discrete element modelling of railway ballast

Granular Matter (2005) 7: 19–29DOI 10.1007/s10035-004-0189-3

ORIGINAL PAPER

W. L. Lim · G. R. McDowell

Discrete element modelling of railway ballast

Received: 15 June 2004 / Published online: 28 January 2005© Springer-Verlag 2005

Abstract The discrete element method has been used to sim-ulate the behaviour of railway ballast under different test con-ditions. Single particle crushing tests have been simulatedusing agglomerates of bonded balls, and the distribution ofstrengths correctly follows the Weibull distribution, and thesize effect on average strength is also consistent with thatmeasured in the laboratory. Realistic fast fracture can be ob-tained if non-viscous damping is reduced. Oedometer tests onaggregates of crushable ballast particles have also been sim-ulated and compared with the results from laboratory tests.Finally, box tests which simulate traffic loading have beensimulated using both spherical balls and 8-ball clumps. It isfound that the 8-ball clumps give much more realistic behav-iour due to particle interlocking.

Keywords Discrete element modelling · Grain crushing ·Railway ballast · Statistics

1 Introduction

Since railway ballast in the track generally comprises largeparticles of typical size approximately 40 mm, it is difficultto treat such a material as a continuum, and the discrete ele-ment method [1] offers a means of obtaining micro mechan-ical insight into its behaviour. For this purpose the programPFC3D [2] has been used. PFC3D is a discrete element pro-gram that has the ability to model entire boundary value prob-lems directly with a large number of particles. The programcontains two entities: a ball and a wall. Crushable particlescan be simulated by forming agglomerates of bonded balls.Contact forces are used to calculate the accelerations of eachball using Newton’s Second Law, and the accelerations areintegrated to give velocities and displacements (and hence

W. L. Lim · G. R. McDowell (✉)Nottingham Centre for Geomechanics,School of Civil Engineering,University of Nottingham,University Park,Nottingham NG7 2RD, UKE-mail: [email protected]

new contact forces via a contact constitutive law), via anexplicit time-stepping scheme. The method has been usedsuccessfully to model single particle crushing tests for sand[3,4] and one-dimensional compression tests on aggregatesof sand [4,5], and also triaxial tests [4,6,7]. McDowell andHarireche [3] applied PFC3D to model the fracture of sil-ica sand particles using crushable agglomerates and wereable to obtain a distribution of strengths which followed theWeibull [8] distribution, and with the correct Weibull mod-ulus, in addition to obtaining the correct size effect on aver-age strength. McDowell and Harireche [5] then used theseagglomerates to simulate one-dimensional compression testson silica sand and showed that yielding coincided with theonset of bond fracture, consistent with the hypothesis byMcDowell and Bolton [9] that yielding is due to the onsetof particle breakage. A preliminary study of triaxial test sim-ulations on an assembly of agglomerates by Robertson [4]found that it was possible to produce yield surfaces similarto those predicted by plasticity models such as Cam Clay [10].

Recent research on railway ballast has shown that theamount of degradation of ballast in oedometer (i.e.one-dimensional compression) tests and box tests which sim-ulate traffic loading, correlates with the tensile strength ofthe ballast as measured by compressing individual ballastparticles between flat platens [11–13]. Lim et al. [14] per-formed single particle crushing tests on a range of ballastsand found that for most ballasts, although a Weibull distri-bution of strengths was obtained, the size effect on averagestrength was inconsistent with that predicted by Weibull sta-tistics. This is in contrast to the results obtained by McDowelland Amon [15] and McDowell [16] for sand particles, whichdid follow the Weibullian size effect. Lim et al. [14] hypoth-esised that this may have been because those sands were eachlargely composed of one mineral.

This paper examines the simulation of single particlecrushing tests on railway ballast in order to obtain the cor-rect distribution of strengths for particles of a given size andthe correct size effect on strength. In particular, the effect ofdamping is shown to play an important role in the fractureprocess. Oedometer tests are also simulated, because until

20 W. L. Lim, G. R. McDowell

recently, the Aggregate Crushing Value (ACV), which is anoedometer test, has been used a standard test for ballast qual-ity [17]. The paper also examines the simulation of ballast ina box test used to study the behaviour of ballast subjected totraffic loading. In this case, the use of bonded balls to makecrushable agglomerates results in so many balls that the sim-ulation is too time-consuming. The paper therefore exam-ines the simulation of the box test using both spherical ballsand clumps to represent ballast particles, to ascertain whetherinterlocking of ballast can be modelled and whether this influ-ences the resilient and permanent deformation of the ballast.

2 Single particle crushing test simulations

2.1 General description

McDowell and Harireche [3] were successful in using PFC3D

to simulate crushable agglomerates having the distribution ofstrengths of silica sand particles and the correct size effecton average strength. For these reasons, the approach used byMcDowell and Harireche [3] is used here to simulate crush-able ballast particles. The strength is measured by compress-ing the particle between flat platens until failure occurs atpeak force when the particle breaks into two or more pieces.The strength is calculated as F/d2 where F is the peakforce and d is the particle size (distance between the load-ing platens) at failure [13,14]. Table 1 [13] shows the resultof the laboratory single ballast particle crushing tests on atypical ballast, denoted here as ballast A, which is a grano-diorite comprising mainly plagioclase (30%), quartz (25%)and alkali feldspar (20%).

Thirty tests were conducted on each agglomerate type,as this is considered to give a statistical representation ofthe average strength and distribution of strengths [18]. Ahexagonal close packed (HCP) agglomerate was taken asthe starting geometry. McDowell and Harireche [3] noticedthat HCP packing gives a regular and maximum possibleball density (74%), which results in an opposite size effecton strength to that observed for real materials. Thus, theyattempted to simulate a correct size effect by using agglom-erates that have a dense random packing where balls in thisconfiguration occupy 64% of the total volume. They achievedthis by removing 13.5% [1- (64/74) = 0.135] balls at randominitially to partly replicate a dense random packing and intro-duce flaws. McDowell and Harireche [3] found that remov-ing 13.5% balls initially was inadequate to simulate the sizeeffect on strength for silica sand, and increased the percent-age of balls removed initially to increase the size effect.They found that removing 30% of balls initially followed

Table 1 Single particle crushing test results for ballast A [13]

Nominal Weibull 37% tensilesize/mm modulus m strength σo/MPa

24 2.82 31.444 3.45 20.7

by a random percentage of the remaining balls in the range0–25% gave the correct size effect for the silica sand testedby McDowell [16]. The same methodology will be used here,where removing 13.5% balls initially at random will be usedfirst, and altered if the size effect for ballast A is not repro-duced. Thus, 13.5% balls were removed initially at randomto replicate a dense random packing, and some balls werethen randomly removed (a random percentage of the remain-ing number of balls in the range 0–25%) to simulate flaws,and then each agglomerate was given a random rotation.Two types of agglomerate were used for this simulation: anagglomerate containing initially 135 balls (giving agglom-erate diameters of 24 mm and 48 mm depending on whetherthe ball diameter is 3.55 mm or 7.1 mm) and a 48 mm diam-eter agglomerate containing initially 1477 balls of diameter3.55 mm. Examples of such agglomerates are shown in Fig-ure 1. Each agglomerate was bonded together with contactbonds. A contact bond can be envisaged as a point of gluewith constant normal and shear stiffness at the contact point.The bond breaks if either the magnitude of the tensile normalor shear contact force exceeds the bond strength specified.

Fig. 1 (a)24 mm agglomerate initially containing 135 balls of diame-ter 3.55 mm; (b)48 mm agglomerate initially containing 1477 balls ofdiameter 3.55 mm and (c)48 mm agglomerate initially containing 135balls of diameter 7.1 mm. In each case 13.5% balls are removed initiallyfollowed by 0–25% and then a random rotation is applied

Discrete element modelling of railway ballast 21

Fig. 2 Typical force-strain plot for 48 mm diameter agglomerate ini-tially containing 1477 balls using different platen velocities

Each agglomerate was stabilised under gravity for 50,000timesteps before compression. In order to reduce compu-tational time, a gravitational field of 9.81×105 ms−2 wasapplied so that the agglomerate stabilised in an acceptablenumber of timesteps. To prevent the agglomerate from shat-tering during this process, the bond strength was temporarilyset to a very high value. Once the agglomerate had stabilised,the gravitational field was reduced gradually to 9.81 ms−2 andthe bond strength reduced to the desired value. Followingstabilisation under gravity, the top platen was located at thehighest point of the agglomerate, and then moved downwardsat a constant velocity to compress the agglomerate betweenthe upper and lower platens.

Figure 2 shows a typical force-strain plot for a 48 mmdiameter agglomerate initially containing 1477 balls, sub-jected to random removal of some balls (e.g. 356 balls in thiscase), then rotation, followed by stabilisation under grav-ity and then loading (note: strain was calculated by divid-ing the platen displacement by the initial distance betweenthe loading platens). Figure 2 also presents three differentplaten velocities used to compress the same agglomerate andit can be seen that a platen velocity of 0.08 ms−1 can beapplied, without significantly affecting the results. It is pos-sible that the sudden loading of the platens could cause elasticwaves in the agglomerate, which could be prevented by load-ing with a zero initial speed but non-zero acceleration untilthe desired strain rate is achieved [19]. However, since theplaten velocity of 0.08 ms−1 appeared to be sufficiently lowto have a negligible affect on the force-strain curve (compar-ing with 0.04 ms−1 in Figure 2), this velocity was consideredacceptable. Thus, all tests, including those on the agglom-erates initially containing 135 balls, were conducted with aplaten velocity of 0.08 ms−1 throughout the test. Tests wereconducted to a strain of 12%.

The initial ball normal and shear stiffnesses kn and ks

were both initially set to 4.97×108 Nm−1, and the normaland shear bond strengths were both set to 2.61×102 N; thesevalues were calculated as first estimates using the proceduresdocumented in the PFC3D manuals, assuming aYoung’s mod-ulus for granite of 70 GPa and a tensile strength of 20.7 MParespectively. The density of the balls was set to 2600 kgm−3,

Fig. 3 Weibull probability plot for 24 mm diameter agglomerateinitially containing 135 balls with stiffnesses and bond strength: (a)unscaled, f = 1; (b) scaled f = 2.96

which is a typical value for granite. These parameters wereused to simulate the 24 mm diameter agglomerate initiallycomprising 135 balls of diameter 3.55 mm, subjected to aninitial random removal of 13.5% balls, then removal of someballs (0–25%) and then random rotation, followed by stabil-isation under gravity and then loading.

2.2 Results

Figure 3(a) shows the Weibull probability plot, which is a plotof ln(ln(1/Ps)) against lnσ , where Ps is the survival proba-bility and σ is the applied value of F/d2 (see Lim et al. [14]for more details). The Weibull modulus m and the 37% ten-sile strength for each set of tests can be simply calculatedfrom this plot, such that the Weibull modulus m is the slopeof the line of best fit, and the value of σo is the value of σwhen ln(ln(1/Ps)) = 0 [14]. It can be seen that the 37% tensilestrength σo (10.6 MPa) is lower than that presented in Table 1for ballast A of nominal size of 24 mm (31.4 MPa). Thus, inorder to reproduce σo of ballast A, the bond strength and balland platen stiffnesses were both scaled by f = 31.4/10.6 =2.96 and the tests were repeated (stiffness and bond strengthare both scaled to give failure at a higher force but at thesame strain [3]). This gives σo = 29.3 MPa in Figure 3(b).For both of these tests, the seed of the random number gen-erator was constant so that the same geometries and flawscould be tested with different bond strengths and stiffnesses.


Fig. 4 Weibull probability plot for 48 mm diameter agglomerate ini-tially containing 1477 balls (f = 2.96)

The size effect was investigated by simulating a 48 mmdiameter agglomerate initially containing 1477 balls ofdiameter 3.55 mm, in the same way as 24 mm diameter 135-ball agglomerates. The parameters used were the same asthe scaled (f = 2.96) parameters for the 24 mm 135-ballagglomerates. Figure 4 shows the Weibull probability plotfor 30 tests on the 1477-ball agglomerates of diameter 48 mmsimulated with the scaled parameters. It can be seen that the37% tensile strength σo (22.4 MPa) is comparable to the valueof σo presented in Table 1 for ballast A of nominal size of44 mm (20.7 MPa). Thus, the use of initially removing 13.5%balls and then removal of some balls (0–25%) randomly isappropriate for simulating the non-Weibullian size effect forballast A. The size effect is non–Weibullian because accord-ing to Weibull statistics, the 37% strength σo should be afunction of size d according to the equation [14]:

σo ∝ d−3/m (1)

In order to reduce computational time for simulatingoedometer tests with agglomerates, 48 mm diameter ballastparticles were simulated using 135-ball agglomerates withlarger balls of diameter 7.1 mm. The normal and shear stiff-nesses were both 9.95×108 Nm−1 and the normal and shearbond strengths were both set to 1.04×103 N. Figure 5(a)shows the result with the unscaled parameters. It can be seenthat the 37% tensile strength σo (10.3 MPa) is lower than thatpresented in Table 1 for ballast A of nominal size of 44 mm(20.7 MPa). Thus, the bond strength and ball and platen stiff-nesses were scaled by f = 20.7/10.3 = 2.01 and the testswere repeated. This gives σo = 20.5 MPa in Figure 5(b). Thisagglomerate was chosen for the oedometer test simulations,which will be presented later.

2.3 Damping

It was noted in some cases that it was not possible to observea diametral fracture of the agglomerate. Hazzard [20] foundthat the stress waves emanating from cracks were capable ofinducing more cracks, and high levels of damping inhibitedthe formation of large clusters or a chain reaction of cracking.

Fig. 5 Weibull probability plot for 48 mm diameter agglomerate ini-tially containing 135 balls with stiffnesses and bond strength: (a) un-scaled, f = 1; (b) scaled f = 2.01

For example, they found that by using a damping coefficientof 0.015 (chosen to represent a granite sample with faults) forthe simulation of the compression of brittle rock using PFC2D

(Particle Flow Code in 2 Dimensions), the peak strength wasreduced by up to 15% and the number of cracks increasedin discrete jumps compared to model runs with high damp-ing (damping coefficient = 0.7). PFC2D and PFC3D providelocal non-viscous damping to dissipate energy by dampingthe unbalanced force in the system [2]. The damping coeffi-cient is a constant that specifies the magnitude of damping.Anattempt was made to make the fracture process more realisticby reducing the damping within the agglomerate. Figure 6shows a force-strain plot for a 24 mm diameter agglomerateinitially comprising 135 balls of diameter 3.55 mm, subjectedto an initial random removal of 13.5% balls, then removal of(0–25%) balls at random, followed by random rotation, sta-bilisation under gravity and then loading. The normal andshear stiffnesses were both 4.97×108 Nm−1, and the nor-mal and shear bond strengths were both set to 2.61×102 N.Figure 6(a) presents the results for two different dampingcoefficients used to compress the same agglomerate. It canbe seen that the agglomerate that was compressed with adamping coefficient of 0.015 failed at a lower force and in amore catastrophic manner compared to the one with a damp-ing coefficient of 0.7. Figure 6(b) shows a plot of the num-ber of broken bonds against strain. It can be seen that theagglomerate that was compressed with a damping coefficientof 0.015 failed by fast fracture – i.e. the agglomerate that


Fig. 6 (a) Force-strain plot and (b) number of broken bonds againststrain for the compression of a 24 mm diameter agglomerate with differ-ent damping coefficients

was compressed with a low damping coefficient failed in amore realistic manner where failure occurred quickly andcatastrophically. However, it was noticed in some cases thatusing a low damping coefficient was still not enough to causediametral fracture of the agglomerate. Although the damp-ing coefficient may not have a large effect on the measuredstrength, it will have an effect on the fragment size distribu-tion which results, and future work examining the degrada-tion of aggregates must take this into account.

3 Oedometer test simulations


An oedometer test was simulated using crushable agglomer-ates with the distribution of strengths of ballast A to comparewith the oedometer test on ballast A conducted in the labora-tory as described by McDowell et al. [11] and Lim [13].Afterthe simulated oedometric compression, the simulated samplewas also unloaded to ascertain the change in horizontal resid-ual stress. The dimensions of the oedometer were 270 mmlong × 270 mm wide × 150 mm deep (the laboratory sam-ple was 300 mm diameter and approximately 150 mm deep[11]). Smaller balls were generated within the oedometer and

then expanded to a final diameter of 48 mm. This procedureis generally used to generate a dense random assembly[4,5,13]. The expanded balls were cycled to equilibrium andthen replaced by agglomerates of approximately the samesize. The agglomerate used in this test was the 48 mm diam-eter agglomerate initially comprising 135 balls of diameter7.1 mm described earlier. Each agglomerate was subjectedto an initial random removal of 13.5% balls, then removalof some balls (0–25%) and then random rotation. The num-ber of agglomerates in the oedometer was obtained throughtrial and error to maintain low contact forces after cycling toequilibrium prior to loading.

The properties of the agglomerates were the same asthe scaled properties of the 135-ball agglomerates of 48 mmdiameter in the previous section. The ball normal and shearstiffnesses were both 2.0×109 Nm−1, and the normal andshear bond strengths were both set to 2.1×103 N. The coeffi-cients of friction for the balls and the walls were set to 0.5and 0, respectively. The stiffnesses of the walls were chosento be the same as for the balls. After the agglomerates werecreated, the sample was cycled for 30,000 timesteps to avoidhigh concentrations of contact force. During this period, thebond strengths were set to a high value (1016 N) to avoidbreakage and the coefficient of friction was set to zero tomake rearrangement easier. The ball properties were thenreset to the final values and the assembly was cycled to equi-librium using the SOLVE command, which limits the ratioof mean unbalanced force to mean contact force, or the ratioof maximum unbalanced force to maximum contact forceto a default value of 0.01 [2]. This assembly was not com-pacted due to high computational time. Figure 7 shows theoedometer sample of 113 agglomerates prior to loading. Therate of loading was chosen to be 0.1 ms−1 because the differ-ence in vertical stress between the top and the bottom wallfor this loading rate was acceptably low: the average verticalstress on the top wall was only 0.15 MPa or 8% higher thanthe vertical stress on the bottom wall, for stress levels cor-responding to 40→41% strain (on the normal compression

Fig. 7 Oedometer sample of 48 mm agglomerates initially containing135 balls prior to loading


Fig. 8 (a) V/Vo against logarithm of vertical stress for oedometer testsimulation using 48 mm 135-ball agglomerates and laboratory oedom-eter test on 37.5–50 mm ballast; (b) total number of bonds against log-arithm of mean vertical stress σmean for oedometer test simulation on48 mm 135-ball agglomerates

line). An 8% difference in vertical stress was considered tobe small because this difference in vertical stress would notchange the one-dimensional compression line significantly[13]. The sample was loaded to a vertical stress of 21 MPa(as in the standard ACV test [21]), and the test took approx-imately 3 weeks to complete on an 800 MHz computer with128 Mb RAM.

3.2 Results

Figure 8(a) shows a plot of volume V normalised by initialvolume Vo against the logarithm of mean vertical stress σ forthe oedometer test simulation on 48 mm 135-ball agglom-erates, and the laboratory oedometer test on 37.5–50 mmballast A. The oedometer test simulation on 135-ball agglom-erates has more initial strain because the sample was not com-pacted, whilst the oedometer test on 37.5–50 mm ballast Awas compacted to maximum density. Yielding (i.e. the pointof maximum curvature on the plot of volume against loga-rithm of applied stress [16]) for the agglomerates appears tooccur at around 30% strain and the yield stress is lower than

Fig. 9 Mean Ko against vertical strain for oedometer test on 48 mm135-ball agglomerates

for the oedometer test on 37.5–50 mm ballastA. This is antic-ipated as the agglomerates were reasonably spherical, whichleads to columns of strong force in the aggregate and yieldingat a lower than expected yield stress. For compacted angularparticles, the average stress on each particle or agglomer-ate would be lower because the load would be transmittedby more force columns [22]. In addition, the agglomeratesin the sample were not compacted, and a lower co-ordina-tion number would lead to fracture at lower stress levels. Itcan be seen that for the oedometer test simulation on 135-ballagglomerates, the compressibility is higher than the oedome-ter test on 37.5–50 mm ballastA at stresses just after yielding,but lower at high stress levels. McDowell and Harireche [5]noted that as agglomerates fracture beyond yield, becauseeach agglomerate is porous and internal voids become exter-nal voids when the agglomerate fractures, this gives a highcompressibility. At high stress levels, however, most of theagglomerates have already fractured and the smallest frag-ments (balls) are unbreakable. Figure 8(b) shows a plot oftotal number of bonds against the logarithm of mean verticalstress σmean for the oedometer test simulation on 48 mm 135-ball agglomerates. It is obvious, comparing Figures 8(a) and(b) in this plot that yielding coincides with the onset of bondbreakage.

Figure 9 shows a plot of the mean value of Ko, that isthe ratio of mean horizontal stress (on all 4 vertical walls)to mean vertical stress, against vertical strain for loadingand unloading in the oedometer test simulation on 135-ballagglomerates. It can be seen that Ko gradually evolves toa constant value of approximately 0.5 beyond yield at 30%strain. If it is assumed that the ball-ball angle of friction (i.e.tan−10.5) is equal to the angle of internal shearing resistance,then Figure 9 is consistent with the equation proposed by Jaky[23]:

Ko,nc ≈ 1 − sin φ′ (2)

where Ko,nc is the value of Ko on the normal compressionline and φ′ is the angle of shearing resistance.


Fig. 10 Ko against OCR for oedometer test on 48 mm 135-ball agglom-erates

3.3 Unloading

The increase inKo during unloading is presented in Figure 10,which is a plot of Ko against overconsolidation ratio OCR(maximum applied vertical stress divided by current appliedstress) for different unloading wall velocities. It can be seenthat the result of unloading the sample with a wall velocityof 0.1 ms−1 differs from the results with the two other wallvelocities and the increase in Ko for all the unloading speedsis not as high as anticipated, comparing with available datafor soils [24].

3.4 Unbalanced forces

It was noted that the stress in the loaded sample couldrelax and approach zero stress rapidly. For example, a further10,000 timesteps were permitted after halting the wall move-ments when the mean vertical stress in the sample reached21 MPa. Figure 11 shows a plot of mean vertical stress σmeanagainst number of timesteps. It can be seen that the mean ver-tical stress in the sample dropped very quickly within 10,000timesteps or 1.25×10−3 seconds. This must be because the

Fig. 11 Mean vertical stress σmean against number of timesteps

Fig. 12 Number of broken bonds against number of timesteps

sample was not in a quasi-static state, with large unbalancedforces within the sample, leading to further bond breakageand rearrangement in the sample. This observation suggeststhat the small increase in Ko during unloading in Figure 10 isbecause the sample is not in a quasi-static state and the con-tact forces in the sample are still changing – i.e. the sampleis not in equilibrium, even though the difference in the ver-tical stress on the top and bottom walls is small. In order toobtain a sample in a quasi-static state, after reaching a meanvertical stress of 21 MPa, the sample was further cycled bymaintaining the vertical stress at 21 MPa until the ratio ofmean unbalanced force to mean contact force Rmean becameequal to 0.001. Figure 12 shows a plot of number of brokenbonds against the number of timesteps. It can be seen that thetotal number of broken bonds increases by approximately10% when the ratio of mean unbalanced force to mean con-tact force Rmean attained a value of 0.001. The rate of increasein bond breakage at this ratio is very small, so the sample canbe considered in a quasi-static state. It should be noted thatan increase in vertical strain of only 1% was necessary forthe sample forces to give this ratio, so the one-dimensionalcompression line in Figure 8 would not change significantly.It can also then be inferred that the elastic stress waves in thesample which may result from the initial non-zero velocity ofthe loading platen, would also not appear to affect the normalcompression line significantly.

The sample that was cycled to a ratio of mean unbalancedforce to mean contact force Rmean of 0.001 was unloaded bymaintaining this ratio at 0.001 at all stages. For example, thesample was unloaded to a chosen OCR and then cycled (i.e.the constitutive equations were solved repeatedly) to a ratioof 0.001 whilst maintaining the desired vertical stress. Figure13 shows a plot of Ko against OCR for the sample unloadedwhilst maintaining this ratio at 0.001. It can be seen that thereis no increase in Ko; in fact, the depth of the sample at anOCR of 10 is 0.1 mm smaller than the depth before unload-ing. It was noted that there were a further 5 bonds whichbroke during unloading, which is negligible compared to thetotal number of bonds broken. However, this could triggerfurther rearrangement in the sample and the use of contactbonds may enhance this effect because contact bonds allow


Fig. 13 Ko against OCR for the sample unloaded maintaining Rmean ≈0.001

Fig. 14 (a) Rolling without slip at a contact bond and (b) constraint pro-vided by surrounding balls which prevent rolling at a contact bond [2]

rolling of balls relative to one another without breaking thecontact bond. A contact bond provides no resistance to roll-ing because a contact bond acts only at a point and not overan area of finite size, and so cannot resist moment [2]. Figure14(a) illustrates a contact bond that allows rolling of ball Arelative to ball B without slipping, thus without breaking thecontact bond. It is noted that this event only occurs for ballsthat are free to roll; additional contact bonds provided by sur-rounding balls will prevent a ball from rolling (Figure 14(b)).

Fig. 15 Box test set-up (front elevation) [11]

Therefore, it is likely that the compressed sample which hasmany fractured agglomerates will result in balls having a sin-gle bond, permitting rolling and further rearrangement. Thesolution to this problem is to have agglomerates which con-tain parallel bonds (which can transmit moment) in additionto using a larger number of balls in each agglomerate (thoughthis may make computational time unacceptable).

4 Box test simulations


Figure 15 shows the laboratory box test set-up [11]. Simu-lated traffic loads are applied repeatedly to a model sleeperon top of the ballast, and properties such as stiffness, per-manent settlement and degradation, can be studied. Due tothe high computational time required to simulate the box testwith the 135-ball agglomerates, balls or clumps need to beused to represent ballast particles in the box. The use of ballsand clumps, both of which are uncrushable, to represent bal-last particles in the box test simulations, is presented here.It should be noted that McDowell et al. [11] found that theamount of breakage in the box test conducted in the labo-ratory was small, which may justify the use of uncrushableballs or clumps for box test simulations.

The size of the box and the sleeper in this simulationare equal to the size of the box and sleeper in the laboratorydescribed by McDowell et al. [11]: 700×300×450 mm and250 × 300 × 150 mm respectively. The diameter of the ballsis 36.25 mm, which is the average size of the ballast usedthe laboratory box test. This leads to estimated normal andshear stiffnesses of 5.08 × 109 Nm−1, following the proce-dure documented in the PFC3D manuals [2]. The stiffnessesof the walls and sleeper were chosen to be the same as for the


Fig. 16 An 8-ball cubic clump

balls. McDowell et al. [11] used a stiff rubber mat to repre-sent typical sub-ballast and subgrade layers in their box test.Since the Young’s modulus of stiff rubber is approximately2,000 times smaller than that of steel [25], the stiffnesses ofthe base were chosen to be 2,000 times smaller than that ofthe value for the walls, namely 2.54 × 106 Nm−1. The ball,wall, and base friction coefficients were all set to 0.5.

As for the oedometer test, smaller balls were generatedwithin the box and then expanded to a final diameter of36.25 mm. The expanded balls were cycled for 5000 time-steps to avoid high contact forces, and for the test with clumps,replaced by clumps of approximately the same size. This pro-cedure has been used successfully by other authors [4,6] toproduce dense random assemblies. It would also be possibleto generate the initial sample by expanding the initial balls insmall increments and cycling to equilibrium after each incre-ment to relax the sample. However, in this case the targetdiameter may not be reached if too may balls are present andthe procedure is likely to be computationally more time con-suming. Since one of the criteria for ballast selection is thatparticles should be equi-dimensional [17], an 8-ball cubicclump was used to represent each ballast particle in the box,as shown in Figure 16.A wall was located at the top of the boxto confine the particles initially. This wall was removed afterthe particles were generated by gradually moving it awayfrom the box, so that particles would not escape from thebox.

In order to reduce computational time, the assembly wascompacted by applying a high gravitational acceleration(9.81 × 103 ms−2) for 50,000 timesteps. After compaction,the gravitational field was reduced gradually to 9.81 ms−2.The assembly in the box was initially loaded by moving thesleeper towards the assembly to give a load equivalent to theself-weight of the sleeper in the laboratory (34 kg). Once theassembly was loaded by the sleeper, the SOLVE commandwas used to reduce the unbalanced forces. The default valueof the ratio of mean unbalanced force to mean contact force,or the ratio of maximum unbalanced force to maximum con-tact force of 0.01, was used. It was shown earlier that limit-ing the ratio of mean unbalanced force to mean contact forceto 0.001, had a small effect on the values of Ko obtainedon unloading in oedometer tests. However, use of a tightertolerance on this ratio greatly increases computational time,and so here the default SOLVE command has been used.The assembly was loaded with a sinusoidal load pulse withminimum load of 3 kN and maximum load of 40 kN, at a fre-

Fig. 17 Box test samples of (a) spherical balls and (b) 8-ball cubicclumps prior to loading

quency of 3 Hz (following McDowell et al. [11]). This wasachieved by modifying the numerical servomechanism de-scribed in the PFC3D manuals [2]. The target stress level inthe original servomechanism is programmed in such a waythat it increases and then decreases in an incremental fashionto follow a sinusoidal curve. Figures 17(a) and (b) show thebox with 1769 spherical balls and clumps, respectively, priorto loading.

4.2 Results

Figure 18 shows a plot of load against vertical deformationfor the first cycle of load in the simulated box test using bothspherical balls and 8-ball cubic clumps. It can be seen that theassembly with the 8-ball cubic clumps is stiffer on loadingthan the assembly with the spherical balls. The clumps givea higher resilient stiffness, and less permanent deformationis produced. This difference must be due to the additionalresistance provided by the interlocking of the non-spheri-cal clumps. The load-deformation plot for the assembly withthe spherical balls differs from that of real granular materialssubjected to repeated load [26], in the sense that further defor-mation occurs after achieving maximum load. This could bedue to the spherical balls allowing the assembly to flow underthe imposed stresses.


Fig. 18 Load - deformation plot for first cycle in the box test on spher-ical balls and 8-ball cubic clumps

4.3 Discussion

It is noted that the stiffnesses of the assembly with the spher-ical balls and 8-ball cubic clumps in the box were very highcompared to those in the laboratory box tests [11], eventhough the normal and shear stiffnesses of the balls weresupposed to correspond to the Young’s modulus of the mate-rial according to the procedure documented in the PFC3D

manuals [2]. This is because the procedure involves calculat-ing the ball stiffness from an elastic beam joining the centresof 2 balls in contact. The stiffnesses of the balls and wallscan easily be reduced, but a Hertzian contact model wouldbe more appropriate such that contact stiffness increases withincreasing contact force. Contact forces in the assembly inthe box are not uniformly distributed (e.g. contact forces forparticles underneath the sleeper are higher than the contactforces for particles near the base of the box), as shown inFigure 19. This highlights further the importance of havinga Hertzian contact law, so that the contact stiffnesses areallowed to vary throughout the box. This would permit anevaluation of the heterogeneous stresses within the box, anda comparison between these stresses and the ballast particletensile strengths to see whether fracture is likely. Future workwill concentrate on establishing the use of suitable clumpsor agglomerates to model ballast in the box, in addition toascertaining whether traffic-induced heterogeneous stressesare sufficient to cause ballast particle fracture.

5 Conclusions

The discrete element method has been used to simulate thebehaviour of railway ballast under different test conditions.Single particle crushing tests have been simulated usingagglomerates of bonded balls, and the distribution of strengthscorrectly follows the Weibull distribution, and the size effecton average strength is also consistent with that measured inthe laboratory. Realistic fast fracture can be obtained if thedamping is reduced, and this must be incorporated into futurework on the study of degradation of aggregates. Oedometer

Fig. 19 Non-uniform distribution of contact forces in the assembly inthe box (contact forces are shown as lines with thickness proportionalto the magnitude of the contact force)

tests on the crushable ballast particles have been simulatedand compared with the results from laboratory tests. It hasbeen found that the yield stress for the agglomerates is lessthan that for the real ballast; this is most likely due to thespherical shape of the agglomerates, which leads to col-umns of strong force in the simulated sample. Minimisingthe difference in vertical stress to produce an acceptable nor-mal compression line does not ensure a quasi-static sample,and further bond breakage and rearrangement is possible ifthe sample is cycled at constant stress level. However, even ifthe sample is maintained in a quasi-static state during unload-ing, further compression can arise; this is thought to be dueto the use of contact bonds, which allow rolling of one ballrelative to another without breaking the contact bond, givingfurther rearrangement during unloading. Future work shouldexamine the use of parallel bonds which transmit momentin addition to the use of agglomerates of larger numbers ofsmaller balls to give more realistic particle shapes.

Box tests which simulate traffic loading have been simu-lated using both spherical balls and 8-ball clumps. It is foundthat the 8-ball clumps give much more realistic behaviour dueto particle interlocking. The Hertzian contact model may bemore suitable in simulating the variability in contact stiff-nesses within the box because of the varying ball contactforces throughout the box. This would permit an evaluationof the heterogeneous stresses within the box, and a compar-ison between these stresses and the ballast particle tensilestrengths to see whether fracture of particles is likely undertraffic-induced stresses.

Acknowledgements The authors would like to thank Lafarge Aggre-gates Ltd and Groundwork Hertfordshire for their financial support forthis project, which was funded through the Landfill Tax Credit Scheme.

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discrete element modelling of railway ballast

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