Dynamic behaviour of high speed railway bridges with ballasted tracks

Marco dos Santos Neves*

Civil Engineering and Architecture Department, Instituto Superior Técnico, Universidade Técnica de Lisboa, Portugal

Abstract

The aim of this article is to discuss the effect of the ballasted track on the dynamic behaviour of high speed railway bridges. The amplification of the dynamic effects is generally associated to resonance phenomena caused by the circulation of trains with speeds exceeding 200km/h. At the present time, there is a comprehensive collection of standards and codes which establish the conditions for passengers’ security stability and comfort criteria that can be guaranteed depending on which railway type is used. For the characterization of the railway effect over the bridge behaviour, simplified models are generally used, allowing the definition of the resilient properties of their component elements or through a simple consideration of the track as a load longitudinally distributed along the deck. In the properties sensitivity study of each type of railway as well as on the comparison of diverse methodologies, a numeric model of finite elements has been developed, representing a continuous viaduct composed by four spans of length 28.40m. The computation of the dynamic response of this case study structure was obtained not only statically but also dynamically. The static analysis was performed using the load model LM71, affected by a dynamic amplification coefficient. The dynamic analyses were performed through moving loads and using various European high speed trains and the HSLM-A models presented on the European codes. Concerning dynamic analysis, the results indicate that the track’s weight influences the structural response; notwithstanding, the resilient properties of the track are shown to influence only the static analyses.

Keywords: Dynamic Analysis; High-speed Railway Bridge; Resonance; Moving Loads; Simplified models of Ballasted Track; Simplified models of Ballastless Track

1. Introduction1

In Portugal, the high-speed railway lines will be a reality in a relatively near future, allowing fast accesses between the various countries in Europe and easing the circulation of people.

Through the future high-speed network, on the top of normal landfills and excavations, various artworks as tunnels and bridges will be set up. On the bridges, the circulating of high speed trains leads to vibration motions that can be prejudicial to the structure if it these effects are not correctly considered at the design stages. Consequently, RAVE, the Portuguese high-speed train network regulator, has to establish support documents that will allow the development of a railway network with better control and performance.

The study of the dynamic effects in bridges has been performed by many authors and the need to guarantee the service limit states on the structure, especially in

——— 1 e-mail: mdsneves@gmail.com

resonance conditions, has pushed forward further research on this area.

The beginning of the research on the dynamic effects in railway bridges is attributed to Willis in 1847 that, through an experimental manner, has formulated an approximate solution of the problem to solve the case of a moving load circulating on a beam and at constant speed. In 1849, Stokes presents one analytic solution to the circulation of a moving mass on a simply supported beam. Afterwards, Timoshenko has realized a notable work, which was the base of many further studies. This author studied the case of a simply supported beam subject to harmonic loads moving at constant speed, which simulated the effects of a multiple axes train circulation. Later on, Inglis presented an innovative approach which characterizes the circulation of a system formed by a suspended and a non-suspended mass, rolling on a simply supported beam.

Frýba [1], presents with great detail a set of analytic solutions which allow the computations of accelerations, displacements and bending moments along a beam with an Euler-Bernoulli behaviour to diverse loading cases, which could vary randomly on time, considering also different support conditions and span lengths.

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More complex formulations came about with the progress of the automatic computations and with the development of the finite element methods. Those formulations started to take into account the interaction between the vehicle and the bridge, allowing a more realistic determination of the dynamic effects, not only on the bridge but also on the circulating vehicle. Nowadays 2D and 3D models are used to model this interaction. 2D models were defined to represent the train made of non-suspended and suspended masses, as presented in studies of authors like Calçada [2], Esveld [3] and by the Specialists’ Committee D214 of the ERRI [4]. In the same context of the 2D space, Yang [5] described models in a 3D space. The same author also presented the conditions for the resonance and cancelling phenomena and defined analytic equations which allow the understanding of these concepts.

Due to the need of establishing the procedures to the design of railway bridges, in the beginning of the 1970s decade, the Office of Research and Experiments (ORE) of the International Union of Railways (UIC) studied the dynamic behaviour in bridges. Later on, the Specialists’ Committee D214, following the studies made by UIC, defined design criteria which allow the adequate dynamic safety of bridges to circulation speeds up to 350km/h. With the aim of standardizing all of these concepts, CEN presented documents such as EN 1991-2 [6].

More recently, various studies have been performed, allowing the identification of the parameters that mostly influence the dynamic behaviour of the structure, also defining the assessment criteria for the consideration of the dynamic effects. Barbero [7], Ruiz [8] and Henriques [9] are examples of such studies.

The circulation of a high speed train requires the compliance of very strict comfort and security criteria and the railway plays a very important role to guarantee these criteria. A traditional ballasted track was adopted in the first high-speed line, later on showing a faster degradation of the track for the circulation of high speed trains. On the occasion of designing the first French high speed line, André Prud’Homme has realized that, to allow an increase of the maximum circulation speeds without the appearance of a important increase in vertical internal forces, with the consequent track degradation, the action should be taken fundamentally in the reduction of two parameters: the non-suspended vehicle mass and the vertical stiffness of the track. The reduction of the first parameter has been obtained with the development of new types of trains. Nevertheless, the second parameter was not always been reduced due to a need to increase the bearing capacity of the infrastructure. In Germany, it has been verified that the increase track stiffness was negative since a faster degradation of the railway was noticed in these cases. Teixeira [10] has made studies trying to determine the optimal parameters which would minimize track degradation and at the same time improve its performance. This author was able to conclude that a

more flexible superstructure should be used to compensate excessive infrastructure stiffness.

In order to understand the effects of the choice of these parameters on the dynamic behaviour of the support structure, the first studies appear in 1867 by Winkler who advocated the modelled track as a supported beam on an elastic ground. Later on, Frýba [1] and Lou [11] used a similar model which also took into consideration the damping of the structure that supports the rail. The fact that the rails were being discretely supported on the sleepers led to the development of simplified models, which modelled the support structure of the track, not in a continuous manner but in a discrete one. Research made by Calçada [2], Man [12], ERRI [4], Hieu [13] and Zhai [14] are good examples of that effort.

With the need to solve some of the ballasted track limitations, in the end of the 1960 decade, a ballastless track and was used for the first time in 1972 along 700 m on the Rheda Station. Nowadays, the Rheda 2000 is a typology largely used in high speed lines.

Similarly to what happened in ballasted track, various authors like Man [12] and Casal [15] used simplified models to characterize the behaviour of this type of track.

2. Design aspects

2.1. Actions

As the train characteristics which circulate on a railway network are not all the same, different dynamic effects come about in each case.

To static loads, research made by the UIC has allowed the elaboration of a load model designed by LM71 and which corresponds to an envelope of different trains. The characteristic values are presented in the Figure 1. This model may also be multiplied by a α parameter which characterizes the traffic type.

Figure 1 – LM71 load model with characteristic load values To dynamic loads, the ERRI [4] presents a series of

models named by HSLM, Figure 2, which were developed taking into account the interoperability of trains with different characteristics, being covered this way, the characteristics of the existing trains, their forecasted evolutions and characteristics that can be demanded to the future trains which will circulate on the high speed lines.

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Figure 2 –HSLM-A model

2.2. Dynamic effects

In the EN 1991-2 [6], various specifications are presented for the design of railway bridges considering the dynamic effects resulting on the circulation of trains.

In a static analysis, the dynamic effects can be considered by affecting the LM71 load model of a dynamic amplification coefficient ( Φ):

For well maintained track

Φ =

1.44퐿 − 0.2

+ 0.82 , com 1.00 ≤ Φ ≤ 1.67 (1)

For track with normal maintenance

Φ =2.16퐿 − 0.2

+ 0.73 , com 1.00 ≤ Φ ≤ 2.00 (2)

In high-speed bridges, the resonance phenomena are not taken into account on a static analysis. To take into account this effect, a dynamic analysis, using the load HSLM models or real trains, should be made. The results are analysed for a range of circulating velocities between 140km/h and 1.2 times the design speed.

Track irregularities can be considered using an interaction model train-track-bridge or considering the coefficient presented on the same norm, that represents the track irregularities (φ'' 2⁄ ).

The structure design should be performed using the most unfavourable analysis.

2.3. Design requirements on a dynamic analysis

An excessive deformation of the bridge can compromise the stability of the train circulation. Such deformation changes the track geometry provoking excessive vibrations on the elements which constitute the bridge and in the train itself, resulting on passengers discomfort. In the EN1990-A2 [16], verifications to the limit states of service are prescribed, defining limits to the deck’s vertical accelerations, displacements on the supports or deformations on the bridge in order to guarantee the circulation safety and the track stability. To guarantee a certain level of passenger comfort, limits regarding the vertical accelerations at the train are also defined.

3. Railway track

In general, the railway allows the guidance of the trains in an economic and safe way and is designed following diverse criteria such as comfort, resistance, construction velocity and maintenance costs.

The railways can be divided into two groups, Ballasted and ballastless tracks, and the better or worse behaviour of these when subjected to train circulation, depends on the layer properties which constitutes them, as well as on their interaction.

The ballasted track, Figure 3, is a solution with more than two centuries of existence and has been used in a vast number of high speed tracks. In the railway bridges, this typology is composed by the superstructure (rails, sleepers and fastening systems) and by the substructure (ballast).

Figure 3 – Ballasted track

The ballastless track was developed in the end of the 1960s and the first solution of this pattern was used along 700m in the Rheda Station in Germany. The Rheda Solution 2000, Figure 4, used nowadays in high speed railway bridges, is made of pre-compressed sleepers embedded on a reinforced concrete layer. The fastening system is made of materials with resilient properties.

Figure 4 – Ballastless track: Solution Rheda 2000

3.1. Ballasted track elements

Besides the fact that this typology is the most commonly used nowadays, the definition of its geometry as well as the properties of the elements used has some variability.

3.1.1. Rails The rails are the first elements in contact with the

vehicle wheels and its main functions are the transmission

Rail

Sleeper Bottom ballast

ShoulderFastening system

RailFastening system

Hidraulically bonded layerReinforced concrete track slab

Bi-block sleeper

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and distribution of the vertical and horizontal forces by the sleepers and guidance of the vehicle wheels. The choice of the UIC60 rail is justified in high speed lines by technical and economical reasons as referred by Gil [17].

3.1.2. Fastening Systems The choice of the type of fastening system depends

essentially of the railway and sleeper type used and the stiffness of the granular layers which support the sleepers. These elements should guarantee a good connection of the rail to the sleeper and the stiffness of the rail pad used has special interest in limiting the stiffness of the railway so as to reduce the dynamic effects resulted by the circulation of the trains. Teixeira [10] states that, in high-speed railways, the stiffness of the rail pads varies between 30 and 500 kN/mm. Despite these values are generally attributed to a certain type of pad, in reality it does not consider the pre-loading effect which has relevant significance in the dynamic properties of the rail pads.

3.1.3. Sleepers The sleepers are elements of considerable stiffness and

allow the distribution of stresses coming from the rail to the layer which supports them. The most common typology is the monoblock sleeper and those elements are generally placed at a distance which can vary between 50 and 70 cm.

3.1.4. Ballast layer The ballast layer is designed in order to guarantee the

capacity of spreading and conveniently transmitting the loads that are transmitted to the supporting structure. In the EN 1991-2 [6], it is referred that, to be assured a good distribution of this stresses without damaging the surface of a bridge or of an eventual ballast mats, it should be used a depth not inferior to 250mm. In general, it is used the depth of 350 mm which allows an efficient maintenance of the track.

3.2. Ballastless track elements

The Rheda 2000 solution, when applied on high speed railway bridges, is made of a Rheda 2000 slab and by a geo-textile. The main function of the geo-textile is to reduce the effects of the interaction between the supporting structure and the Rheda 2000 slab, preventing the degradation between them, due to the cyclic impacts resulting from the train circulation. Casal [15], presents a value of 2E10 KN/m to the vertical stiffness of the geo-textile.

4. Modelling of the railway

The numeric or analytic modelling of the railway has been used together with field experiments to the study of its behaviour as well as its element characterization, its properties and for the comprehension of the vehicle-track--bridge interaction.

Facing the wanted goals, more or less detailed models can be used depending on the needs and its importance to the results which are pretended. In this article, the study was made with 2D simplified railway models.

4.1. Modelling the ballasted track

Two simplified models, based in studies of previous authors, were used aiming at the understanding of the modelling effects of various elements belonging to the ballasted track:

Non-vibrating Ballast model: Calçada [2] and Man [12] used on their studies a model which considers the rails modelled as a beam with an Euler-Bernoulli or Timoshenko behaviour and with such a length that the edge restrictions do not affect the structural behaviour. The sleepers are modelled as suspended masses connected to the beam, in the top, through parallel systems of spring-damper which represents the pads’ properties, and to the subsoil/bridge, in the bottom, through parallel spring-damper systems which represent the ballast properties. In this model, the distance between elements is defined by the spacing between sleepers. The Figure 5 shows the model.

Figure 5 – 2D simplified model of a ballasted track without consider the ballast as a vibrating mass

Vibrating ballast model: The Specialists’ Committee D214 of the ERRI [4], presents a similar model to the one presented beforehand, but which considers the Ballast modelled as suspended masses. These masses are connected to the sleepers, in the top, and to the subsoil/bridge, in the bottom, through systems in parallel of spring-damper which respectively represent the pads’ properties and the connection between the ballast and the bridge subsoil.

Subsoil

Rails dr

Kb Cb

CpKp

Bridge

Sleeper

Ballast

Pad

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Figure 6 – 2D simplified model of a ballasted track and considering the ballast as a vibrating mass

4.1.1. Definition of the properties of the ballast Zhai [14] presents various equations which allow the

calculation of the vibrating mass and the vertical stiffness as a function of the geometry of the ballast layer and the distance between sleepers.

The computation of each vibrating mass that corresponds to the influence of a half-sleeper is related to the attenuation angle (α) as presented in the Figure 7 and expressed by the following equation.

푀 = 휌 ℎ 푙 푙 + (푙 + 푙 )ℎ 푡푔훼 +

43ℎ 푡푔 훼 (3)

In which ρb is the ballast density, hb is the depth of ballast, le is the effective supporting length of half sleeper and lb is the width of sleeper underside.

The vertical stiffness for each vibrating mass of the ballast is calculated following equation (4):

퐾 =

2(푙 − 푙 )푡푔훼푙푛(푙 푙⁄ ) (푙 + 2ℎ 푡푔훼) (푙 + 2ℎ 푡푔훼)⁄ 퐸 (4)

where Eb is the elastic modulus of the ballast. These equations assume that there is no overlapping over the adjacent cone regions of ballast.

Using the equations (3) and (4) the vibrating mass value and the vertical stiffness of the ballast are computed as presented in Table 1, corresponding to the influence of a sleeper.

Table 1– Vibrating mass and vertical stiffness of the ballast corresponding to the influence of a sleeper

Designation Notation Value Effective supporting length of half sleeper le [m] 1.30

Width of sleeper underside lb [m] 0.30

Density of ballast ρb [kg/m] 2.04E+03

Elastic modulus of the ballast Eb [Pa] 1.10E+08

Ballast stress distribution angle α [°] 14.04

Depth of ballast hb [m] 0.35

Vertical stiffness of the ballast Kb [N/m] 3.30E+08

Vibrating mass of ballast Mb [kg] 770.94

Figure 7 – Model of the ballast under one rail support point

4.2. Modelling of the ballastless track

Casal [15], uses in his studies the bi-dimensional model represented in Figure 8 which pretends to model the behaviour of the ballastless track for Rheda 2000 typology. In this model, the rails are represented as beam elements with an Euler-Bernoulli behaviour. The Rheda 2000 slab is modelled as a segmented beam of 6.5 m length and with a Timoshenko behaviour. The connection of this element to the rails and to the bridge’s deck is considered discretely through systems of parallel spring-damper, with the resilient properties of the pads and of the geo-textile, respectively.

Figure 8 – 2D Model representative of the ballastless track used on the

analysis made by Casal [15]

5. Dynamic analysis of a hypothetical high speed bridge

This section focus on the numeric modelling of a hypothetic high speed railway viaduct which roughly corresponds to S. Martinho railway viaduct located in Alcácer do Sal, Portugal. This viaduct, presently under construction, will accommodate rail traffic to speeds of 220 km/h.

The study here presented comprised the numerical (FE) modelling of a segment of the viaduct, made of four spans with individual lengths of 28.4 m and a total length of 113.6 m.

The deck, 13 m wide and with dimensions to contain a dual railway line, is pre-stressed and is connected to two longitudinal beams. This viaduct presents a ballasted track; nevertheless it was considered ballastless on Casal’s research [15].

Subsoil

Rails dr

Kb Cb

CpKp

Bridge

Sleeper

Ballast

Pad

Connection ballast--bridge/subsoil

CbpKbp

KGt CGt

CpKp

Slab Rheda 2000Rails Pads

Geo-textileSubsoil/bridge

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Figure 9 – Cross section of the S.Martinho viaduct

5.1. Finite element model

Considering the interest in comparing the dynamic effects resulting from the adoption of different types of railway tracks, the same numerical model was used on the dynamic analysis performed by Casal [15] with the ballastless track.

The grid model developed to represent the viaduct is made of two longitudinal alignments of frame elements which represent the deck beams, connected to frame elements cross directed which represent the support deck. The diaphragms were also modelled using the same type of elements. In the absence of experimental data, the calibration of the model was realized using another, supposedly more refined, FE model with volume elements. The final model presents 3002 nodes and 3243 frame elements.

The natural frequencies of the viaduct (considering just its self weight) were computde and two vibration modes are graphically represented on Figure 10 and Figure 11.

Figure 10 – First mode shape of the viaduct (f= 4.03 Hz)

Figure 11 – Fourth mode shape of the viaduct (f= 7.04 Hz)

5.2. Dynamic analysis of the viaduct

The dynamic analysis of the viaduct was made considering the ten models HSLM-A defined on the EN1991-2 [6] and real trains ICE2, ETR-Y, EUROSTAR, TGV, TALGO, THALYS and VIRGIN. The analyses were made using the modal superposition method, considering the contribution of all mode shapes up to 30 Hz, as defined on the EN 1991-2 [6]. The envelope of the response was computed using a speed step of 10km/h to all trains and a more refined step on the resonance zones. The range of speeds considered was 140 up to 420 km/h. The irregularities of the track were considered as noticed in the same code.

5.2.1. Dynamic analysis of the viaduct without consideration of the railway models

On this type of analysis, the track was considered as a

uniformly distributed load along the deck. To a ballasted track, the EN1991-2 [6] refers that the

dynamic analyses should be performed considering the minimum density of a clean ballast with a minimum depth and a maximum saturation density to a dirty ballast taking into account the probable increase on depth of ballast layer. On the research made, it was considered a depth of ballast on the sleepers of 350 mm and two different ballast densities ρballast=17 kN/m3 and ρballast = 20kN/m3. On Table 2, the weight considered on the viaduct is presented (double track) to the ballasted track and to the ballastless track considered by Casal [15]. It is noticed that the weight of the ballasted track and of the ballastless track corresponds respectively to 30.7% and 12.3% of the total structures’ weight. The usage of the ballasted track, for being much heavier, may have high costs on the verification to the ultimate limit states.

Table 2 – Weight of the different typologies of railway considered on

the modelled viaduct

Typology Properties Weight (kN/m)

Ballasted track Model 1 (ρballast=17 kN/m3) 95 Model 2 (ρballast=20 kN/m3) 112

Ballastless track Rheda 2000 35

Figure 12 and Figure 13 present the maximum

accelerations and the vertical displacements obtained for the model 1 due to the circulation of the HSLM-A load models. It is verified that the maximum acceleration peak occurs for the HSLM-A7 load model to a circulation speed V=412km/h and with the av=4.38 m/s2. This value is exceed the 3.50 m/s2 limit presented on the EN1991-2 [6] for a ballasted track. On the analysis made with real models, the maximum acceleration occurs for the Virgin train, at a speed of V=415km/h and with a value of av=3.21m/s2 which is lower to the maximum acceleration obtained due to load models HSLM-A. Relatively to the

0.20

0.35

1.00

3.10 1.40 2.00 2.00 1.40 3.10

2.00 2.25 2.15 4.40 2.00

1.970.40

1

13.00

13.15

1.435

2.00

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maximum vertical displacements obtained, it was noticed that the maximum peak occurs on the HSLM-A10 load model to a speed of 365km/h and with a value of δv=11.1 mm. The maximum displacement obtained is lower than the δlim=L/600 limit presented on the EN 1990-A2 [16]. Concerning real trains, the maximum vertical displacement of the structure occurs for the ICE-2, at a speed of 360 km/h, and with a value of δv=7.2mm which is lower to the one obtained to the HSLM-A load models. It was thus concluded that the accelerations and the maximum displacements do not occur necessarily due to the same load model, not even with the same resonance speed and the obtained results using the HSLM-A load models are an envelope of those obtained for the real trains.

Figure 12 – Maximum accelerations along the load path of model 1, due to the circulation of HSLM-A trains to different circulation speeds Figure 14 and Figure 15 show the accelerations and

the maximum vertical displacements obtained in model 2 due to the HSLM-A load models circulation. It was noticed, that comparatively to the results obtained for model 1, the consideration of a heavier track corresponds to a reduction of the maximum vertical accelerations, whereas the maximum vertical displacements remained approximately unchanged.

Figure 13 – Maximum displacements along the load path of model 1, due to the circulation of HSLM-A trains to different circulation speeds

Figure 14 – Maximum accelerations along the load path of model 2, due to the circulation of HSLM-A trains to different circulation speeds It would be expected that the use of a heavier track

would allow a reduction of the maximum accelerations. Notwithstanding, as the resonance peaks occur at lower speeds, it is noticed the appearance of a new resonance peak on the HSLM-A8 load model in a speed of V=416km/h, with an acceleration value of av=4.69 m/s2, which is higher than the maximum value obtained in model 1. The use of a heavier track is not always the best solution to reduce the maximum accelerations on the bridge.

Figure 15 – Maximum displacements along the load path of model 2, due to the circulation of HSLM-A trains to different circulation speeds

Casal [15], on his research, obtained the maximum

vertical accelerations peak with the HSLM-A10 model. As shown in Figure 16, the use of a ballastless track

brings some advantages, since a lighter railway leads to an increase of the speed of the resonance peaks. The variation of the resonance speed between model 2 and the ballastless track corresponds to ∆V = 40 km/h. In the same HSLM-A load model, maximum vertical deck accelerations are increased on the ballastless track. To the same resonance maximum peak, this type of track presents an acceleration of av=4.79 m/s2, which, relatively to the obtained value to model 2, corresponds to an increase of 1m/s2. Despite this fact, it is possible to guarantee the 5m/s2 limit, presented on the EN1991-2 [6].

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

140 180 220 260 300 340 380 420

Acel

erat

ion

(m/s

2 )

Speed (km/h)

A10A9A8A7A6A5A3A4A2A1

2.50 3.50 4.50 5.50 6.50 7.50 8.50 9.50

10.50 11.50

140 180 220 260 300 340 380 420

Dis

plac

emen

ts (m

m)

Speed (km/h)

A10A9A8A7A6A5A4A3A2A1

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

140 180 220 260 300 340 380 420

Acel

erat

ion

(m/s

2 )

Speed (km/h)

A10A9A8A7A6A5A4A3A2A1

2.50 3.50 4.50 5.50 6.50 7.50 8.50 9.50

10.50 11.50

140 180 220 260 300 340 380 420

Dis

plac

emen

ts (m

m)

Speed (km/h)

A10A9A8 A7A6A5A4A3A2A1

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Figure 16 – Comparison of the maximum vertical accelerations obtained on the ballasted tracks and on the ballastless model made by Casal [15], along the load path on the deck and due to the circulation of HSLM-A10

load model On Figure 17, the vertical displacements obtained to

the same HSLM-A load model are compared and it is possible to notice that the use of different railways typologies does not influence the maximum computed values, notwithstanding their occurrence to different circulation speeds.

Figure 17 – Comparison of the maximum vertical displacements obtained on the ballasted tracks and on the ballastless model made by

Casal [15], along the load path on the deck and due to the circulation of HSLM-A10 load model

5.2.2. Dynamic analysis of the viaduct considering the railway models

Table 3 present the parameters used on the study of

the influence of the various belonging elements of the ballasted and ballastless track on the dynamic behaviour of the structure. The geometry of the ballasted track as well as the ballast properties, are related to the properties of model 2.

Table 4 correspond to figures of the 10 first natural frequencies of the structure with and without the consideration of the ballasted track models. Minimum variations on the values of each vibration mode were noticed and on the first mode shapes this variation is barely perceptible. Such results indicate that the consideration of more refined railway models do not significantly affect the global structural behaviour.

Analysing Figure 18 and Figure 19, it is of clear understanding that exists a superposition of the results obtained not only for the maximum vertical accelerations but also for the maximum vertical displacements along the load path to the considered speed range. It is concluded, though, that the consideration of the ballasted track models does not change the structural response when dynamic analysis are limited to lower than 30 Hz mode shapes. To the viaduct modelled, with a span length of L=28.4m, the distribution of the loads along various sleepers resulting from the track models, does not contribute to the acceleration reduction nor to the structures’ displacements.

Figure 18 – Comparison of the maximum accelerations with and without railway models, along the load path on the deck, due to the circulation of

an HSLM-A8 load model

Figure 19 – Comparison of the maximum displacements with and without railway models, along the load path on the deck, due to the

circulation of an HSLM-A8 load model

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

140 180 220 260 300 340 380 420

Acel

erat

ion

(m/s

2 )

Speed (km/h)

Model 1Model 2Ballastless track

3.50

4.50

5.50

6.50

7.50

8.50

9.50

10.50

11.50

140 180 220 260 300 340 380 420

Dis

plac

emen

ts (m

m)

Speed (km/h)

Model 1Model 2Ballastless track

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

140 180 220 260 300 340 380 420

Acel

erat

ion

(m/s

2 )

Speed (km/h)

Model 2

Vibrating ballastNon vibrating ballast

2.50

3.50

4.50

5.50

6.50

7.50

8.50

9.50

140 180 220 260 300 340 380 420

Dis

plac

emen

ts (m

m)

Speed (km/h)

Model 2

Vibrating ballastNon vibrating ballast

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Table 3 – Adopted parameters on the ballasted track models and on the ballastless track model used by Casal [15]

Railway parameters Notation Value

Non vibrating ballast

Vibrating ballast

Ballastless track

Rail UIC60

Section Sr [cm2] 76.86 76.86 76.86 Mass mr [kg/m] 60.34 60.34 60.34 Elasticity modulus Er [GPa] 210.00 210.00 210.00 Poissons’ coefficient µr [-] 0.30 0.30 0.30 cross-sectional inertia XX Ixx [cm4] 3050 3050 3050 cross-sectional inertia YY Iyy [cm4] 515.6 515.6 515.60

Pad Vertical stiffness Kp (MN/m) 651 65 100 Vertical damping Cp (kNs/m) 5.501 5.50 15

Sleeper Mass Ms [kg] 300 300 - Sleeper spacing ls [m] 0.60 0.60 0.65

Ballast Vertical stiffness Kb [N/m] 3.30E08 3.30E08 - Vertical damping Cb [Ns/m] 1.20E052 1.20E05 - Vibrating mass Mb [kg] - 770.94 -

Connection Ballast/deck Vertical stiffness Kbp [N/m] - 1000E062 - Vertical damping Cbp [Ns/m] - 50E032 -

Slab Rheda2000

Specific weight ρL [kN/m3] - - 26.37 Width x height Lxh [mxm] - - 2.60x0.24 Elasticy modulus EL [GPa] - - 34.00 Poissons’ coefficient µL [-] - - 0.20

Geo-textile Vertical stiffness KGt [MN/m] - - 1.3E10 Vertical damping CGt [kNs/m] - - 1.3E07

Table 4 – Comparison of the vibration frequencies of the viaduct with and

without considering the ballasted track models

Vibration mode

Vibration frequency [Hz] 2 i

Model 2 Vibrating ballast

Non vibrating ballast

1 3.058 3.059 3.059

2 3.600 3.600 3.600

3 4.653 4.653 4.654

4 5.132 5.146 5.147

5 5.222 5.236 5.237

6 6.078 6.079 6.079

7 6.383 6.397 6.397

8 6.736 6.749 6.750

9 9.719 9.719 9.720

10 10.412 10.411 10.411

Table 5 shows the reduced variation of the vertical

acceleration along the various belonging elements of the ballast track. The maximum variation of the accelerations obtained between the rail position and the deck position is lower than 0.5%.

——— 1 Values adopted by Kaewunruen [18] for a 6.5 mm rubber pad 2 Values adopted by ERRI [10] in a simplified 2D ballasted

model

The representation of the response of the viaduct on the frequency domain allows a better understanding of the frequencies and modes that are more relevant to the final response of the structure. The results presented in Figure 20 have been obtained using the FFT (Fast Fourier transform) algorithm. The value of the excitation frequency which mostly contributes to the overall behaviour corresponds to 4.66 Hz and originates a resonance phenomenon with the third vibration mode. The consideration of the vibrating ballast model does not contribute to the response dissipation for a range of frequencies up to 100 Hz.

Table 5 – Comparison of the accelerations with and without the railway models towards various nodes, at loading structure position at 102.40m,

and due to the circulation of the HSLM-A8 load model at a speed of V=415km/h

Track’ position

Aceleration

Model 2 Vibrating model

Non vibrating model

[m/s2] [m/s2] [m/s2]

Rail - 4.749 4.736

Sleepers’ mass - 4.746 4.733

Ballast’ mass - 4.740 -

Deck 4.693 4.733 4.726

In fact, calculating the vibration natural frequencies of

the ballasted track, assuming a simplified SDOF model

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graphically represented on Figure 21, a fundamental frequency of vibration of 115Hz has been computed, thus largely exceeding the 30 Hz limit of the vibration mode shapes to be considered in the dynamic analyses. Barbero [7] has reached the same conclusions and notices that the railway vibrated jointly with the bridge in case the vibration frequencies of the track would be much superior to the bridge vibration frequencies.

Figure 20 – Comparison of the accelerations in the frequency domain with and without the railway model, at the loading structure position at 102.40m, and due to the circulation of the HSLM-A8 load model at a

speed of V=415 km/h

Figure 21 – Model with three degrees of freedom representative of the behaviour of the Ballasted track

The dynamic analyses considering a more flexible

ballasted track were also considered. Nevertheless, the obtained results did not show different variations.

Casal [15], has reached very similar conclusions on the dynamic analyses performed with ballastless track. As shown in Figure 22, the acceleration values computed using a ballastless track model present slight variations in the resonance peaks relatively to the results obtained on the viaduct without considering the track model. Such variations are not owed to the track vibration but to the stiffness increment of the global structure due to the increased flexural stiffness resulting from Rheda 2000 slab and rail.

Figure 22 – Comparison of the maximum vertical accelerations with and without the ballastless track model, along the loading path and due to the

circulation of a HSLM-A10 load model, Casal [15]

5.3. Static analysis of the viaduct considering the railway models

The verification of the passengers comfort may be performed using the LM71 load model without the consideration of the dynamic amplification factor computed through equations (1) and (2).

Table 6 shows that in a static analysis, the railway consideration does influence the obtained results. To the dynamic properties of the ballasted track considered in the dynamic analysis, it is noticed a vertical displacement variation between the rail and the deck position of Δδ=13.4% and the pads are the elements which more influence this variation. If a vertical stiffness Kp= 25 MN/m is considered at the pads and a more flexible ballast, the displacement difference, between the rail and the deck corresponds to Δδ= 23%.

Considering a design speed of 350km/h and using the abacus presented in the EN1990-A2 [16], a L/δlim=1558 relation is computed indicating a “very good” comfort level. In this case, to a continuous structure, this value should be multiplied by 0.9. The relation L/δ at the rail level corresponds to a figure of 2653 which is superior to the recommended, being thus guaranteed, the comfort level considered.

Table 6 – Maximum displacements obtained at the middle of the 4th span, on the loading path, due to the LM71 loading model affected out of the

dynamic coefficient φ

Track’ position

Vibrating ballast model

Value ∆δ L/δ

[mm] [%] -

Rail 10.70 13.4 2653

Sleepers’ mass 9.87 4.6 2876

Ballast’ mass 9.55 1.1 2975

Deck 9.44 0.0 3009

-90

-70

-50

-30

-10

10

30

0 10 20 30 40 50 60 70 80 90 100

Ampl

itude

(dB

)

Frequency (Hz)

Model 2

Vibrating ballast

q1

q2

q3

Rail

Sleeper

Ballast

M3

M2

M1

K3

K2

K1

0.00

1.00

2.00

3.00

4.00

5.00

6.00

140 180 220 260 300 340 380 420

Acel

erat

ion

(m/s

2 )

Speed (km/h)

Without railway model

With railway model

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6. Conclusions

At the present work, some issues related to the safety verification of high speed railway bridges were presented as well as the effect of considering railway models in its dynamic behaviour.

The amplification of the railway bridges response due to the dynamic effects resulting from resonance phenomena has been proven to be determinant to bridge design. The main parameters involved on the structural dynamic behaviour correspond, on the one hand to the type of train (axles, axles’ spacing and loads) and the corresponding circulation speed range, and on the other hand to the structural characteristics, namely the vibration frequencies, mode shapes and damping.

The consideration of various HSLM-A models when compared to the various existing real trains has allowed the conclusion that these high speed load models cover the range of effects induced by existing real trains, correctly leading to envelopes of all relevant effects (e.g.: deck acceleration and vertical displacement).

The choice of a certain railway typology influences the dynamic behaviour of the structure. Considering a heavier track modelled as a load distributed along the deck, the maximum computed accelerations are lower and the resonance peaks occur at lower speeds. The fact that the first structural resonance phenomena occur at lower speeds, may generate the occurrence of new resonance peaks within the considered speed range, possibly leading to other negative effects. The use of a ballasted track, generally leads to lower resonance speeds (and dynamic effects on deck), which does not necessarily imply an advantage, since the permissible deck accelerations are also lower due to the possibility of the instability of the ballast layer.

The use of bi-dimensional railway models, has pointed to the conclusion that the explicit consideration of diverse typologies of railway track does not significantly change the results relevant to bridge deck design and verification, when these are compared to more simplified models without the explicit consideration of the track, particularly when the 30 Hz limit is considered for the mode shapes to be included in the analyses. A variation of 0.5% only in the vertical deck acceleration has been obtained in the ballasted track models, thus showing that there is a negligible attenuation of the dynamic effects within the ballast layer.

Pertaining only to the viaduct under analysis (span length of L=28.4m), it has also been concluded that the distribution of the load per axis along the sleepers (computed with the use of explicit railway track models) does not contribute to the reduction of the structural response.

The fact that no significant differences have been noticed in these ballasted models can be explained by the railway track behaving as a SDOF vibrating sub-system. Considering this assumption, the first vibration mode shape of the track as a whole corresponds to a frequency of the

115 Hz, which is much higher than the 30 Hz limit considered on the dynamic analysis, implying that railway track vibrates jointly with the bridge and that these (local) mode shapes of the track are not excited.

The static analysis performed for the verification of the passengers comfort level show that the stiffness of the railway influences the computed displacements at the rail level. It has also been noticed that the stiffness of the pads was the property which greatly influences the vertical track displacements, and that the use of excessively flexible pads may reduce the comfort level.

Acknowledgments

The studies reported in this article were developed within the activities of the Civil Engineering and Architecture Department (DEcivil) of the Instituto Superior Técnico (IST), TU Lisbon, Portugal.

The author would like to express the sincere acknowledgments to Professor Jorge Miguel Proença by the supervision and all the support given throughout the research, to Hugo Casal due to all the elements supplied throughout the comparative study of the different railways typologies, and to Professor António Reis and Eng. Nuno Lopes due to the supply of all the design elements regarding S. Martinho’s viaduct.

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