Predicting Settlements of Ballasted Tracksdue to Voided SleepersKira Holtzendorff, Technical University BerlinUlf Gerstberger, Technical University Berlin

1 Introduction

On high speed lines, quasi-periodic void pattern (see Fig. 1) due to ballast settlement can beidentified where a predominant wavelength of about 4 m can sometimes be observed. Precisefield measurement data that confirm the existence of those void patterns are not available.However, laboratory experiments conducted by Gudehus et.al. [4] could explain themechanism of how the “track waves” grow. Their results show that irrespective of the initialirregularities in the vehicle-track system such as varying ballast densities, different subgradeproperties, out-of-round or flat wheels, certain wavelengths develop that seem to be inherentfor given system parameters.

Quasi-periodic void pattern of a ballasted track

A model of vehicle-track interaction in the frequency domain is presented which enables thedetermination of the dynamic wheel loads that are generated by the quasi-periodic voiddistribution. The results obtained from different types of ballasted tracks – conventional,broad and frame sleepers – are discussed.After some information about the settlement mechanism of ballast due to long term dynamicloading of numerous train passages, the predominant influencing parameters are introduced.Based upon that, an equation predicting differential ballast settlement is proposed which canbe considered as a modification of one of the classic phenomenological equations given inreferences by Hettler [7], Shenton [18] and summarised in [9].Finally, the development of the settlement below each sleeper after N load cycles iscalculated. First examples are presented where the wavelength is varied.

At present, measurement data from real tracks are rarely available. Hence the resultspresented here are based on a detailed literature review. Therefore we do not claim theprediction of settlements presented here to be quantitatively realistic. We have chosen areference track with a reference excitation and have scaled other results to it in order to beable to assess either different track structures or the type of excitation.

If not mentioned otherwise, the following parameters are used for the dynamic simulations:

Vehicle velocity: 250 km/hStatic wheel load 75 kN (ICE car body)Subgrade stiffness modulus 432 MN/m2 (stiff, cs=300m/s)Ballast layer thickness 30 cmPad stiffness kp 1.5·10

8 N/m (Zw 700, stiff)Pad structural damping ·kp = 0.13·kpRail UIC 60Load distribution angle (ballast) 15 ° degreesMaximum void z0 1mm

2 Chosen track types

Three different types of sleepers for ballasted tracks are selected:

1. conventional sleepers (B75)2. broad sleepers (BSS)3. frame sleepers (RS 95),

where the first type is chosen as a reference track for the prediction of settlement in chapter 9.The track structures are depicted in Fig. 2 and the governing geometry data are given in Table1. For further information about frame sleepers see [3,16]

Types of ballasted tracks, [11]

Track type 1. B75 2. BSS 3. RS 95

Sleeper length [m] 2.80 2.40 2.60

Sleeper width [m] 0.33 0.57 -

Spacing [cm] 0.6 0.6 1.0

Bearing area [m2] 0.759 1.026 ~ 1.197

Bearing area [m2/m] 1.265 1.71 ~ 1.197

Table 1: Geometry data for three different types of ballasted tracks

3 Model description

Initially the dynamic wheel loads due to the excitation by voided sleepers are determined.Here a complete vehicle-track model is used containing a vertical model of an ICE car bodythat is described in Fig. 3.

Twodimensional vehicle model with 10 degrees of freedo

The track model with conventional or broad sleepers which is shown in Fig. 4 consists of therail (modelled as a Timoshenko beam), sleepers (rigid masses), ballast (ballast blocks, linearor nonlinear material behaviour) and the subgrade (homogeneous halfspace). The overlappingcoupling over two neighbouring sleepers is taken into account.

Model of a conventional ballasted track according to Knothe/Wu[12,13]

The model of the third track type can be seen in Fig. 5. The rail is continuously supported andmodelled as a Timoshenko beam. The frame sleeper is modelled as a rigid mass neglectingbending and torsional deformations, allowing a vertical and a pitching degree of freedom. Thecoupling between two neighbouring sleepers via the subgrade is taken into account. Themodelling of the ballast and subgrade is similar to that of conventional sleepers.

Model of a ballasted track with frame sleepers [3]

0 50 100 150 200 250 300

0,0

2,0x10-9

4,0x10-9

6,0x10-9

8,0x10-9

1,0x10-8

1,2x10-8

1,4x10-8

1,6x10-8

1,8x10-8

B75

BBS

RS95

rece

ptan

ce [m

/N]

frequency [Hz]

In order to give an idea of the different dynamic properties of the three track models, theirreceptances are plotted in Fig. 6. The frequency range of main interest in this paper is between8 and 60 Hz, that is equivalent to L = 1.2 – 9.0 m. At low frequencies the receptancesbasically stay constant as the inertia forces of the track are not yet relevant. The higher valuesof the receptance of the frame sleeper track is due to the sleeper pads.

Fig. 6: Dynamic receptances of all three track types

The linear vehicle-track-interaction analysis is carried out in the frequency domain whereasthe nonlinear sleeper/ballast contact condition as well as a possible nonlinear deformationbehaviour of the ballast are taken into account during a preceding filtering process. It can beshown that nonlinear ballast behaviour, the vehicle velocity as well as the wavelengh of thequasi-periodic void distribution significantly influence the magnitude of the dynamic wheelforces (cf. ref. [8]).

4 Analysis of dynamic wheel forces

In Fig. 7 the dynamic wheel forces Q for the chosen track types are plotted with respect tothe wavelength of the irregularity. The wavelengths are multiples of the sleeper spacing. Thewheel forces for type 1 and 2 appear to be almost exactly the same. Q for track type 3 iscomparatively low due to the fact that the track receptance in the low frequency range issignificantly higher than for type 1 and 2. Hence the track structure is softer which in turngenerates lower dynamic forces. Here wavelengths L < 2 m are omitted as the spacing is 1 mand therefore the minimum L equals 2 m meaning that every second sleeper is voided.In contrast to the excitation by out-of-round wheels the frequency range considered here isvery low; it varies between 8 and 60 Hz. Consequently the dynamic wheel forces arerelatively low and do not differ much from each other.

0 1 2 3 4 5 6 7 8 9 10

2

4

6

8

10

12

14

16

18

20

B75

BBS

RS95Q [k

N]

wavelength L [m]

Fig. 7: Dynamic wheel forces Q over wavelength L for three track types

5 Mechanism of ballast settlement

After the built-in of the ballast layer and the following stabilising procedure, the ballast can beconsidered as a stabilised material where the particles are locked against each other. Wedistinguish between two basic mechanisms - compaction and particle rearrangement. Initiallysettlement occurs due to a compaction process during the first load cycles. However, after aprecise built-in procedure one can expect this plastic ballast deformation to be minor. Thesecond effect of particle slip (lateral flow) can occur during the whole time of operation. Herethe layer becomes destabilised by exceeding certain levels of stress or vibrations induced bythe passing train. Both mechanisms are described in Fig. 8. According to Selig/Waters [17]the range of slip strongly depends on the horizontal stress 3 that had been built up earlier, butgradually diminishes during the vibration impact. The lower the horizontal stress, the morelikely it is that the particles will slip against each other.

Settlement mechanism in the ballast layer

6 Parameters influencing ballast settlement

6.1 Deviatoric stressFor a certain level of the deviatoric stress 1- 3 the friction resistance at the contact pointsbetween the ballast particles will be exceeded. Consequently, the particles start to move androtate against each other. A tendency of horizontal spreading of the ballast layer can beobserved. According to Selig/Waters [17] this effect occurs when the applied deviatoric stressexceeds a limiting value that is found in the vicinity of the stress ratio at failure of the ballastfabric.

6.2 VibrationsSelig/Waters [17] claim the horizontal stress 3 to be crucial with respect to the amount ofballast settlement. Initially, the degree of the horizontal stress below the sleeper is relativelyhigh as the layer is confined from the cribs and shoulders and the material adjacent to theloaded area. When dynamically loaded by the passing trains, the stress decreases due to thevibrations and the particles start to flow in lateral direction.

6.3 DegradationParticle abrasion and breakage is caused by particle slip, tamping actions and from differentenvironmental sources. The voids of a fouled ballast become progressively filled with fineparticles. As a consequence, the damping degree of the material increases which in turn leadsto an increased radial receptance of the material, cf. ref. [6]. Hence the stability of the ballastfabric decreases and the particles start to spread horizontally.

6.4 Subgrade stiffnessThe stiffness of the supporting subgrade clearly influences the quantity of ballast settlement.The more the subgrade deforms elastically the more the ballast particles are able to move intodifferent directions. Raymond/Bathurst [15] have shown results from experiments where theballast layer in a box is put on an elastic support with different degrees of compressibility. InFig. 9 a dramatic increase of plastic deformation of the ballast layer on soft subgrade can beidentified. This effect was confirmed by Guerin et. al. [5]. Based on their experiments atreduced scale they developed a settlement equation that depends on the elastic deflection ofballast together with the elastic support during the loading.

Influence of ballast support compressibility on ballast deformation, [15 ]

7 Settlement equation

The prediction of ballast settlement means the prediction of the plastic deformation of theballast layer. As the ballast is a coarse material, the application of a plastic constitutive lawfor soils - modelled as a continuum - is questionable. In the past many authors (cf. [9]) havedeveloped empirical formula to predict the approximate mean settlement of a track. Thephenomenological equation proposed here is a modification of the formulae in the literature.All of them depend on the number of load cycles or gross tonnage. Some depend on thedynamic forces applied in the system, the stresses in the ballast layer or e.g. the initialporosity, vehicle velocity and subgrade stiffness.Fröhling [2] developed a phenomenological settlement equation that predicts the individualsettlement of the substructure (ballast and subgrade) below each sleeper. For the settlementcalculation measurement data of the stiffness of the substructure is required. In addition, thegoverning dynamic wheel forces are calculated. The idea of calculating the individualsettlement below each sleeper is adopted here. It is aimed to study how voids of different sizesdevelop in the long term. However, we do only consider the ballast settlement not includingsettlements of the subgrade.As measurement data is rare, a settlement equation is required that is not necessarily basedupon that, but still allows us to learn about the evolution of differential settlements withrespect to

type of excitation (wavelength, amplitude), track type, different influencing parameters.

Likewise experimental or measurement data about the evolution of horizontal stress in theballast layer is hardly available and the applied vehicle-track interaction tools are unable tocalculate them. For that reason some simplifications are made with respect to thedetermination of the parameters in chapter 6.

The settlement equation will be of the form

Esubdegdynbe0iiN I,I,I,,Nlogfzz . (1)

It describes the settlement of the ballast below sleeper i depending on the given voidamplitude zi0 at the time considered, the number of load cycles N, the vertical, equivalentstress be in the ballast layer, the dynamic factor Idyn, the degradation factor Ideg, and thesubgrade stiffness factor IEsub.Applying the vertical stress in the ballast layer b instead of the stress ratio 1/ 3 implies thatit can be considered as a measure of stress at failure as well. At the same time the dynamicfactor Idyn takes account of the fact that dynamic effects such as the ballast accelerationreduces the horizontal stresses. be is the equivalent vertical stress that relates b to a realisticload collective of a track. It is based on the fact that the heaviest wheel load Q0 is the one thatpredominantly influences the quantity of settlement. The equation for be is given in chapter8.1.The degradation factor Ideg refers to the present quality of the ballast material.

8 Determination of influencing parameters

8.1 Vertical stress in ballast layerBased on the dynamic wheel loads that are calculated with the vehicle-track interaction modelin the frequency domain, the force S between the sleeper and the ballast can be determined.The vertical stress b is calculated by dividing S by the bearing area of the sleeper. In Fig. 10

b is plotted against the wavelength of the periodic irregularity for the three track typesconsidered. The vertical stress is plotted either for sleepers with maximum or zero void.Obviously the smaller the wavelength is, the lower the stresses at voided sleepers and thehigher the stresses at sleepers with no voids. By comparing b for type 1 and 2, it can bestated that the stress level of type 2 is reduced due to the increased bearing area of the broadsleepers. The frame sleeper track (type 3) guarantees a higher stress distribution as it behavesmore softly in the low frequency range, cf. receptances in Fig. 6.

1 2 3 4 5 6 7 8 90

10

20

30

40

50

60

b) max. void

B75

BBS

RS95

b [k

N/m

2]

wavelength L [m]

1 2 3 4 5 6 7 8 930

40

50

60

70

80

90

100

110

120a) zero void

B75

BBS

RS95

b [k

N/m

2

]

wavelength L [m]

Fig. 10: Vertical stress in ballast at sleeper of a) zero and b) maximum void (z0 = 1mm)

The results show that not only the level of vertical stress in the ballast layer affects theevolution of differential settlements below the sleepers as then the voids would “move”towards the sleepers where the maximum stresses appear – these are generally the onesinitially not being voided. However, this has not been confirmed in practise and gives rise tofurther investigate the influence of the horizontal stresses, see chapter 8.2.Shenton [18] developed an equation where the equivalent vertical stress be can be related to arealistic load collective:

2,0

321

35

3b25

2b151b

be ...NNN...NNN

(2)

8.2 Dynamic factor IdynAn analysis with a vehicle-track-interaction model in the time domain [10] was carried out toexamine the correlation of the parameters

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

B75

B75 z0=1mm

BBS

BBS z0=1mmat [m

/s2]

wavelength L [m]

Acceleration of the sleeper at Amplitude z0 and wavelengh L of the periodic track irregularity.

The influence of at on the level of horizontal stresses is assumed to be crucial as described inchapter 7. Frame sleepers have not yet been modelled in the time domain including nonlineareffects.

Fig. 11: Acceleration at sleeper with maximum and zero void, track type 1 and 2

As expected Fig. 11 shows that the existence of voids significantly affects the level ofacceleration of the sleeper. In the frequency range between 15 and 70 Hz (equalsL = 1.2 – 9.0 m) the track receptances (see Fig. 6) differ to an extent resulting in differentcurves for type 1 and 2. Fig. 12 confirms that the maximum acceleration increases inproportion to the increasing voids’ amplitude z0. The amplitude of at clearly varies over thewavelength. This must not necessarily result from dynamic properties of either track orvehicle but from the fact that the sleeper where at has been calculated lies either exactly at the“bottom” of the wave or shortly before or after. Further investigation is required.

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

at (z0max=1mm) at (z0max=1.5mm) at (z0max=2mm)

at [m

/s2]

wavelength L [m]

Fig. 12: Sleeper acceleration at over wavelength L for varying maximum voids amplitudes z0

The factor Idyn can be determined as a function of the above mentioned parameters at, L andz0. Further information will be given in the lecture.

8.3 Ballast degradationGuldenfels [6] investigated the influence of the degree of ballast fouling on mechanicalproperties of the ballast. ERRI defines the degree of ballast fouling as the percentage of thematerial passing the 22.4 mm sieve. According to Guldenfels the behaviour of the ballastsignificantly changes when approaching 50 to 70% of degradation. Generally, the replacementof the ballast material on high speed lines is proposed when exceeding 30% of degradation.Very little information about the deterioration of the ballast with respect to the passing grosstonnage is given in the literature. Therefore the data given here can only be considered as anexample of how to incorporate the quality of the ballast material into a settlement equation.Eisenmann and Mattner [1] conducted laboratory tests of cyclic loaded ballast and determinedthe gradation before and after. The graph presented in Fig. 13 is derived from there. As theyonly applied 1.75·106 load cycles (equivalent to about 18 month operation on German highspeed lines) an extrapolation had to be done assuming that the increase of degradation isnonlinear and the ballast will be replaced at 30% of degradation after about 15 years. Theseassumptions roughly coincide with measurement data from Deutsche Bahn tracks given in ref.[14].

0

5

10

15

20

25

30

35

40

45

0 50 100 150 200 250 300

gross tonnes [x 106]

Deg

rad

atio

n [

%]

Fig. 13: Idea of ballast degradation with respect to passing gross tons

The ballast degradation does not only depend on the passing gross tons but also on themagnitude of the void at the sleeper considered. The ballast fouling process accelerates atlarger voids. Once the required measured data of ballast degradation over time is available, anamplification factor Ideg could be formulated.

8.4 Subgrade stiffnessA first proposal of the subgrade stiffness factor IEsub is as follows:

stiff Medium soft1.0 1.1 1.2

Table 2: Stiffness factor IEsub

From Raymond & Bathurst [15] a nonlinear relationship between subgrade stiffness andballast settlement can be derived. However, in Table 2 we propose a linear relationship as theeffect of subgrade stiffness cannot be investigated without taking into account the othersettlement parameters. For example, the calculation of the vertical stresses b as well as theassessment of the dynamic factor Idyn is also based on the subgrade stiffness. Here the factorIEsub accounts for the additional effect of either restricted ballast particle rearrangement onstiff subgrade or increased rearrangement on soft subgrade.

9 Prediction of settlements - examples

In the preceding chapters a model for settlement prediction was presented. For a reasonabledetermination of the settlement parameters more measurement data are required. Thereforethe stiffness of subgrade as well as the degree of ballast degradation is held constant in thefollowing examples. At first, we choose a reference track with a reference void pattern:

sleepers of type B75 (track type 1) max. void amplitude z0 = 1mm wavelength L = 3.6 m stiff subgrade no ballast degradation (clean)

load collective => only ICE car bodies

In the following Figures 14 to 17 the influence of the wavelength of the quasi-periodic voidexcitation on the development of differential ballast settlement is studied.In Fig. 14 the development of settlements below each sleeper after 1·106 load cycles for thereference case is presented. In the plot the uniform settlement of the whole track is omitted sothat only differential settlements are depicted. At a wavelength of L = 3.6 m the influence ofthe sleeper acceleration dominates compared to that of the vertical stresses. This can bederived from the fact that the settlement ziN accumulates most at the sleepers with maximumvoids.

Initial and differential settlement after N=1·106 cycles, L = 3.6 m (reference case)

A contrary case is depicted in Fig. 15 where the settlement for a given wavelength ofL = 1.2 m is plotted. As the voided sleepers do not hit the ballast surface while the trainpasses, vibrations are not induced into the ballast material. Hence only the vertical stressestransmitted at the sleepers with no voids generate the settlement. In this case the amount ofdifferential settlements decreases. This leads to the assumption that this void pattern woulddisappear during the operation.

Initial and differential settlement after N=1·106 cycles, L = 1.2 m

The next example is given for L = 4.8 m in Fig. 16. The same effect as for L = 3.6 m can beidentified. Similarly, the voids grow where the maximum sleeper acceleration is induced intothe ballast layer, but as a whole less differential settlement develops.

Initial and differential settlement after N=1·106 cycles, L = 4.8 m

Finally, the development of differential settlement for L = 6.0 m is given in Fig. 17. Only anegligible amount of differential settlement has developed after 106 load cycles. Little sleeperacceleration is generated for long wavelengths irrespective of the voids’ amplitude. At thesame time the level of vertical stress transmitted by the sleepers does not vary much from onesleeper to the next. Therefore mainly uniform settlement of all sleepers is generated.

Initial and differential settlement after N=1·106 cycles, L=6.0m

10 Conclusions

The vehicle-ballasted track system is excited by a quasi-periodic void distribution. Based on ashort term dynamic analysis of vehicle-track-interaction, the vertical stress in the ballast layerand the dynamic wheel forces are compared for three different types of sleepers. The level ofvertical stress depends on the specific void pattern along the track and the frame sleeperguarantees the best distribution of stresses among the sleepers. Vertical stress, sleeperacceleration (as a measure of horizontal stress), ballast degradation and the subgrade stiffnesshave been identified as the predominant parameters influencing the differential settlement ofballast. It could have been shown that the sleeper acceleration strongly depends on thewavelength of the excitation.A phenomenological settlement equation is proposed that predicts the long term differentialsettlement below each sleeper. Generally it can be confirmed that at shorter wavelengths(L 1.8 m) the influence of the vertical stress be onto ballast settlement becomes morerelevant as the voided sleeper hardly hits the ballast surface - then sleeper acceleration at islow - due to the bending stiffness of the rail. Increasing L we can identify a range ofwavelength (1.8 L 4.8 m) where at reaches its maximum (assuming that the horizontalstresses reach their minimum at the same time) and therefore dominates the development of

ballast settlement at voided sleepers. Further increasing of the wavelength leads to a betterdistribution of the vertical stress along the track as well as the acceleration or horizontalstress, respectively, so that differential settlement diminishes. In Fig. 18 the growth or thereduction of differential settlement after N=106 depending on the wavelength is indicated. ForL 3.0 m differential settlement diminishes and increases for 3.0 L 5.4 m whileremaining the same for L 5.4 m.

Normalized differential settlement with respect to the initial differential settlement afterN=106 for wavelengths 1.2 L 6.0 m

More measurement data referring to

the evolution of horizontal stresses in the ballast layer, the degradation of the ballast material with respect to the number of load cycles, void distributions and amplitudes,

is required. Then the influencing parameters could be quantified and the procedure ofpredicting differential ballast settlement be adjusted.

11 References

[1] Eisenmann, J., Mattner, L.: Settlement behaviour of ballast at high vehicle speed (inGerman), Research Report No. 1245 of Prüfamt für Bau von Landverkehrswegen,Technical University of Munich, 1988

[2] Fröhling, R.D.: Deterioration of railway track due to dynamic vehicle loading andspatially varying track stiffness. PhD-Thesis, University of Pretoria, 1997

[3] Gerstberger, U.: On the dynamics of frame sleeper tracks (in German), slidesof a lecture given at the Technical University of Berlin, February 2001,http://www.ice.fb12.tu-berlin.de

[4] Gudehus, G.: Evolution and tackling periodic track waves (in German), Proceedings ofBaugrundtagung, Stuttgart 1998

http://www.ice.fb12.tu-berlin.de

[5] Guerin, N., Sab, K., Moucheront, P.: Experimental identification of a settlement lawfor ballast (in French), Canadian Geotechnical Journal, Vol. 36, No.3, June 1999

[6] Guldenfels, R.: Ballast degradation with respect to soil mechanics (in German), PhD-Thesis, Publications of the Geotechnics Institute at ETH Zürich, Vol. 209, 1996

[7] Hettler, A.: Triaxial tests for ballast - static and cyclic loading (in German),Eisenbahntechnische Rundschau, Issue 36, 1987, p. 399-405

[8] Holtzendorff, K.: Dynamic investigations of ballast loading with modified modellingof ballast behaviour (in German), Der Eisenbahningenieur, Issue 3, Vol. 52, p. 44-47,2001

[9] Holtzendorff, K.: Investigation of ballast settlement and modelling (in German), InProceedings of a IFV-Symposium „Schotteroberbau (ballasted tracks)“. Berlin,February 1999

[10] Knothe, K., Wu, Y.: Simulation of a train passing over a discrete ballasted track (inGerman), VDI-Fortschritt-Berichte, Reihe 12, No. 412, Düsseldorf 2000

[11] Knothe, K.: Track dynamics (in German), Ernst & Sohn, Berlin, 2001[12] Knothe, K., Wu, Y.: Modelling subgrade as a halfspace for the calculation of vehicle-

track-interaction (in German), Mitteilung aus dem Institut für Luft- und Raumfahrt, TUBerlin, 316 (1997)

[13] Knothe, K., Wu, Y.: Receptance behaviour of railway track and subgrade, Archive ofApplied Mechanics, 68 (1998), p. 457-470

[14] ORE: Assessment criteria for ballast quality and ballast layer condition, QuestionD 128, Report No. 2, Utrecht, 1991

[15] Raymond, G.P., Bathurst, R.J.: Performance of large-scale model single tie-ballastsystems, Transportation Research Record, Issue 1131, 1987, p. 7-14

[16] Rießberger, K.: The frame sleeper track – an innovative ballasted track (in German),Eisenbahntechnische Rundschau 49 (2000), Issue 3, p. 126-136

[17] Selig, E.T., Waters, J.M.: Track geotechnology and substructure management, ThomasTelford, London, 1994

[18] Shenton, M.J.: Ballast deformation and track deterioration, in track technology.Proceedings of a Conference organised by the Institution of Civil Engineers and heldat the University of Nottingham, 11-13 July 1984, p. 253-265, London, 1995, ThomasTelford