DETERMINATION O TRAIN SPEED LIMITS ON RENEWED TRACKS USING TAMPING MACHINE AND NUMERICAL OPTIMISATION V.L. Markine, C. Esveld Section of Road and Railway Engineering aculty of Civil Engineering, Delft University of Technology Stevinweg 1, NL-2628, CN Delft, The Netherlands E-mail: v.l.markine@citg.tudelft.nl, c.esveld@citg.tudelft.nl ABSTRACT The aer resents a methodology for analysis of lateral resistance of a railway track and its alication to determination of train seed limits for newly laid tracks or tracks after renewal. Ballast arameters and safe seed limits of trains are determined using the same 3-D finite element model. irst, the model arameters related to the lateral resistance of ballast are determined on the basis of measurement data. Lateral track load/dislacements diagrams have been obtained by shifting a real track using a taming machine (which is normally used to lift a track). The measurements are also simulated using the 3-D E model of the track. Based on the erimental data and the results of numerical simulation, ballast arameters are determined using an otimisation technique (a Sequential Quadratic Programming method). The track model with the obtained ballast arameters is then used to determine maximum allowable temeratures for tyical train velocities according to safety criteria given in UIC Leaflet 720. inally, by comaring the maximum allowable temeratures with combined equivalent ones, safe train seed limits are determined. The resented rocedure is alied to determine seed limits for a railway track using track lateral resistance data measured at different moments after track renovation. The results are resented and discussed. INTRODUCTION Sufficient lateral strength of a railway track is imortant for safe track oerations. High lateral forces acting on a track are caused by a moving train as well as by temerature variation. The lateral strength of a railway track is to a large tent defined by ballast lateral resistance. or a newly laid track or a track after full maintenance, ballast articles are not good enough consolidated and Consolidated ballast therefore lateral resistance of such a track is oor. This can be observed from igure 1 where a tyical lateral behaviour of wellconsolidated and tamed ballast l during a single sleeer test is shown. The eak strength of Tamed ballast tamed ballast is substantially lower than the one of consolidated ballast. In order to revent track buckling, the lateral Lateral force Dislacement igure 1 Lateral behaviour of ballast in a single sleeer test (not to scale)
forces occurring during oerations on newly laid tracks or tracks after maintenance should be restricted. Usually, this is achieved by temorary restriction of train seeds on renewed sections of such a track. As the number of train asses grows, ballast becomes more consolidated and its lateral resistance increases. Accordingly, the restriction of the seed limits can be released. Obviously, an introduction of temorary seed restrictions deteriorates oerational efficiency of a railway track. One way to reduce the oerational hindrance is to use dynamic stabilisation directly after track maintenance. Jet, it is still necessary to know whether the imosed seed limitations are safe or not. To aly roer seed limits, knowledge about lateral resistance of a track and its interretation for determination of safe seed limits are required [8]. In ractice the lateral resistance of a track is measured using taming machines. A taming machine, which is normally used to lift u a track, shifts a track in the lateral direction. Lateral track resistance can then determined from load/dislacements diagrams obtained during measurements. Although such information could be used for estimation of ballast condition, it was not ossible to directly use it for assignment of temorary seed limits for renewed tracks. On the other hand, UIC Leaflet 720 [6] gives safety criteria for tracks with continuously welded rails based on a rail temerature. Using these criteria it is ossible to determine maximum allowable rail temeratures with no risk of buckling. The temeratures are obtained using a finite element model, which requires knowledge about lateral resistance of ballast. That is why, at Delft University of Technology a technique for identification of ballast lateral resistance arameters basing on measured lateral resistance of a real track has been develoed, so that the UIC safety criteria could be alied for determination of temorary seed limits [8]. Measurement of Track Lateral Resistance Using Taming Machine Ballast arameters identification (TU Delft) Determination of Temorary Seed Limits (TU Delft) Determination of Tallowable, Safety criteria (UIC Leaflet 720) igure 2 lowchart of rocedure for determination of temorary seed limits on renewed tracks In this aer a technique develoed at Delft University of Technology that rovides a link between measurement of lateral resistance of a track and assignment of oerational train seeds is resented (igure 2). The 3-D finite element model of a railway track is used for both identification of ballast lateral resistance arameters and determination of the corresonding seed limits. Ballast resistance arameters are determined using load/dislacements diagrams obtained by shifting a real track in the lateral direction. The field measurements are simulated using the finite element model of a track. Ballast resistance arameters are then determined by matching the results of the numerical simulation with measurement data. The identification of ballast arameters is erformed using a modern otimisation technique. The obtained ballast arameters are then used in the numerical model of a track for determination of train seed limits. The seed limits are determined by comaring the maximum allowable temeratures calculated according to the UIC safety criteria for a number of tyical train seeds with combined equivalent ones. Both rocedures (ballast arameter identification and determination
of seed limits) have been imlemented in a software ackage BLATRES (TU Delft). inally, the technique has been alied to determine train seed limits on a track after full maintenance. NUMERICAL MODEL O TRACK The 3-D inite Element (E) model of a railway track used here is shown in igure 3. The model can be used for analysis of both straight and curved tracks [8,14]. It also accounts for geometry misalignments which are aroximated by a sine function (igure 3a). Rails are modelled by 3-D elastic beam elements whereas for fasteners and ballast, linear as well non-linear sring elements are used. The choice between linear and non-linear sring elements was based on the erimental data of various tracks available from the literature [4,5,13,14]. Thus, the vertical behaviour of the ballast is described by linear sring elements (igure 3b), which rovide a suort to rails also known as Winkler foundation [7]. A contribution of fastenings and ballast to the longitudinal resistance of a track is assumed to be linear and therefore they are also modelled using linear sring elements reresenting the total longitudinal stiffness of a track (igure 3b). Combined torsional stiffness of a track which deends on a fastening system, track gauge and sleeer sacing, is aroximated here by linear torsional srings (igure 3a). 2x (Half Wave Length λ ) Peak amlitude ( δ ) a. Bogie Centre Sacing Train Centre Sacing (TCS) b. igure 3 E model of track: a. To view; b. Side view It is known from eriments (e.g. from a single sleeer test [14]) that the lateral behaviour of ballast is non-linear (igure 1) to model which an elasto-lastic sring element is used. The element can describe two tyes of non-linear behaviour, namely bi-linear or non-linear with softening as shown in igure 4. The element behaves as linear-elastic until the alied lateral load s has reached the eak value s = (in case of no vertical loads) and corresonding eak dislacement W. After that the element begins to yield, that is the deformations are increasing with no increase of the force
s aroaching its limit value elastic stage is The softening branch is described by the function l as shown in igure 4. The stiffness l K el of the sring during the K el = ( 1 ) W l w / W l s = + ( ) 2 ( 2 ) The rate of softening is defined by the limit value of the arameter W l > W so that the force s corresonding to the dislacement w = Wl is equal to the average between and l (igure 4a). rom ( 2 ) it can be seen that if the limit strength l is taken equal to the eak strength the model describes bi-linear ballast behaviour as shown in igure 4b. s Model with softening s Bi-linear model s max s max loaded Lateral resistance l Kel loaded unloaded ulift w Lateral resistance Kel unloaded ulift w W Wl W Lateral dislacement Lateral dislacement a. b. igure 4 Lateral ballast behaviour (model): a. with softening; b. bi-linear rom erimental data it is known that ballast behaviour deends on vertical loads. The model accounts for the effect of vertical forces by introducing the Mohr-Coulomb criterion. The yielding now occurs at the force s max (loaded and ulift branches on igure 4) which is evaluated as s = s tanϕ ( 3 ) max where is the eak strength of ballast for unloaded track; v v s is the vertical load er unit length; tan ϕ is the friction coefficient between the sleeer and ballast. Thus, if no vertical load is alied, ( s = 0), the ballast yields at the lateral force s = v max.
The stiffness of ballast Kel also deends on the vertical load and can be obtained from ( 1 ) and ( 3 ) according to the formula sv tanϕ Kel = ( 4 ) W where W is the eak elastic dislacement for unloaded track (reference case). To aroximate the softening branch one can notice that in case of loading the eak strength of the ballast is changing by the factor b = s /. Assuming linear deendence of the lateral resistance from vertical max loads, the ballast limit strength Introducing the arameter l is also increased by the factor b and becomes equal to s l max. s = l / and taking into account ( 2 ) the softening branch in resence of vertical load can be described by the formula lim s = s max [ s lim + (1 s lim )2 w/ W l Thus, the bi-linear ballast model is characterised by the two elastic arameters, W and the friction coefficient tan ϕ whereas the model with softening is defined by five arameters, namely, l, W, W l and tan ϕ ] ( 5 ). The model with softening can accurately describe ballast behaviour but its use requires a non-linear static resonse analysis, which can be time consuming. On the other hand, the resonse quantities of a track can be obtained very fast if a bi-linear model of ballast is used, since the resonse analysis in this case is linear. An amle of simulation of the lateral behaviour of a track using bi-linear ballast model is given in igure 5 a. In this amle, a track is loaded by the lateral force that uniformly increases from 0 to 17 kn. Additionally, the vertical load v = 100kN has been alied to the track (the same alication oint as the lateral load). The simulated lateral dislacements of the track lotted for the load alication oint are shown in igure 5 b. The corresonding ballast behaviour in the lateral direction (bi-linear model) is shown in igure 5 a. rom these figures it can be observed that lastic deformations of the track W occur when the yielding of ballast ( W ) begins, i.e. W = W = mm ( 6 ) 2 Thus, the lateral behaviour of a railway track can be simulated using the above described numerical model roviding that ballast lateral resistance is known. On the other hand, if the lateral resistance of a whole track is known one can try to solve an inverse roblem to determine the arameters defining the lateral behaviour of ballast. A rocedure for determination of ballast arameters using an otimisation technique is resented below. a b igure 5 Lateral behaviour of ballast (a) and whole track (b)
DETERMINATION O BALLAST LATERAL RESISTANCE Lateral resistance of ballast can be estimated by erforming a test with a searate sleeer or a track anel [15]. However, it can be very ensive or even imossible. In this section a rocedure for tracting of the ballast characteristics from total lateral resistance of a track is resented. Measurement of lateral resistance of track One technique for measurement of lateral resistance of a track emloyed by Nederlandse Soorwegen (Netherlands Railway) is resented here. The technique makes use of a taming machine, which, instead of lifting, shifts a track frame in the lateral direction. To amlify the effect of vertical loads on the lateral resistance of a track, an additional vertical load can be alied as well. The resulting lateral dislacements of the frame are measured and recorded. The alied lateral force and resulting lateral dislacements of the track are then combined in a forcedislacement diagram. A tyical force-dislacement diagram is shown in igure 6. An alied lateral force is slowly increased that results in movement of a track frame (ath O-A in igure 6). As the lateral dislacements have achieved a rescribed maximum value (oint B in igure 6) shifting gris are released so that the track frame is moving back (ath B-C-D in igure 6). It should be noted that the maximum dislacements of the track during the test are relatively small (max. 5mm) so that the residual dislacements of the track are not very large and the original track geometry can easily be restored. The elastic characteristics of the track can be obtained by considering the residual dislacements of the track (ath O-D in igure 6). The maximum elastic dislacements W of the track are then estimated by the dislacements of the track after the lateral load has been released (ath B-C-D in igure 6), i.e. W =. ( 7 ) A B O W D C W igure 6 orce-dislacement diagram of track measured emloying taming machine
The maximum elastic force elastic roerties of the track machine here for ballast arameter identification. can be obtained from the force-dislacement diagram as well. The and BALLAST PARAMETER IDENTIICATION W measured emloying a taming will be used General arameter identification roblem In tyical arameter identification roblem there is an object (or rocess) to be investigated and there is a numerical model that can describe behaviour of such an object. The investigated object is considered as a black box that roduces resonses deending on values of some inut arameters X. The numerical model is characterised by the object arameters X as well as by several tuning T arameters a = [ a 1,..., a N ]. The goal of the identification roblem is to find a set of the arameters a such that the obtained numerical model most accurately describes the object behaviour, in other words the difference between the object and model resonses for the same set of inut arameters should be minimal. Thus, a general arameter identification roblem can be formulated as the following minimization roblem: Minimize subject to the constraints where P ~ G( a) ( X ) ( X, a) ( 8 ) = 1 Ai ai Bi, i =1,..., N ( 9 ) T a = [ a 1,..., a N ] is the vector of the design variables (tuning arameters of the numerical model) that can be varied during the otimization; is a set of resonses obtained from eriments with the object; ~ is a set of resonses of the numerical model; A i, B i are the uer and lower bound of the tuning arameter a i; X is the -th set of inut arameters of the original object; P is the total number of such sets. * A solution of the roblem ( 8)-( 9 ) a can be found using a conventional method of mathematical rogramming. Here an otimisation technique based on a Sequential Quadratic Programming (SQP) method is used [9]. The method is iterative. It is based on successive linearizations of P non-linear differences in ( 8 ), combining a first order trust region method with a local method which uses aroximate second order information. The first order derivatives are aroximated by finite differences. Identification of ballast arameters The method for model arameter identification described above has been used for determination of ballast arameters, W,, W and tan ϕ. Since the arameter characterising the elastic roerties of the ballast ( and l l W ) do not affect the lastic behaviour of a track and to seed u
an otimisation rocess, the arameters and W can be determined searately from the other arameters. Moreover, due to a small shift of a track during measurements and therefore lack of erimental data about lastic deformations of a track it is difficult to obtain the values of the arameters l, W l which characterise the lastic behaviour of ballast. That is why determination of these arameters is not discussed here although if the erimental data is available the same technique (as for the elastic arameters and W ) can be alied to determine the arameters l and W l. Using the methodology of the revious subsection one can try to find values of the ballast arameters and W such that the dislacement of a track in a reference oint of the model W and of a real track W due to the elastic limit force are close as much as ossible. Thus, the following otimisation roblem similar to the roblem ( 8 )-( 9 ) is to be solved Minimise ~ G ( a) ( X) ( X, a) ( 10 ) with T a = [ W ], = [ ] ~ X, ( X) = [ ], ( X) = [ ]. ( 11 ) W W Here the vector of the resonses of the track structure (and consequently of the model ~ ) consists of the dislacement only one oint on the track W (and model dislacement W ), namely the alication oint of the lateral force. If dislacements of other oints of a track are available they can be taken into account as well. Because of the linear relation between the force and dislacement of the track, the dislacements corresonding to one loading ste s = are considered (i.e. P = 1 in ( 8 )). When non-linear behaviour of a track is analysed, more loading stes (comonents of the vector X ) should be used. Since the track dislacements are obtained erforming a static linear analysis the otimisation roblem( 10 )-( 11 ) can be solved relatively fast. To seed u the otimisation it can also be assumed that yielding of the whole track occurs when ballast begins yield, i.e. W = W = W. The vector of the design variables of the roblem ( 10 )-( 11 ) then has only one comonent, namely T a =[ ]. If no tra vertical load is alied, the obtained otimal arameters limits of ballast, i.e. tra vertical load force reads max * * = and * * * T a = [ W ] are the elastic * W = W. In case the measurement data have been obtained with v the obtained otimal arameter * is considered as the maximum yielding s = defined in ( 3 ). The eak strength of ballast can then be calculated from ( 3 ) that * = + s tanϕ ( 12 ) where s v is the vertical load er sring element. The corresonding limit deformation is evaluated as follows v W
* W =, el * Kel W K = ( 13 ) It should be noted that the rail temerature during the measurements can also be taken into account in the ballast arameter identification roblem. DETERMINATION O SPEED LIMITS As the ballast arameters have been obtained they can be used in the numerical model to determine maximum allowable seed limits. The calculation of seed limits is based on the safety criteria given in UIC Leaflet-720 [6]. According to these criteria, the maximum allowable temerature of the rails T allow is defined by two critical buckling temeratures T b, min and T b, max (igure 7) as follows if T > 20 C, T = T + 0. 25 T allow b, min, if 5 C < T < 20 C, T allow = T b, min, if T 0 C, T = T 5 C where allow b, min, T = T b T., max b, min Temerature (above normal) lastic and irreversible deformation line Tbmax Tbmin elastic and reversible deformation line certainly buckling risk for buckling no risk for buckling Temerature (above normal) lastic and irreversible deformation line Tbmax (lastic branch) Tbmin Tbmax PB (rogressive buckling) elastic and reversible deformation line certainly buckling risk for buckling no risk for buckling Lateral dislacement Lateral dislacement a b igure 7 Critical buckling temeratures of rails The critical buckling temeratures deend on a number of loading factors, namely the temerature loads, vertical train axle loads and lateral loads due to curvature of track and rail misalignments. The lateral forces deending on the train velocity, size of the misalignment and wheel quality are aroximated here by the formula where V is the train velocity; αmv =, ( 14 ) R lat 2
2 2π δ R = is the equivalent radius of misalignments (igure 3); 2 (2λ) α is a coefficient characterising wheel quality, viz. α =1.0 for a assenger train and α =1.5 for a freight train. The track model (igure 3) is used to calculate the maximum allowable temerature Tallow according to the safety criteria given above. Obviously, the track structure with no lateral load corresonding to the train velocity V = 0km/ h gives the maximum value of T allow whereas calculations for higher train velocities result in lower temeratures T allow. The maximum allowable temeratures ( Tallow ) i, i = 1,..., N calculated for N tyical velocities of the train are to be comared with a combined equivalent temerature T equiv [6]. The combined temerature T equiv deends on the following factors: actual track temerature relative to neutral temerature; effects of (eddy current) braking/accelerating; interaction with civil structures; etc. The articular velocity of a train ositive. The maximum train velocity can be used as the train seed limit on a given track. NUMERICAL EXAMPLE V i is considered to be safe if the difference T allow T is V i equiv V for which the difference Tallow Tequiv is still ositive V Verification of arameter identification rocedure To verify the arameter identification rocedure erimental data measured on a ballast track with wooden sleeers have been used [10]. The data has been obtained emloying taming machines. An average value of the measured maximum elastic lateral force and dislacement of the track and W (igure 6) have been chosen as inut values of the identification model (Table 1). A zero temerature of the rails during the eriment was chosen and the friction coefficient for the wooden sleeers ϕ = 1. 1 have been used. Other arameters of the model, which have been used to determine ballast arameters, can be found in [8]. Since there was no tra vertical load alied to the track structure during the measurements * * ( = s = N ), the values of the arameters and W obtained during the identification have v v 0 been taken as ballast roerties (see Eq. ( 12)), viz. (Table 1). Inut [ N] W [m] T [ N / m / m' ] = * = 44680 N m, W = W * = 0. 0018 m / Outut [ N / m] W [m] [ N / m / m] 64800 0.0018 0 12471 44680 0.0018 6928333 Table 1 Results of identification roblem during verification K el
Ballast (sring element) and track lateral behaviour using the obtained ballast arameters when no tra vertical load is alied are shown in igure 8. To verify the results of identification, lateral resistance data of the same track measured with an tra vertical load v = 100kN (alied at the alication oint of the lateral force [10]) have been used. The measurements with the tra vertical load have been simulated using the model with the obtained ballast arameters and W. The results of the simulation are shown in igure 9. igure 8 Model obtained using measurement data without vertical load; case with no vertical load igure 9 Model obtained using measurement data without vertical load; case with vertical load 100kN Comaring these results with the erimental data it can be seen that the simulated and measured (average) values of the maximum elastic force, = 132 kn in igure 9 and = 132 kn in [10] resectively, are very close. rom this comarison it can be concluded that the numerical model rovides correct results. It should be noticed that the measurement results with tra vertical force can also be used to determine the friction coefficient between the ballast and sleeer. Determination of seed limits Now the rocedure resented in the revious sections has been alied for determination of seed limits for a track after slier renewing and taming of the ballast. The lateral resistance of the track has been measured at different moments after the maintenance [11]. Here three tyes of measurement data have been used, viz. directly, 3 and 12 days after the maintenance. Using these data ballast elastic limits have been obtained using the arameter identification rocedure [8]. They are collected in Table 2. Since the measurement data rovide not enough information about lastic behaviour of a track, the arameters characterising lastic behaviour of ballast have been defined based on very conservative assumtion that reads
= 0. 8, W l = 0. 02 m. ( 15 ) l Using the ballast arameters from Table 2 maximum allowable temeratures have been found for tyical train velocities, namely V = 60, 90, 110, 130, 180 km / h. It should be noted that the set of tyical velocities in the software BLATRES can easily be adjusted if other velocities are to be investigated. The maximum allowable temeratures have been determined for a freight train (axle load = 225 kn), for a straight track with misalignments λ = 5 m and δ = 30 mm. The calculated maximum allowable temeratures are lotted in igure 10. rom this figure it can be seen that lateral resistance of ballast directly after renovation is oor. If the actual equivalent temerature is T equiv = 40 C as it shown in igure 9 then the maximum allowable seed of train is V * = 60km/ h. The large imrovement of ballast quality after 3 days can be lained by the use of stabilisation after the track maintenance. Measurements Ballast arameters e [N] We [mm] [N/m] W [mm] l [N/m] Wl [mm] 0 days 76500 0.0011 9993 0.17 7994 20 3 days 90260 0.0011 26584 0.35 21267 20 12 days 120030 0.0014 55643 0.69 44514 20 Table 2 Ballast arameters after track renewing [8] The lots shown in igure 10 obtained using the rocedure described in the aer can be effectively used in assignment of seed limits for renewed tracks. Maximum allowable temeratures Temerature [C] 160 140 120 100 80 60 40 20 0 60 90 110 130 180 Seed [km/h] 0 days 3 days 12 days T equivalent igure 10 Maximum allowable temeratures for track after renovation
CONCLUSIONS A rocedure for determination of lateral ballast resistance based on measurement result with a real tack has been resented. The ballast arameter identification is erformed using an otimisation technique. The rocedure has been verified using the measurement results with and without tra vertical load. A technique for determination of seed limits is resented as well. The technique reresents a link between measurements of track lateral resistance and UIC safety criteria. It can also rovide a ground for assignment of temorary seed limits on tracks after renovation or after full maintenance. The ballast arameter identification and seed limit determination techniques have been imlemented in the software BLATRES that can be installed on isting taming machines. REERENCES 1. CEN (1995) Draft EN BBB-2 - Railway alications - Track Test Methods for astening Systems, Part 2, Determination of Torsional Resistance. CEN/TC 256/SC 1/WG 17 N 160, December 1995. 2. ERRI-D202 (1994) Imroved Knowledge of orces in CWR Track (including switches), Euroean Rail Research Institute, ERRI D 202/RP1, Utrecht, August 1994. 3. ERRI-D202 (1995) Imroved Knowledge of orces in CWR Track (including switches), Review of Existing Exerimental Work on Behaviour of CWR Track. Euroean Rail Research Institute, ERRI D 202/RP1, Provisional Reort, Utrecht, ebruary 1995. 4. ERRI-D202 (1996) Lateral Resistance Test. Euroean Rail Research Institute, Utrecht, ERRI D 202/WG3, Draft Reort Deutsche Bahn 50548, Setember 1996. 5. ERRI-D202 (1996) Lateral Resistance Test. Euroean Rail Research Institute, Utrecht, ERRI D 202/WG3, Draft Reort BR research Limited, reort RR-TCE-81, November 1996. 6. ERRI-D202 (1996) Laying and Maintenance of CWR Track. International Union of Railways Leaflet 720 R, October 1996. 7. Esveld, C. (1989) Modern Railway Track. MRT-Productions, Duisburg. ISBN 90-800324-1-7 8. Markine, V.L., Esveld, C. (1999) Analysis of Lateral Behaviour of a Railway Track Structure Using an Otimization Technique. Reort 7-99-124-1, Laboratory for Road and Railway Engineering, Delft University of Technology. ISSN 0169-9288 9. Hald, J, Madsen, K. (1985) Combined LP and Quasi-Newton Methods for Nonlinear L1 Otimization. SIAM J. Numer. Anal., Vol. 20, 68-80. 10. NS/CTO (1990) Zijdelingse Weerstandsmetingen van Drie Proefvakken een Referentievak van Elk 300 m o het traject Sittard-Roermond km 22.7-23.9 uitgevoerd in week 44 1989. Raort CTO/6/10.256/010. 11. NS/CTO (1991) Zijdelinkse weerstandmetingen o het traject Weert-Roermond km 74.430-78.250 uitgevoerd in week 35 van 1990, betreffende een nieuwe werkwijze van dwarsliggers verwisselen met behul van de Mechanische Dwarsligger Verwisseling (MDV). Raort CTO/6/10.473/0005
12. Samavedam, G., Kish, A., Purle, A., Schoengart, J. (1993) Parametric Analysis and Safety Concets of CWR Buckling, U.S. Deartment of Transortation, John A. Vole National transortation Systems Center, Cambridge. Reort DOT-VNTSC-RA-93-25, December 1993. 13. Samavedam, G. (1995) Theory of CWR Track Stability. oster-miller, Inc. Walthman. 14. Van, M.A. (1997) Stability of Continuous Welded Rail Track. Ph.D. Thesis, Delft University Press. ISBN 90-407-1485-1 15. Zand, J. van t and Moraal, J. (1997) Ballast Resistance under Three Dimensional Loading. Delft University of Technology, aculty of Civil Engineering, Laboratory of Roads and Railways, reort 7-97-103-4, ebruary 1997.