Rail weld geometry and assessment concepts weld geom… · assessment concepts. In this article,...

Rail weld geometry and assessment conceptsM J M M Steenbergen� and C Esveld

Railway Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft,

The Netherlands

The manuscript was received on 15 September 2005 and was accepted after revision for publication on 22 February 2006.

DOI: 10.1243/09544097JRRT38

Abstract: Worldwide, metallurgical rail welds are being geometrically assessed by the principleof vertical deviations satisfying given tolerances, measured with steel straightedges or occasion-ally with digital/electronic straightedges. In this approach, the geometrical shape of the weld inlongitudinal direction has no real influence, although it has a direct relation with the dynamicwheel–rail interaction forces, which are responsible for track deterioration. In this article, differ-ent new assessment methods for rail welds are proposed and evaluated in practice, after which achoice is made for the best method. This is done in line with the situation in the Netherlands,where the chosen new method was recently introduced and standardized (2005). The proposedmethod is based on a limitation of the gradient of the discrete measurement signal, implying alimitation of the wheel–rail dynamic contact force.

Keywords: rail welds, welding, weld geometry, rail joints, weld assessment

1 INTRODUCTION

The step from railway tracks with bolted rail joints totracks with continuously welded rails (CWRs) was animportant improvement of the geometry of the inter-face between wheel and rail. However, rail weld sur-face irregularities (especially in situ made thermitand flash butt welds) in CWR tracks remain asource of high-frequency excitations of both thetrack and the vehicle, as the quality of the verticalgeometry along the rail is always less than that ofnew, straightened rail or ground rail. Especially, onhigh-speed lines, the continuity of the rail surfacein longitudinal direction is of great importance.

With respect to the track, high-frequency exci-tations are most detrimental for concrete sleepers,rail fastenings, and other track components andresponsible for non-uniform ballast settlement orlocal decompaction. Concerning the vehicle, high-frequency excitations may introduce non-linearitiesin wheel–rail contact, causing non-continuous

railhead damage and wheel flats, as the wheel actsas a grinding element on both rim and railhead.These high-frequency excitations finally lose theirenergy by dissipation (plastic deformation, irrevers-ible creep, and other processes related to thegrowth of track damage) and migration to lower fre-quency vibrations of other train and track com-ponents. From the above point of view, it is crucialto realize a vertical weld geometry as close as poss-ible to a straight line. This asks for appropriateassessment concepts. In this article, different newassessment methods for rail welds are discussed.One of them is proposed for practice and evaluated.This is done in line with the situation in theNetherlands, where this new method has beenrecently standardized (2005) [1, 2].

2 PRACTICAL REQUIREMENTS ON ASSESSMENTMETHODS

The geometrical shape of the rail weld has a directrelation with the magnitude of the dynamic wheel–rail interaction force. Both the frequencies and thecorresponding magnitudes of this force are relatedwith the growth rate of damage and deteriorationof the different track components.

�Corresponding author: Railway Engineering, Faculty of

Civil Engineering and Geosciences, Delft University of Technol-

ogy, Section of Road and Railway Engineering VBK, PO

Box 5048, NL 2600 GA Delft, The Netherlands. email:

[email protected]

257

JRRT38 # IMechE 2006 Proc. IMechE Vol. 220 Part F: J. Rail and Rapid Transit

It is presupposed that for geometrical assessment, adigital, electronic straightedge is used to measure theweld, so that the geometrical data are available aftermeasurement. Then, an optimal approach to weldassessment would be the evaluation of the spectrumof the contact forces occurring for theweld under con-sideration, where the spectral amplitudes should notexceed predefined values. This would require thecomputation of the dynamic contact force betweenwheel and rail weld, followed by a spectral analysisin the frequency domain. If the considered welddoes not satisfy the requirements, then the weld geo-metry should subsequently be changed in such away that it would satisfy the norms. As these normsare expressed in terms of forces and not of geometry,an additional computational effort is required todetermine the most efficient way to arrive at anacceptable geometry in an iterative process.

A simplified approach would be to calculate thedynamic contact force in the time domain for theconsidered weld, where the maximum force shouldnot exceed a predefined value.

These methods have two disadvantages, whichmake themunsuitable in practice. First, themagnitudeof wheel–rail contact forces depends on the propertiesand the configuration of the track structure to a largeextent. Therefore, the influence of the local trackproperties should be accounted for, which makes auniform assessment, independent of the track typeand component properties, impossible. A secondproblem is of a computational nature. Basically, thesolution of the dynamic wheel–rail contact problemin a linear form asks for the solution of a set ofdifferential equations (of second order). It is notfeasible in practice to perform these calculations foreach separate rail weld each time it is beingmeasured.

However, in general, it may be stated that the‘smoothness’ of the geometry is a measure for theoccurring dynamic forces between wheel and rail,and therefore, for the rate of track deterioration.This concept will be elaborated in the following sec-tions. The main objective is to develop a practicaltool for the assessment of the geometry of railwelds, by directly relating the vertical rail geometryto the magnitude of dynamic contact forces whichoccur for that geometry, without having to performcomplex dynamic calculations for each separategeometrical measurement.

3 WHEEL–RAIL DYNAMICS FOR SHORTLENGTH-SCALE DISTURBANCES IN THEINTERFACE GEOMETRY: MODELSIMPLIFICATIONS

The velocity for trains generally ranges between 40and 300 km/h, with operational speeds for

conventional passenger trains of �140 km/h andfor freight trains of �80–100 km/h. This yields amost relevant velocity range of �20–40 m/s. Givenrelevant wavelengths in a range of 0.1–2 m (as willbe discussed in section 4), the corresponding mostrelevant frequency range of excitation of thewheel–rail system can be found within 10–400 Hz(which is the low- and mid-frequency range).

In the train–track dynamics, several stiffness fac-tors and masses play a role in this frequency rangefor the determination of the wheel–rail contactforces. The most important masses to be mentionedare the unsprung mass Munsprung and the equivalenttrack mass Mtrack; the most important stiffnesses arethe primary suspension stiffness k1, the wheel–railHertzian contact stiffness kH, and the equivalenttrack stiffness ktrack. Damping is not of primaryimportance at the wheel–rail interface. In a mostgeneral form, the dynamic wheel–rail contact forceas a function of time may be written as

Fdyn(t) ¼ f z(x),V , Munsprung, Mtrack, k1, kHertz, ktrack� �

(1)

where z(x) is the vertical rail geometry in thelongitudinal direction and V is the train speed.

For each frequency in the frequency domain, a cer-tain mass in combination with a related stiffness willhave a dominating role in absorbing the excitationenergy and determine the magnitude of wheel–railcontact forces. For the band with the lowest frequen-cies, the combination of wheel mass on track stiff-ness will be dominating; for increasing frequency,the role of the wheel mass as the dominating masswill be taken over by the equivalent track mass; andfor the highest part of the frequency range, theHertzian stiffness will replace the track stiffness.

Now, a quasi-static response of the dominatingmass–stiffness combination is assumed. This willallow for a reduction of the dynamic contactproblem, in terms of second-order differentialequations, to an algebraic problem, as has beendiscussed in section 2. If the minimum relevantwavelength L in the rail irregularity is taken as0.1 m (section 4), the excitation frequency at a trainvelocity V (m/s) equals f ¼ V =L ¼ 10V (Hz), fromwhich the requirement for f can be derived directly.A quasi-static response occurs for f , 0:5f0approximately, where f0 is the natural frequencyof the mass–spring system under consideration, tobe determined from the dominant mass M andstiffness K. The resulting requirement for quasi-static response is given by (using SI units)

V ,1

40p

ffiffiffiffiffiK

M

r(2)

258 M J M M Steenbergen and C Esveld

Proc. IMechE Vol. 220 Part F: J. Rail and Rapid Transit JRRT38 # IMechE 2006

To which extent this requirement is satisfieddepends on the specific track properties andmust be verified by measurements. The train velo-city V herein is given by the line-section speed inquestion.

In general, the unsprung axle mass is �1800 kg forconventional trains [3]. In general, near weld irregu-larities in ballasted track, the ballast is not well-compacted below the sleepers (the problem of‘fluttering’ sleepers with voids underneath arisesafter some track use). Therefore, the track stiffnessmay be assumed – for small vibration amplitudes –to be determined by the rail bending stiffness,where the rail deflection is largely quasi-static andthe contribution of high-frequent modes can beneglected. Assuming now an equivalent track massconsisting of two fully effective concrete sleepers ofeach 300 kg [4] and 4 m of effective rail mass(60 kg/m; UIC 60), the equivalent track mass can beestimated at 850 kg. This is ,50 per cent of theunsprung axle mass. For this reason, the wheelmass is considered as fixed in the vertical directionduring the time interval that the wheel passes theweld irregularity.

In this context, also the traditional concept of P1

and P2 forces, which was originally developed in asomewhat different context, for insulated or bolted

rail joints [5], can be used (Fig. 1). Owing to thedifference in inertia between unsprung wheelmass and equivalent track mass, the response ofthe wheel to the relatively short-length excitationwill show a certain delay relative to the response ofthe track (rail and sleepers). The P1 force, which isa quasi-instantaneous amplification of the wheel–rail contact force, originates from the reaction ofthe track and is mainly determined by the trackequivalent stiffness and the rail equivalent mass.This reaction may be quasi-static for irregularitieswith a somewhat longer length scale and/orrelatively low velocities or dynamic for shorterlength scales and/or higher velocities. In this context,‘quasi-static’ refers to the system governed by theHertz contact stiffness and the track equivalentstiffness in parallel and the equivalent track mass(Fig. 2(a)). The time scale of the peak is determinedby the length scale of the irregularity and the actualtrain speed or, alternatively, by the frequency of thegoverning mass–stiffness combination.

The P2 force results from the reaction of the axlebox to the excitation. Its magnitude is determinedmainly by the unsprung wheel mass and the equival-ent track stiffness, and the time scale of the peak islarger because of the lower eigenfrequency of thedominating mass–spring system when comparedwith the system consisting of equivalent track massand stiffness. Owing to the P2 force, the sleepersdirectly after the weld are often observed to be wellfixed in the ballast bed, as they are pressed into theballast by all passing axles. P1 and P2 may be con-sidered to be proportional, i.e. the magnitude of P2

is determined by P1.Figure 2(b) shows the dominating mass–stiffness

combination resulting from the earlier assumptions,disturbed by the weld irregularity z(x) moving attrain speed V and valid during the time interval ofpassage of the weld. As the Hertz contact is farmore stiff than the equivalent track stiffness, it isassumed to be rigid in the model.

Fig. 2 Mass–stiffness combinations in the wheel–rail system (a) and disturbance by a rail

irregularity (b)

Fig. 1 Qualitative behaviour of P1 and P2 dynamic

wheel–rail interaction forces at an interface

irregularity on ballasted track [5]

Rail weld geometry and assessment concepts 259


The equation of motion of the system in Fig. 2(b) isgiven by

Mtrack €u(t)þ ktracku(t) ¼ 0 (3)

A quasi-static response of the equivalent trackmass (in the system depicted in Fig. 2(a)) wasassumed, yielding

u(t) ¼ z(t) (4)

The dynamic contact force in the model ofFig. 2(b)) is then given by

Fdyn(t) ¼ ktracku(t) ¼ ktrackz(t) (5)

and it is directly proportional to the excitation as afunction of time. Using equation (4), this expressionmay be written alternatively in terms of the secondderivative of the excitation as a function of time

Fdyn � Mtrack €z(t) (6)

Thus, the dynamic component of the P1 contactforce between wheel and track is approximated asthe inertia force originating from the track mass,which follows the vertical irregularity. The introduc-tion of a dimensionless calibration factor b yields

Fdyn ¼ bMtrack €z(t) (7)

where b accounts for dynamic influences which werenot modelled, inaccuracies introduced by assump-tions on the response and non-linearities; its valuecan be chosen as a discontinuous function of theline-section speed and should be determined fromthe validation measurements. Using the equalityd2z=dt2 ¼ V 2(d2z=dx2), this expression for thedynamic contact force can be transformed to thespace domain

Fdyn ¼ bMtrackV2 d

2z

dx2(8)

The maximum of this expression, for a certain vel-ocity, is found for the maximum of the spatial secondderivative of the disturbance z(x)

Fdyn,max ¼ bMtrackV2 d2z

dx2

��max

(9)

Using a quasi-static approach, the second deriva-tive of the geometry thus becomes a direct measurefor dynamic contact forces.

In Fig. 3, finite-element simulation results(obtained with the DARTS-NL software [6]) areshown for a train vehicle, passing an artificial weldirregularity in the track. The irregularity is a smooth‘wave’ with an amplitude of 1 mm and a length of1 m. The train speed V equals 140 km/h. The follow-ing parameter values have been adopted in the

Fig. 3 Response of the wheel–rail system to a short-length irregularity



calculation: a wheel mass (half unsprung mass) of970 kg; a sleeper mass of 300 kg; UIC 54 rail;a primary suspension stiffness of 1:8 � 106 N=m (perwheel); a rail pad stiffness of 1:2 � 109 N=m, and aballast stiffness of 30 � 106 N=m per sleeper, wherethe latter value is rather small to account for thegenerally bad compaction of the ballast underneaththe sleepers close to the irregularity. A non-linearHertz contact model has been used. The followingquantities are shown in Fig. 3, where all parametersare functions of time.

1. The geometry of the irregularity z.2. The vertical wheel displacement uwheel of the first

wheel of a passing bogie, relative to the displace-ment resulting from the quasi-static verticaltrack deflection because of the passage of thestatic axle load; the displacement is defined posi-tive in upward direction and calculated in amoving coordinate frame.

3. The dynamic displacement utrack,dyn of the rail/track relative to the quasi-static deflection calcu-lated at the centre of the irregularity. This displa-cement is defined positive in downwarddirection and calculated in a fixed coordinatesystem.

4. The wheel–rail contact force Fdyn for theconcerned wheel (the dynamic force is superim-posed on the static value) in a moving coordinateframe.

5. The axle box acceleration aaxle in a movingcoordinate frame.

The repetitive pattern in the track displacement inFig. 3 is due to the passage of the second axle of thetrain bogie. From Fig. 3, it can be observed that theassumptions that have been made in this paragraphcorrespond well to the real system response. Thetrack mass directly follows the irregularity in analmost quasi-static manner, whereas the responseof the axle mass shows a delay relative to the irregu-larity. It can be shown that this effect is even strongerfor a non-smooth irregularity with a shorter lengthscale. The P1 force, resulting from the instantaneousreaction of the track to the irregularity, as well as theP2 force, resulting from the reaction of the unsprungmass, can be clearly distinguished. The magnitude ofboth peaks follows from the hatched area in Fig. 3.Finally, it is observed that the dynamic contactforce is proportional to the axle box acceleration.

4 PRELIMINARY OPERATIONS ONMEASUREMENT DATA

Usually, the longitudinal rail weld geometry ismeasured with a straightedge with 1 m basis [7].

Other (mostly longer) bases exist, but 1 m is auniform choice. Further, it is measured at thecentre of the top of the railhead in the cross-sectionalplane. For connections between worn andnew rails, there are other possibilities, as a wornrailhead is no longer symmetric; however,because a uniform choice has to be made for stan-dardization, commonly the centre of the railhead istaken.

When a digital straightedge is used, the samplingof the rail geometry is a discrete process. Therefore,a minimum wavelength exists that can be registeredfor rail welds. This minimum wavelength isdescribed by aminimum of five sampled coordinatesor four sampling intervals (Fig. 4).

Measurement devices sample the vertical railgeometry with an interval of 5 mm. Before startingoperations on the measured signal, it is averagedover a distance of 25 mm (including five datapoints); the resulting data points have an interval of25 mm. The reason of this averaging process is toavoid very noisy signals (especially for the deriva-tives) because of irregularities on microscale. Theseirregularities, with length scale in the same order ofmagnitude as the width of the contact patch on therailhead, are physically not relevant because of thelocal plastic railhead deformation and wear aftersome track use. An example of this filtering processis given in Fig. 5.

Using a 25 mm interval, as explained earlier, leadsto a minimum full wavelength for welding irregulari-ties of 0.1 m. This corresponds well to the situation inpractice. Smaller wavelengths may exist in theory,but in reality, these components have such smallamplitudes (Fig. 5) that they will have disappearedfrom the signal by both wear and plastic deformationafter some wheel passages.

In this way, a global range of 0.1–2 m for the rel-evant wavelengths in the weld irregularity can beestablished. The largest wavelength of 2 m in thisrange corresponds to geometrical deviations of theend parts of rail sections (resulting from straighten-ing by the rollers in the production process), whichare welded together, or the ‘set-up’ of both railends by the welding crew or the welding machinebefore welding them together.

Fig. 4 Minimum number of sampling points to

describe a single wavelength



5 GEOMETRICAL ASSESSMENT METHODS,COMPARISON, AND ASSESSMENT PROPOSALFOR CONVENTIONAL TRACKS

The ‘smoothness’ of the sampled geometry of a weldcan be quantified in a number of ways. To investigatewhich parameter is most suitable to describe andassess its quality, the following methods aredistinguished:

(a) zeroth derivative-based method, based on alimitation of the sampled original geometry;

(b) first derivative-based method, based on a limit-ation of the first derivative or gradient of thesampled geometry;

(c) second derivative-based method, based on alimitation of the second derivative of thesampled geometry.

Method (a) is the usual approach, where verticaltolerances have to satisfy certain requirements.These requirements are usually independent of theline-section train speed. It is clear that the contactforce corresponding to the weld geometry is anincreasing function of V. This means that for aspecific weld geometry satisfying the requirements,in a part of the network designed for 140 km/h, theallowable contact forces are much higher than forthe same weld in a marshalling yard, which isinconsistent.

A typical example of the result of the current railwelding methods (especially for in situ welding)and their assessment are shown in Fig. 6. Both railends are ‘set-up’ under a small angle in verticaldirection (usually the overlift is 1–2 mm on a

1–1.2 m basis [7]). After welding, the resulting‘roof’, which does not fall within the maximum toler-ance, is ground off. The result is often far fromsmooth and introduces large dynamic amplificationsin contact forces.

Assessment method (a) can only be enhanced bydifferentiating on the line-section speed. Therefore,this method is not discussed any further, and in thefollowing, the focus is on methods (b) and (c).

In Fig. 7, an example is given of the application ofboth methods (b) and (c) for a weld geometrymeasurement. Similarly as in section 4, the measure-ment data are averaged and filtered before calculat-ing the derivatives. The averaged measurementsignal and the first and second derivatives areshown, each normalized by the respective maximumof their absolute value. It is clear that both methodsprovide a good evaluation of the ‘smoothness’ ofthe wheel–rail interface geometry.

The maximum dynamic contact force for a weld isa function of the train speed (equation (9)), andtherefore, a differentiation on line-section velocityis introduced. The following speed intervals arechosen (according to the standards in TheNetherlands [7]): 0–40, 40–80, 80–140, and 140–200 km/h.

To enable a comparison between methods (b)and (c), the so-called weld quality index (QI) isintroduced as a relative measure of the ‘dynamicquality’ of a certain rail weld geometry. This QI is a

Fig. 6 Typical geometry example of an in situ made

rail weld, resulting from current methods and

standards

Fig. 7 Averaged discrete measurement signal, first and

second derivatives, each normalized with their

maximum absolute value

Fig. 5 Measured longitudinal rail geometry and averaged signal



non-dimensional number, which is defined as theratio of the calculated maximum of the absolutevalue of the derivative (first or second) per weld tothe corresponding intervention level for the con-sidered train velocity range. Thus, a QI less than orequal to 1 leads to acceptance of the weld and avalue larger than 1 leads to rejection.

On the basis of experience from practice, empiricalintervention levels have been determined for bothapproaches (b) and (c), by performing an analysisof a representative weld population (a sample of 72rail welds) for the different speed regimes [1].Implementation of the quadratic influence of thetrain speed appears to be not feasible, as it leads toexcessive values; therefore, an influence of thevelocity with a power smaller than 2 has beenadopted. The adopted intervention levels aregiven in Table 1, based on which the QI can be deter-mined for a specific weld, for a given line-sectionspeed.

A comparison between both assessment methods(b) and (c) for the considered weld populationshows that, in general, approach (b) leads to lessextreme values than approach (c). This also followsfrom the average scores for the analysed population.According to approach (c), average QIs were 1.2, 1.8,3, and 3 for the four considered velocity ranges, and1.2, 1.6, 2, and 2.2 according to approach (b), respect-ively, which is much more moderate. Further,method (c) turns out to be most sensitive to verysmall, short-length irregularities (with a lengthscale of some centimetres) and, on the contrary,not very sensitive for longer irregularities (with aglobal length scale of 0.5 m). This is not the case forapproach (b).

Very small, short-length irregularities (indenta-tions) often occur after rail welding at both sides ofthe weld material because of both shrinkageafter grinding and further cooling down and differ-ences in steel hardness along the rail surface withlocal minima in the heat-affected zones at bothsides of the weld metal [7]. In Fig. 8(a), an exampleis given of an almost perfectly straight weld, showing

these indentations. The weld in Fig. 8(a) has QIs1.3, 1.7, 3.3, and 3.3 (for the four speed intervals)according to (c) and indices 0.6, 0.8, 1, and 1.1according to (b). In Fig. 8(b), an example is shownof the geometry of a weld with an irregularity witha longer length scale (the weld is of bad quality,with maximum height of 0.8 mm). The weld hasQIs 0.6, 0.9, 1.4, and 1.4 according to (c) and 1.1,1.4, 2, and 2 according to (b). As the assessmentconcept (c) has these drawbacks, concept (b),based on first derivatives or geometrical gradients,proves to be most appropriate for practicalimplementation.

In Figs 8(c) and (d), examples are shown of thepractical differences between the current assessmentmethod (a) and the proposed method (b). The weldgeometry in Fig. 8(c), which shows an aggressivestep, is accepted according to many currentstandards (which allow a vertical tolerance ofþ0.3 mm on a 1 m basis for many conventionaltracks and on a 1.6 m basis for the French highspeed (TGV) track [7]). However, according to thenewly proposed method based on the gradient ofthe signal, the weld has indices 1.1, 1.7, 2, and 2.5for the four speed intervals and should be rejected.In Fig. 8(d), an example is shown of an almost perfectweld. However, as it has some negative heightcoordinates, it is rejected according to the currentstandards, which do not allow negative values.According to the newly proposed method, the weldhas indices 0.5, 0.7, 0.9, and 1 and is, therefore,accepted for all speed intervals.

Fig. 8 Measurement examples of longitudinal weld

geometries (a) with indentations due to non-

uniform shrinkage after welding and grinding;

(b) with an irregularity with longer length

scale; (c) with a step; and (d) with negative

height coordinates

Table 1 Empirical intervention levels for first and second

derivative of the longitudinal rail weld geometry

differentiated on line-section speed

Intervention levelfor the discretizedweld geometry Speedinterval (km/h)

Gradient(mrad)

Secondderivative[1023 m21]

0–40 3.20 202.5040–80 2.40 101.2580–140 1.80 82.65140–200 0.90 81.00



6 GEOMETRICAL CORRELATION BETWEENTHE ORIGINAL GEOMETRY AND THEFIRST AND SECOND DERIVATIVES OF ADISCRETE SIGNAL WITH A LIMITEDEXTREME GRADIENT

In the previous, the establishing of interventionlevels for the original weld geometry z(x) and itsderivatives dz=dx and d2z=dx2 were considered asmethods of assessment, where z(x), dz=dx, andd2z=dx2 were treated as independent variables (inthe sense that standardization of the maximum ofone of them does not influence extreme values ofthe other variables). However, limitation of theextremes of the first or second derivative of a discretespatial signal with limited length implicitly leads to alimitation of the geometry itself and the remainingderivative. This phenomenon will be the subject ofthis paragraph.

To determine the relations between the originalgeometry and the derivatives of the sampled geometrywhen their extreme values are limited, the followingquantities are introduced, along with their definitions:

(a) u: the angle of the signal with the horizontal axisor the slope in an arbitrary sample interval; itsstandardized maximum is denoted as a;

(b) tan u: the discrete gradient of the signal in sampleinterval i according to tan ui ¼ ziþ1 � zið Þ=d; itsstandardized maximum equals tana;

(c) Du: the relative angle (slope) of the signal in twosuccessive intervals at the sample position;

(d) Dtan u: the relative discrete gradient of the signalin two successive intervals or the discrete tran-sition in the gradient at the sample position;

(e) dz=dx: the first derivative of the signal at sampleposition i according to dzi=dx ¼ ziþ1 � zi�1ð Þ=2d;

(f) d2z=dx2: the second derivative at sample positioni according to d2zi=dx

2 ¼ ziþ2 � 2zi þ zi�2ð Þ=4d2.

To be able to work with continuous expressions forthe contact force in time in a later stage, also the ver-tical rail geometry must be defined in the intervalbetween the sampling positions. The most basiclinear interpolation of the signal is chosen to obtaina continuous signal.

In Figs 9(a) and (b), the above quantities are shownand compared for a sampled harmonic signal withthe minimum wavelength 4d. Comparison ofFigs 8(a) and (b) shows that, for a harmonic signal,the values of the gradient and the first derivative areequal in the sampling positions, which is not truefor the relative gradient and the second derivative.

Now, the special case is considered that the discre-tely sampled rail geometry, obtained after the aver-aging and filtering process, is a purely harmonicsignal with a gradient in each interval at the interven-tion level (Figs 10 and 12). In this particular case, onewavelength is fully described by five sampling points(the wave is quadri-linear). The sample interval isd ¼ 25 mm and the signal length is l ¼ 1 m ¼ 40d.The minimum wavelength is given by Lmin ¼ 4d( ¼ 100 mm); the maximum equals Lmax ¼ 80d( ¼ 2000 mm). The amplitudes corresponding tothese wavelengths are given by ymin ¼ d tana ¼

25mm � tan a and ymax ¼ 20d tana ¼ 500mm � tana,respectively. In general, the amplitude correspondingto wavelength Ln can be written as

yn ¼ 14Ln tan a ¼ nd tan a; Ln ¼ 4nd (10)

where n is the number of considered harmonics, withn ¼ 1 chosen to correspond to the minimum wave-length; 1 4 n 4 20. Figure 10 shows the relationbetween the wavelength and the amplitude of the dis-crete signal.

Next, the relation between the gradient and theoriginal geometry is considered. The extreme value

Fig. 9 Discrete harmonic signal with minimum wavelength, discrete gradients and relative

discrete gradients (left), and derivatives in the sampling positions (right).



of zi(x) corresponding to a standardized gradienttana is reached for a harmonic signal with maximumwavelength Lmax ¼ 80nd (2 m) and equals (Fig. 10)

zmax ¼14Lmax

dz

dx

��norm

; 14Lmax ¼ 0:5m (11)

where dz=dx��

norm¼ tana. In Table 2, the values of zmax

are given for the intervention levels for the gradient,as given in Table 1. The values obtained withequation (11) are theoretical maxima correspondingto the norm. For practical relevance, in Table 2, theextreme values are also given assuming a more rea-listic harmonic waveform (described by 80 samplingpoints) instead of a quadri-linear waveform withwavelength 2 m. Therefore, the values in the fourthcolumn of Table 2 give an indication of the relationbetween traditional standards in terms of verticaltolerances and the proposed assessment in terms ofdiscretized gradients.

Finally, the relation between the first and secondderivatives of a discrete signal is addressed. InFig. 11(a), the relation is shown between the discretegradient and the relative discrete gradient for a dis-crete signal with limited gradient tana. A linear and

directly proportional relation exists, with a band-width 2tana (this bandwidth is symmetric or thepossible deviation equals the limit value). It may beconcluded that for a discrete signal, a limitation ofthe gradient implies a limitation of the relative gradi-ent at the sampling positions. This is due to thefact that a linear interpolation between the samplingpositions is used, thus limiting the transition in angleto the sum of the minimum and maximum angles ortwice the allowed angle.

A similar relation exists between the first andsecond derivatives of a discrete signal, when its firstderivative is limited. This is shown in Fig. 11(b).Given a maximum value for the first derivative y=d,the limits of the second derivative, in absolutesense (i.e. irrespective of the actual slope), aregiven by

d2

dx2zi ¼

(ziþ2 � zi)=2d � (zi � zi�2)=2d

2d

¼ +y=d + y=d

2d¼ +

y

d2ð12Þ

For a standardized maximum slope a and anactual slope u, the maximum relative slope isgiven by u+ a (Fig. 11(a)) or the bandwidth istwice the norm. The second derivative has a smallerbandwidth. For a standardized maximum valueof the first derivative y=d (corresponding to a)and an actual value dz=dx of the first derivative,the maximum value for the second derivative isgiven by

d2

dx2zi ¼

dz=dx+ y=d

2d( ¼

dz=dx+ tana

2d) (13)

Or, the bandwidth is inversely proportional tothe sample interval and directly proportional to thestandardized slope. Where a doubling of the relativegradient may occur, a reduction of the secondderivative with 1/d is found (Fig. 11(b)).

Fig. 10 Geometry development possibilities for a

harmonic signal with gradients in each

interval at the intervention level

Table 2 Relation between an assessment method based on gradients and the traditional method

based on tolerances

Speed interval(km/h)

Gradientintervention

leveldz

dx

��norm

(mrad)

Theoretical toleranceon 1 m basis

zmax ¼ 0:5dz

dx

��norm

(mm)

Tolerance on 1 mbasis for a ‘single-wave

form’ zmax �1

p

dz

dx

��norm

(mm)

0–40 3.20 1.60 1.0240–80 2.40 1.20 0.7680–140 1.80 0.90 0.57140–200 0.90 0.45 0.29



7 RELATIONSHIP BETWEEN THEINTERVENTION LEVEL FOR THEGRADIENT OF THE RAIL GEOMETRY ANDTHE LOCAL MAXIMUM DYNAMICWHEEL–RAIL CONTACT FORCE

Equation (8) described the dynamic wheel–rail con-tact forces in terms of the second spatial derivative ofthe discrete signal of the rail geometry. Substitutionof equation (13) yields the following expression forthe dynamic contact forces in terms of the firstspatial derivative of the discrete signal

Fdyn(x) ¼ bMtrackV2 1

2d

dz

dx+ tana

� �(14)

Thus, a linear relationship is found between thedynamic contact force and the first spatial derivativeof the weld geometry, however, within a certainbandwidth. The maximum of expression (14) isfound (with dz=dx

�� norm

¼ tana) by

Fdyn;max ¼ bMtrackV2 1

d

dz

dx

��norm

(15)

An important result has been obtained. Equations(14) and (15) establish a relationship (within a certainbandwidth) between the gradient of the discretelongitudinal contact geometry and the correspond-ing force between wheel and rail, when the gradientis standardized.

In equation (15), the full bandwidth was used todetermine the maximum (in Fig. 11(b), the bound-aries of the band were used). From a statisticalpoint of view, it can be argued that the discrete gra-dients have a distribution which is, over a large partof the gradient domain, close to normal with zeromean (as will be shown in section 9). Therefore, azero bandwidth will be closest to reality when maxi-mum contact forces for a large number of welds areconsidered. Using the central relation in Fig. 11(b)with zero bandwidth yields

Fdyn;max ¼ bMtrackV2 1

2d

dz

dx

��norm

(16)

Equation (15) gives the deterministic maximumcontact force corresponding to a specific sampledgeometry. The statistical uncertainty introduced by

Fig. 11 Implicit limitation of relative gradient by limitation of the gradient of a discrete signal

(left) and implicit limitation of second derivative by limitation of the first derivative of

the signal (right)

Fig. 12 Linear relation between the wavelength and the

amplitude of a quadrilinear harmonic signal



the bandwidth can be accounted for by validation ofa new calibration factor that will be introduced in thefollowing; this way, it makes no difference whetherexpression (15) or (16) is used.

Now, again the particular case of a longitudinalweld geometry consisting of a purely harmonicsignal with a discrete gradient in each samplinginterval corresponding to the intervention level isconsidered (Figs 9 and 10). The maxima of first andsecond derivatives are then given by (Fig. 9(b))

ynnd

¼ tana (17)

and

ynn2d2 ¼ tana

nd(18)

respectively, where the linear relation betweenamplitude and wavelength (10) was used. Substi-tution of equation (17) in equation (14) and equation(18) in equation (9) yields

Fdyn;max ¼ bMtrackV2 tana

d(19)

Fdyn;max ¼ bMtrackV2 tana

nd(20)

A comparison of expressions (19) and (20) showsthat the expression for the maximum dynamicforce obtained from the second derivative dependson the wavelength (or, alternatively, on n), whereasthe expression obtained by substitution of therelation between first and second derivatives is inde-pendent of the wavelength. This result is also validfor a harmonic signal described by more samplingpoints per wavelength. However, results for a harmo-nic signal cannot be extended automatically to anarbitrary geometry. The sampled geometry (withfinite record length) can be decomposed into afinite number of harmonics using the discrete Four-ier transform (DFT). Because the Fourier transform isa linear operation, the total dynamic response can befound as the superposition of the responses to eachcomponent. However, a constraint on the gradientof the signal does not imply a constraint on the gra-dient of each of the components in the summation.Therefore, result (15) is more general.

Equations (15) and (16) may be summarized in ageneral form, with a dimensionless calibration factor g

Fdyn;max ¼ gMtrackV2 1

d

dz

dx

��norm

(21)

g is the dimensionless calibration factor (–) (which

may vary per interval for the line-section speed),Mtrack the equivalent track mass (kg), V the trainspeed (m/s), d the sampling distance in the averagedand filtered rail geometry signal (m), and dz=dx

�� norm

the standardized gradient of this signal (–).Thus, a final relation is established between the

maximum dynamic wheel–rail contact forces andthe standardized gradient of the discrete rail geometry.

8 VALIDATION OF THE INTERVENTION LEVELSAND EXTENSION TO HIGH-SPEED ANDHEAVY-HAUL LINES

There is an important difference between axle loadsfor conventional trains and tracks and high-speedand heavy-haul tracks, respectively. For high-speedtrains, the static component of the vehicle–trackinteraction force is much lower (�170 kN), whereasrunning speeds are much higher (�300 km/h) thanfor conventional tracks (usually ,225 kN withspeed 140 km/h). For heavy-haul tracks, the oppositeholds: axle loads are much higher (�350 kN) andrunning speeds are much lower (,100 km/h).

Therefore, allowable dynamic amplifications aremuch lower for heavy-haul lines than for high-speed lines. However, as the dynamic componentof the vehicle–track force (according to equations(9) and (21)) increases quadratically with the speed,weld geometry requirements for high-speed linesneed to be much stricter in a consistent approach.Also from a practical point of view, for heavy-haullines, the rail welds are not a real issue, as rails onthese tracks are often ground. The earlier relationsamong geometrical weld quality, static axle load,and running speed are expressed in the following for-mula, where Q represents the axle load and equation(21) has been used for its dynamic component

Qtot ¼ Qstat þ gM

dV 2 dz

dx

��max

��!

dz

dx

��max

, DQ1

g

d

M

1

V 2ð22Þ

On the basis of this expression, an interventionlevel for heavy-haul tracks has been proposedin reference [8]; the focus in this article will be onrequirements for high-speed tracks and thevalidation of intervention levels.

Requirements for high-speed lines can easily bederived on the basis of the geometrical quality of thesurface of new rails in longitudinal direction. Thereis no need for rail welds to have a better geometricalquality than the rails themselves. To determine thisquality, geometry measurements have been carriedout on new straightened rails; a population of 100



segments of 1 m length has been measured [9]. Thedistribution function of the calculated rail QI forthese segments is given in Fig. 13. This rail QI hasbeen calculated assuming an intervention level of0.7 mrad for the discretized rail gradient.

Assuming this intervention level, 95 per cent of therail measurements satisfy the requirements and havean index less than 1. Therefore, 0.7 mrad may betaken as an appropriate value for the interventionlevel of the weld geometry gradient for runningspeeds up to 300 km/h (and higher). In fact, thisaccuracy is very close to the maximum obtainableaccuracy in welding; often, special grinding equip-ment (e.g. a grinding train) is necessary to realizethis accuracy [9]. In Fig. 14, cumulative distributionsare shown of rail weld geometry measurements onthe Dutch high-speed line after manufacturing andafter grinding with a grinding train (one single pas-sage with two types, leading to a QI less than 2 for99 and 96 per cent of the populations, respectively).

9 STATISTICAL ANALYSIS OF WELD AND RAILMEASUREMENT DATA: PARAMETERDISTRIBUTIONS

The consequences of the proposed new geometricalassessment of rail welds have been studied by per-forming some statistical analyses on weld measure-ment data. A sample has been taken with a set of110 rail weld measurements (these welds have beenmanufactured with an assessment based on theprinciple of tolerances).

In Fig. 15, the cumulative distribution functions forthe weld QI are shown for the earlier introduced trainspeed intervals; also the high-speed norm has beenadded. The norm value 1 for the QI is also shown.The QI range less than 1 is shaded; parts of thefunctions situated in this area are ‘approved’. Someadditional statistical information is given inTable 3. It is observed that the differentiation of stan-dards to the line-section speed has an importantinfluence on the relative percentage of acceptance.

In Fig. 16 at the left-hand side, the QI is shown forthe weld population, plotted against the standardnormal variable. If the resulting function is a straightline, this implies that the QI has a normal distri-bution (with mean and standard deviation followingfrom the graph). In Fig. 16 (left), the best linear fit isshown; this is shown for 300 km/h only (using theleast-squares method), as all graphs are basicallyderived from the same data set and have the same

Fig. 13 Cumulative distribution function of the QI (at

300 km/h) for new vertical rail geometry (100

segments of 1 m)

Fig. 14 Influence of grinding with a grinding train on

the cumulative distributions of the QI (at

300 km/h) for welds on high-speed track

Fig. 15 Cumulative distribution function of the QI for

a weld population of 110 welds at different

line-section speeds

Table 3 Statistics for a population of 110 vertical geome-

try measurements of rail welds

Line speed (km/h) 40 80 140 200 300

Mean QI 0.87 1.16 1.55 3.10 3.98Standard deviation 0.43 0.57 0.76 1.53 1.97Coefficient of variation 0.49 0.49 0.49 0.49 0.49Acceptance: QI in two digits 74% 46% 23% 2% 2%QI with one digit id 55% 33% id id



Fig. 17 QI of 100 rail segments versus standard normal variable on linear (left) and logarithmic

(right) scales

Fig. 16 Standard normal variable versus the weld QI. Linear scale (left) and logarithmic scale

(right) for a population of 110 rail welds and different line-section speeds

Fig. 18 Left: cumulative distribution functions of discretized weld gradients. Right: discretized

weld geometry gradient versus standard normal variable



‘shape’. It is clear that a linear fit is a bad approxi-mation. The same procedure has been repeated forthe right-hand plot in Fig. 16 using a logarithmicscale for QI. A linear relation between the standardnormal variable and ln(QI) implies a log-normal dis-tribution of the QI. The result is much closer to alinear relation; neglecting the ‘tails’ (which containonly very few data points), the approximation by a

linear function is rather accurate. Furthermore, inthe lower tail, the QIs with a low value, or thealmost perfect welds, are found; these are not somuch of interest.

Comparison of the left and the right plots in Fig. 16shows that the distribution of the QI (and thus themaximum slope in the weld geometry) can beapproximated as log-normal. This could have beenexpected, as the QI follows directly from the maxi-mum slope in the weld geometry. When consideringthe discretized gradient itself as a normal randomvariable, this maximum should have an extremevalue distribution, which always has some expo-nential form. If the gradient would be exactlynormally distributed, its maximum would have atype I – Gumbel distribution. The fact that theabsolute values of the gradient are taken and thatthe gradient itself does not have an exactly normaldistribution (which is shown later in this section)disturbs this picture.

In Fig. 17, the same procedure has been repeated,but now for the QI of the rail segments. Clearly,

Fig. 20 Possible screen output of a measuring device

for the assessment of the geometry of rail

welds

Fig. 19 Left: cumulative distribution functions of discretized rail gradients. Right: discretized rail

geometry gradient versus standard normal variable

Fig. 21 Semi-mechanical grinding of the rail weld surface in longitudinal direction, ground rail,

and electronic geometry measurement



the log-normal distribution is a rather accuratedescription for this variable.

In Figs 18 and 19, the behaviour of the discretizedweld and rail gradients are analysed (not theirrespective maxima). Distribution functions areshown in the left plot; in the right plot, their valueshave been plotted against the standard normal vari-able. A log-normal distribution is not possiblebecause of the presence of negative gradient values.It can be concluded that for the discretized gradientsof the welds, the normal distribution is accurate, withmean value exactly at zero, when the tails (whereextreme values occur) are disregarded; in the interval[20.003, 0.003 mrad], the normal distribution fitswell. This conforms the expectations, as the gradientis a random variable. The limitation of the validitydomain of the normal distribution is probably dueto an implicit limitation of this random variableintroduced by the vertical tolerances, which causesthe deviating ‘tails’. This is confirmed in Fig. 19; thediscretized rail geometry gradient itself has analmost perfect normal distribution with a meanvalue exactly at zero.

10 PRACTICAL IMPLEMENTATION OF THEPROPOSED ASSESSMENT METHOD

In practice, after welding of rails, the quality of thegeometry can be measured using an electronicdigital straightedge. In its processor, the routinesfor the calculation of the QI can be programmed.The device then samples the rail geometry andplots the QI on a screen. With the help of thisoutput, the geometry can be optimized such thatthe standards are met. An example of such a screenoutput is given in Fig. 20. At the location wherethe longitudinal rail geometry shows a relativelylarge angle, the gradient, normalized with thenorm value for 40 km/h, exceeds the norm (the QIis 1.7), and the weld should be ground beforeacceptance.

In Fig. 21, the steps for the assessment of rail weldsare shown in practice.

11 CONCLUSIONS AND CONSIDERATIONS ONFURTHER RESEARCH

An assessment method for the geometrical assess-ment of rail welds has been proposed, based on alimitation of the maximum of the gradient of the dis-crete measurement signal. The limitation of the gra-dient implicitly limits the maximum dynamicwheel–rail contact force at the weld. This assessmentmethod has been standardized in the Netherlands(2005).

However, there remain several open questions andaspects, which need further experimental and theor-etical research, which will be carried out in the nearfuture. This will allow for the validation of the newstandards. The subjects to be investigated are asfollows.

1. The level of acceptance, which was chosenempirically on the basis of experience (at leastfor conventional tracks), should be coupled withlife-cycle analyses, where the most critical com-ponent(s) in a track system should be determined.In this way, an optimization of the interventionlevels can be performed.

2. Measurements should be carried out to verify andvalidate equation (21), the basic tool for new weldassessment, with respect to the coefficient forobtaining quantitative results for the dynamicwheel–rail interaction force.

3. Monitoring of the initially realized weld geometryas a function of the passed tonnage (improvementversus deterioration of the weld quality as a resultof plastic deformations and wear) is an importantissue.

4. This holds also for the monitoring of the overalltrack quality (the other track components suchas rail fasteners, sleepers, and ballast) versusinitial weld quality as a function of passedtonnage.

REFERENCES

1 Steenbergen, M. J. M. M. and Esveld, C. Voorstelvoor normering van lassen in spoorstaven. Report 7-04-220-7, ISSN 0169-9288, Delft University of Technology,Delft, 2004.

2 Steenbergen, M. J. M. M., Esveld, C., and Dollevoet,R. P. B. J. New Dutch assessment of rail welding geome-try. Eur. Railway Rev., 2005, 11, 71–79.

3 Anon. Manchester benchmarks for rail vehicle simu-lation. Veh. Syst. Dyn., 1998, 30, 295–313.

4 De Man, A. P. Dynatrack, a survey of dynamic railwaytrack properties and their quality, 2002 (Delft UniversityPress, Delft, The Netherlands).

5 Jenkins, H. H., Stephenson, J., Clayton, G. A.,Morland, G. W., and Lyon, D. The effect of track andvehicle parameters on wheel/rail vertical dynamicforces. Railway Eng. J., January 1974, 2–16.

6 DARTS-NL. Dynamic analysis of a rail track structure,available from www.esveld.com, 2005.

7 Esveld, C.Modern railway track, 2nd edition, 2001, p. 56,316, 570 (MRT-Productions, Zaltbommel, TheNetherlands).

8 Esveld, C. and Steenbergen, M. J. M. M. Force-basedassessment of weld geometry. In Proceedings of 8thInternational Heavy Haul Conference, Rio de Janeiro,Brazil, 14–16 June 2005, pp. 51–58.

9 Esveld, C. Measurements of rail and weld geometry onHSL-South, 2005 (ECS, Zaltbommel, The Netherlands).



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