The railway source has been modelled applying in the surface of the embankment below the ballast a stress having a value equal to the dynamic stress produced in the contact surface ballast-embankment.

In particular, according to [^{2} (Figure

Loading conditions.

Time history of the rail force.

The applied load function gives a significant contribution for the frequencies included in the range from 20 to 35 Hz, with a central frequency of about 27 Hz. This range, in the present study-case, is representative of the frequencies produced by the passage of a freight train at the velocity of 60 km/h, and having a boogie axle distance of 9 m. The load can appear overabundant but it takes into account the dynamic amplification factor by the shortest wavelength [

The definition of the model mesh is in general a compromise between the necessity to restrict the number of the elements and that to limit their dimensions. However, the element’s dimension must be accurately chosen in order to adequately represent the deformed shape associated to the wave lengths

If

In the light of this, the maximum dimension of the element has been defined in the following way:

on the basis of the material properties, there has been calculated the minimum wave propagation velocity for the considered domain; that is, for the superficial elements there has been considered the Rayleigh’s velocity

there has been defined the maximum value of the appropriate frequency

there has been calculated, using the expression (

By applying this procedure to the two types of soil medium, there has been obtained

By considering these dimensions, the mesh has been made as follows:

in the part of the model where the response is significant to the aim of the study, corresponding to a strip, large about 40 m and 40 m deep, closest to the rail track, a maximum dimension of 0.5 m has been assigned to the elements; this mesh has been designed to be denser in this focal area in order to achieve a good precision;

in the part of the model included between the quoted part and the lower edge a maximum dimension of 0.5 m has been assigned when the clayey soil has been analysed, while in the case of sandy soil the dimensions of the element gradually increase until 1.5 m near the lower edge;

externally to the above mentioned areas, the dimensions of the elements increase gradually reaching the value of 3 m in proximity of the vertical boundary far 300 m from the symmetry axis.

The linear variation of the mesh dimensions, from the side

The scheme of the finite element mesh is shown in Figure

Scheme of FEM mesh.

Rayleigh’s parameters determination.

It can be observed that outside the zone of interest (

The time step integration has been assigned taking into account the Courant condition, that defines the maximum time step as [

It should be noted that in the problems involving different type of waves, like the propagation of the

Having in mind this, in this task there has been assumed that

In both cases of clay and sand the FEM simulations have been performed in a time of

Boundary conditions have been imposed in the model in consideration of the geometric and mechanical conditions of symmetry and taking into account the need to limit the errors, namely, the difference existing between the theoretical value of the stress in a point of the boundary and the value obtained in presence of the restraints.

FE calculations need a self-consistent simulation area to operate on. Especially in the case of wave propagation, special boundary conditions have to be incorporated into the calculation scheme [

In the light of this, there has been applied

in the symmetry edge, symmetry restraints like bi-pendulum in

in the lower edge, fixities at the bedrock;

in the right-side edge, Lysmer’s dampers in the directions

The right-side edges have been applied at a distance of 300 m from the source; the errors obtained in the boundary have been

In an FEM model the damping behaviour can be modelled using Rayleigh’s method. The damping matrix can be obtained from the following relation:

The

For each 2D model, a frequency analysis has been carried out and the first resonance frequency

There has been calculated the frequency

Rayleigh’s parameters value has been calculated by establishing that, for frequencies equal to

For the clayey soil Rayleigh’s parameters have been calculated assuming that a damping ratio

In Figures

Damping ratio

Damping ratio

In order to validate the model, results by Beskos et al. [

In particular the comparison between the results given by the proposed 2D FEM model and the Beskos one has been performed considering the trend of the attenuation ratio

In each point of both vibration-sensitive site and site close to the trench the attenuation ratio

Since the sites are not points but large zones and considering the evolution of the attenuation ratio increasing distance (Figure

As it can see in Figure

Model results versus Beskos results.

From the numerical FEM simulations of the 540 scenarios the values of the displacements and velocities have been obtained; the minimum and the maximum values reached in the significant points are summarised in Table

Maximum and minimum values of displacements and velocities.

Site close to the trench | ||||

Soil type | Velocity dir. | Velocity dir. | Displacement dir. | Displacement dir. |

Sand | 0.13 | 0.09 | 0.00098 | 0.00056 |

0.35 | 0.25 | 0.00252 | 0.00619 | |

Clay | 0.16 | 0.13 | 0.00564 | 0.00369 |

0.29 | 0.48 | 0.01084 | 0.02509 | |

Site vibration-sensitive | ||||

Soil type | Velocity dir. | Velocity dir. | Displacement dir. | Displacement dir. |

Sand | 0.052 | 0.035 | 0.00078 | 0.00009 |

0.21 | 0.20 | 0.00198 | 0.00341 | |

Clay | 0.035 | 0.028 | 0.00177 | 0.00040 |

0.27 | 0.30 | 0.01084 | 0.01714 |

In Figure

Samples of results in a point in case of sandy soil.

When the horizontal propagation of in-plane waves is considered, the Rayleigh waves and the

At quite distance (

If an open trench is excavated near the source, the incidence of the waves on this obstacle (discontinuity field) gives rise to reflected and transmitted body waves. Behind the open trench, as the distance increases, the transmitted

According to this phenomenon the results show a great attenuation of displacement and velocity passing from the site close to the trench to the vibration sensitive site (see Table

To evaluate the effectiveness of the trench there has been adopted the very strict following criterion: a trench is effective in screening the ground borne vibration if the value of the attenuation ratio is lower than 0.4 for every considered displacement and velocity.

The analysis of the FEM results for the two types of ground has been performed having in mind this criterion.

The results give some indications for the design of the open trenches in both cases of site close to the trench and vibration-sensitive site (see Tables

Design parameters for trench—site close to the trench.

Clayey soil layer | Sandy soil layer | |||||

1/5 | — | — | — | |||

1/2 | 1.25 | all | all | |||

2/3 | 6 | all | all | all |

Design parameters for trench—vibration-sensitive site.

Clayey soil layer | Sandy soil layer | |||||

1/5 | — | — | — | 0.2 | 0.05 | |

1/2 | 1.25 | all | 0.5 | 0.8 | 0.05 | |

2/3 | 0.5 | all | all | 0.2 | all |

The effect of frequency is fully taken into account by normalizing the open trench dimensions with respect to the Rayleigh wavelength.

The obtained results are very strict in regard to the dimensions and the position of the trench because these features depend on the ratio

For instance, in clayey soil results show that the trench is effective if it is 5 m deep and far away at least 12 m from the source, for thickness layer equal to 10 m or 2 m deep and far away at least 24 m from source, and for layer thickness equal to 3 m.

Some results of the parametric studies carried out are plotted in Figure

Some results of parametric study (

Artificial neural networks (ANNs) are a tool that simulates the biological processes and, as demonstrated by the wide applications in engineering field, they are able to solve functional mapping problems. ANNs are an assemblage of mathematical simple computational elements called neurons.

In particular the back propagation neural network (BPNN) is a collection of neurons distributed over an input layer, that contain the input variables of the problem, one or more hidden layers with a certain number of nodes, and an output layer with a number of nodes equal to the output variables. The nodes between layers are connected by the links having a weight that describes quantitatively the strength of the connection.

A significant effort is required in the selection of the ANN architecture, particularly, as it is obvious, in the definition of the hidden layers and the corresponding nodes.

The learning process utilized for this type of ANN is the “back-propagation learning” which consists in an error minimization technique [

To ensure an efficient convergence and the desired performance of the trained network, several parameters are incorporated in the training phase. These parameters include the learning rate, the momentum term, and the number of training iterations. The learning rate is a factor that proportions the amount of the adjustment applied at each time the weight is updated. The use of a momentum term could carry the weight change process through one or more local minima and get it into global minima. The early stopping method and the number of training epochs determine the training stop criteria [

For the present application the neural networks have been trained and tested with the data given from the FEM modelling of the 540 scenarios.

In details, the input data chosen for the neural analysis have been 6 and precisely: width of the trench (_{s}

To this purpose a great number of trial ANNs with one, two, and three hidden layer have been trained to evaluate the performance of different network architectures in the comprehension and generalisation of the problem.

The training procedure has been tested using a test set of examples having a percentage of 10% of the training set; the prediction performance of the developed model has been exhibited in the test set.

The error term, represented by the mean square error (RMS error), computed for the validation input-output pairs, has been monitored during the training process of the networks. The error normally decreases during the initial phase of training. However, when the network begins to overfit the data (a situation arising when ANN works well only with the training data), the error on the validation set will typically begin to increase. When the validation error increased for a specified number of iterations, training was stopped and the weights at the minimum of the validation error were saved. This point is the point of maximum generalization [

Considering this, networks with errors comprising in the following ranges have been selected: RMS error training set: 0.02–0.1; RMS error test set: 0.02–0.12; Correlation training set: 0.5–1. It has been stated that the neural networks having the training parameters included in the quoted ranges provide acceptable results.

The analysis of the training and the testing phases of these neural networks allowed to evaluate the percentage of contribution of each input factor in the output determination. In other terms the study allowed to assess qualitatively and quantitatively the influence of the features of the trench and the soil medium in the attenuation process of ground-borne vibrations.

The results obtained for both vibration-sensitive site and site close to the trench are explained in the following.

Regarding the attenuation ratio of the horizontal and vertical displacements, the neural network analysis has given the percentages of contribution summarised in Tables

Percentage of contribution to the attenuation of the displacements—site close to the trench.

Attenuation ratio for horizontal displacements | Attenuation ratio for vertical displacements | ||

Parameters | Percentage of contribution | Parameters | Percentage of contribution |

28.00% | 31.00% | ||

26.00% | 28.00% | ||

17.00% | 14.50% | ||

15.50% | 13.00% | ||

10.50% | 9.00% | ||

3.00% | 4.50% |

Percentage of contribution to the attenuation of the velocities—site close to the trench.

Attenuation ratio for horizontal velocities | Attenuation ratio for vertical velocities | ||

Parameters | Percentage of contribution | Parameters | Percentage of contribution |

31.00% | 27.00% | ||

23.00% | 26.00% | ||

17.00% | 15.00% | ||

14.00% | 13.50% | ||

11.00% | 11.50% | ||

4.00% | 7.00% |

By analysing the results it can be deduced that for both vertical and horizontal displacements the percentage of contribution to the attenuation ratio of the depth of the trench

Regarding the attenuation of the velocities the hierarchy of influence of the parameters seems to be the same of the displacements: the influence of the depth of the trench

The percentages of contribution to the attenuation ratio of the horizontal and vertical displacements are summarised in Tables

Percentage of contribution to the attenuation of the displacements—vibration-sensitive site.

Attenuation ratio for horizontal displacements | Attenuation ratio for vertical displacements | ||

Parameters | Percentage of contribution | Parameters | Percentage of contribution |

36.00% | 42.00% | ||

30.00% | 26.00% | ||

13.00% | 12.00% | ||

11.00% | 11.00% | ||

7.00% | 7.00% | ||

3.00% | 2.00% |

Percentage of contribution to the attenuation of the velocities—vibration-sensitive site.

Attenuation ratio for horizontal velocities | Attenuation ratio for vertical velocities | ||

Parameters | Percentage of contribution | Parameters | Percentage of contribution |

34.00% | 35.00% | ||

27.00% | 22.50% | ||

17.00% | 16.00% | ||

14.00% | 15.50% | ||

5.00% | 8.00% | ||

3.00% | 3.00% |

In this site the influence of the depth of the trench

Regarding the attenuation ratio of the horizontal and vertical velocities the trend is the same of the displacements but the contribution of the terrain’s parameters (

In order to intercept the elastic waves and in particular Rayleigh’s ones generated by the moving source from reaching buildings between the railways and the sensitive site, vibration screening may be established by means of the open trenches. Several factors determine the effectiveness of the open trenches in the screening of the ground borne vibrations in soil medium.

In the present work, as outcomes of an FEM modelling of the problem and of an analysis carried out by means of the neural network, the weights of the main geometric features of both the trench and the soil medium have been established.

The main conclusions of this application are the following:

first of all, the screening performance of the trench is mainly affected by the depth of the trench, in both cases of site close to the trench and vibration-sensitive site. From ANN analysis the percentage of contribution of this geometric parameter in reducing displacements and velocities of vibration reached the 25%–35%;

layer thickness increasing, the attenuation ratio increases according to the depth of the trench; to this reason, it appears very noteworthy in the topic not only the depth

to the aim of evaluating the effectiveness of the trench we have to analyse all the kinematic parameters and not only the displacement in one direction, generally the direction