Wheel Load Contact Area

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44 International Journal of Civil Engineerng. Vol. 8, No. 1, March 2010

1. Introduction

Ballasted railway tracks mainly comprise twomain parts: superstructure and substructure. Steelrails, various types of rail fasteners, timber, steel,or concrete sleepers, and granular ballast, sub-ballast, and subgrade materials are majorcomponents used in railway track construction.Historically, the understanding of track structuralbehavior has been facing difficulties. This is dueto different mechanical properties of trackcomponents from one side, and complexinteraction between track components from theother side.

Wide range of track configurations could bedesigned and constructed. This makes railwaytrack structure to be subjected to regular changes.Consequently, based on theoretical andexperimental investigations, large number andsometimes contradictory design criteria havebeen suggested by railway authorities andpractitioners. Such a diversity of design criteriausually makes railway track design procedure adifficult task. This highlights the necessity of aneed for a thorough review of the currently usedrailway track analysis and design methods.

This paper focuses on a wide range of railwayrelated design codes as well as a great deal ofrecent technical researches the suggestions ofwhich are usually utilized in the analysis anddesign of railway tracks. Results of this review ispresented and discussed in order to indicate therequired improvements needed to be consideredin the current railway track analysis and design.

2. Conventional Methods of Analysis and

Design of Railway Tracks

Generally, railway track systems are designedto provide a smooth and safe running surface forpassing trains. They also serve to sustain theloads imposed to track structure mainly as aresult of trains passages and temperaturechanges.

Due to great variety of structural elementsused in track system, it is usually more practicalto perform the analysis and design procedure foreach element as a single structural unit. Such anapproach, subsequently, includes the interactionbetween track components through the definitionof suitable boundary conditions and load transferpatterns. Furthermore, since the dynamicresponse characteristics of the track are notsufficiently well understood to form the basis ofa rational design method, current practicesgreatly relies on relating the observed dynamicresponse to an equivalent static response. This iscarried out by making use of various load factors.

Evaluation of conventional methods in Analysis and Design ofRailway Track System

J. Sadeghi1,* and P. Barati1

Received: September 2009 Accepted: February 2010

Abstract: Current practices in railway track analysis and design are reviewed and discussed in this paper. Themechanical behavior of railway track structure comprising of various components has not been fully understood dueto the railway track structural complexity. Although there have been some improvements in the accuracy of currenttrack design methods in recent decades, there are still considerable uncertainties concerning the accuracy andreliability of the current methods. This indicates a need for a thorough review and discussion on the current practicesin the analysis and design of railway track systems. In this paper, railway design approaches proposed by variousstandards along with the results of a wide range of technical researches are studied and necessary suggestions aremade for the improvement of current practices in the analysis and design of railway tracks.

Keywords: railway track, track components, analytical models, design criteria, track design procedure.

* Corresponding Author: Email: [email protected]

1 Associate Professor, Iran University of Science andTechnology, Narmak, Tehran, Iran

2 Research Assistant, Iran University of Science andTechnology, Narmak, Tehran, Iran

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45J. Sadeghi and P. Barati

This method is being widely accepted and used inanalysis and design of railway track systems.

The track is designed utilizing load bearingapproach [1] to ensure that the concentrated loadsof the wheels are transferred to the formationwhile ensuring that the strength of thecomponents are not exceeded. Several importantcriteria are defined to secure this objective. Thesecriteria mainly include limits on rail and railfastener stresses and deflections, sleeper strength,pressure distributed in sleeper-ballast interface,and the stresses distributed on granularsupporting layers underneath the track.

2.1. Rail

Rail as the most important track elementsubjected to wheel loads must be able to securelysustain these loads applied in vertical, lateral, andlongitudinal directions and subsequently transferthem to the underlying supports. Rail is the trackelement which is in direct contact with the rollingstock. It is therefore very necessary, in particularfrom a safety point of view, to ensure the properfunctioning of rails in the track system.

The most important recommended criteriaused in the conventional rail analysis and designprocedure are shown in Figure 1. As it isillustrated in this figure, rail design criteria aremainly divided into two categories. Structuralstrength criteria include wheel-rail contactstresses and rail bending stresses. Havingsatisfied the structural strength criteria, theserviceability requirements should be completelymet for a specific rail section to ensure its properstructural and operational performance. In fact,aside from the calculation and controlling of rail

stresses, it is critical that the design engineer hasdeep understanding about the real operatingconditions the railway track may experience.

Current practices in the calculation of railbending moments and vertical deflection aremainly based on “beam on elastic foundation”model. This model proposed for the first time byWinkler in 1867 and thereafter developed byZimmerman in 1888 [2]. The basic assumption inthe Winkler model is that the deflection of the railat any point is proportional to the supportingpressure under the rail (see Figure 2).

The corresponding equations for thecalculation of rail bending moment and raildeflection are as follow:

(1)

(2)

where, y(x) and M(x) are the verticaldeflection and the bending moment of the rail atthe distance “x” from the load point, respectively.The parameter is defined with the followingequation:

(3)

Winkler model is basically developed for acontinuosly supported beam on an elasticfoundation. This approach neglects some realconditions of railway tracks. First, theassumption of continous support under the raildoes not reflect the effects of actual discretesupport provided by cross sleepers. Second, thismodel does not include the interaction betweensupport materials (i.e. ballast, sub-ballast, andsubgrade materials) and simply uses a Bernoulli-Euler beam theory to calculate rail deflectionsand bending moments. Moreover, different tracksupporting layers are not clearly distinguished,

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Rail Design Criteria

Structural Strength Serviceability

Wheel-RailContact Stresses

Rail Bending Stresses

Rail Deflection

Rail Wear Life

Fig. 1. Recommended rail design criteria

Fig. 2. Beam on elastic foundation model

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and track support is considered as a one-layercomponent. Finally, it is assumed that supportingsleepers fastened tightly to the rail would resistagainst rail bending through their rotationalstiffness. The latter is another area of deficiencyin Winkler model.

Some researchers have questioned thereliability of the Winkler model. As a result, morerealistic approaches are proposed and someimprovements are made in the basic “beam onelastic foundation” model. For instance, “beamon discrete support” model has been developedand more recently analyzed using a practicalenergy approach [3] to compensate for the errorscaused by the assumption of continuous supportunder the rail. “Pasternak foundation” and“double beam” models are also introduced [2] totake into account the interaction between tracksupporting layers and multi-layer nature of tracksupport, respectively. Kerr [1] has reported theresult of a research in which the effect of therotational stiffness of the sleepers in calculationof rail deflection and bending moment areconsidered.

Design procedure for a specific rail sectionalways starts with the calculation of the designwheel load. This load is defined as the product ofstatic wheel load and a corrective factor known asdynamic impact factor to compensate fordynamic as well as impact effects of wheel loadresulted from wheel and rail surfaceirregularities. Taking into account variousparameters which affect the magnitude ofdynamic impact factor, several researchers alongwith railway authorities has been proposeddifferent relationships for the estimation of thisparameter. Some of these equations aresummarized in Table 1.

The magnitude of vertical rail deflectioncalculated using Equation (1) is greatlydependent upon track modulus. Track modulus isdefined as the load required producing unitvertical deflection in unit length of the rail. Fortypical tracks with light to medium rails,AREMA [5] recommends a value of 13.8 MPawhen calculating vertical rail deflection.

The rail bending stress is usually calculated atthe center of the rail base assuming the purebending conditions to be applicable. The bending

stress at the lower edge of the rail head also maybe critical if the vehicles impose high guidingforces between wheel flange and rail head whilepassing around the curves. Having calculated themagnitude of rail bending stress, comparisonshould be made between this stress and theallowable limit. AREMA [5] has recommended apractical methodology for calculation andcontrolling of rail bending stress based uponfatigue consideration and through thedetermination of several safety factors.According to this method, the allowable bendingstress is defined as:

(4)

where, y is the yield stress of rail steel and tis the longitudinal stress due to temperaturechanges and can be calculated using thefollowing equation:

(5)

The parameters A, B, C, and D in Equation (4)are safety factors to account for rail lateralbending, track condition, rail wear and corrosion,and unbalanced superelevation of track,respectively. Some of the recommended values ofthe above safety factors are presented in Table 2.

Wheel-rail contact stresses mainly includerolling and shear stresses. The magnitude of thesestresses is greatly dependent upon the geometry

tEt �� ..��

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Developer Equation

AREMA DV21.51��

Eisenmann t..1 ��

ORE �� 1

BR2/1

21 )(784.8 �

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India 5.014.581

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South Africa DV92.41��

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Sadeghi 264 10108098.1 VV ��

Table 1. Recommended relationships for the calculation ofdynamic impact factor [4]

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of ellipsoidal wheel-rail contact patch. Manyinvestigations have been carried out to developreliable formulations for the calculation of thesestresses. The most applicable formulas are thosesuggested by Eisenmann [7]. He conducted ananalysis of rolling and shear stress levels inwhich the simplifying assumption of uniformdistribution over the wheel-rail contact area wasmade. In this analysis wheel and rail profileswere also represented by a cylinder and a plane,respectively (see Figure 3).

Based on Hertz’s theory, Eisenmann [7]suggested the following formula for thecalculation of the mean value of the rollingcontact stress:

(6)

where, 2b (mm) is the breadth of wheel-railcontact area. Eisenmann [7] adopted the value of2b=12 mm. The contact length (2a) is alsocalculated from the following formula:

(7)

The values of wheel loads transferred to therail head through the contact area often exceedthe yield limit of the contacting materials. In thissituation, the resulting surface plasticdeformations jointed with wear processes acts to

flatten out the contact area. Therefore, the contactsurface can be approximated by a rectangle of thelength of 2a and the breadth of w based upon theassumption of contact between a plane (rail) anda cylinder (wheel). For such a condition, Smithand Liu [8] suggested the following formula forthe calculation of the contact length:

(8)

Considering required fatigue strength for railsteel, Eisenmann [7] proposed the limit value formean rolling contact stress as a percentage of theultimate tensile strength of rail steel. Based onthis assumption, subsequent criterion issuggested:

(9)

Shear stress distribution is chiefly occurs in therail head area and is in a close relationship withthe magnitudes of normal principal stresses.Eisenmann [7] observed that the values of majorand minor stresses do not follow the samereduction patterns with increasing depth from therail head surface. Such a discrepancy results inthe appearance of a maximum value of shearstress at a depth corresponding to half of thecontact length. Maximum shear stress value issimply interrelated to mean rolling contact stressvalues and is given with the following equation:

(10)

As indicated earlier, the magnitudes of shearand rolling contact stresses are interrelated.Using the theory of shear strain energy appliedfor the condition in which the two principalstresses are compressive, the subsequent criterionfor the shear stress limit could be obtained:

(11)

Criteria related to the performance of the railsunder operating conditions mainly include rail

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Table 2. Values of rail bending stress safety factors [6]

Fig. 3. Uniform distribution of wheel-rail contact stress [2]

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vertical deflection and rail wear life. Thesecriteria are presented hereunder.

AREMA [5] has proposed a limiting range forthe magnitudes of vertical rail deflections.According to this recommendation, extremevertical rail deflections should be kept within therange of 3.175 to 6.35 millimeters. Lundgren andhis colleagues [9] has incorporated thisrecommendation and proposed the diagrampresented in Figure 4 as the limit values ofvertical rail deflection. This diagram is basedupon the capability of the track to carry out itsdesign task.

Domains indicated in Figure 4 are described asfollows:

A: Deflection range for track which will lastindefinitely.

B: Normal maximum desirable deflection forheavy track to give requisite combination offlexibility and stiffness.

C: Limit of desirable deflection for track of lightconstruction (with rails weigh < 50 kg/m)

D: Weak or poorly maintained track which willdeteriorate quickly.

It should also be noted that values of deflectionin Figure 4 do not include any looseness or playbetween rail and pad or pad and sleeper. Inaddition, these values represent deflectionsdirectly under the wheel load.

The other serviceability criterion is the railwear life. Although many investigations havebeen carried out to develop a rational method forestimation of this parameter, the results at bestare still empirical and have no theoreticalsupport.

The University of Illinois has conducted aresearch on some U.S. railway tracks in order toinvestigate the rail wear rate and develop amethod for the estimation of rail wear life. The

following formula is suggested by the researchersof this university for the estimation of annualabrasive rail head area wear (mm2/year) [10]:

(12)

Having estimated Wa and considering themaximum rail head limit , the rail wear lifecould be calculated from the following formula:

(13)

Danzig and his colleagues [11] from theAREMA association also carried out extensiveinvestigations to find a proper formulation for theestimation of rail wear life. Based on the resultsobtained, they suggested the following equationwhich represents the rail wear life in terms of MGTpassed the track over a specific period of time:

(14)

aWA

yT �

( A )

A.Dc.DwK.tWaW )1( ��Fig. 4. Track deflection criteria for durability [9]

Fig. 5. Envelope of rail wear limits for loss of rail headhight and width [17]

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Allowable rail wear limits have been proposedby many researchers and almost each railwayauthority has established its own specific criteria.For example, an envelope of maximum rail wearvalues for different rail sections is proposed byCanadian National Railway as presented inFigure 5. Using the diagrams outlined in thisfigure, maximum allowable rail head height andwidth loss could be determined.

The values indicate acceptable rail wear lifeusually range from 20 to 50 percent in rail headarea reduction. The weight of unit length of therail, the amount of MGT passed over the trackduring its service life, and the train speed are themost important parameters which determine theproper values of allowable rail wear limits to bechosen. The more weight of unit length of therail, the greater amount of rail head areareduction would be allowed. Oppositely, thegreater amount of MGT and higher values oftrain speed call for more limited rail head areareduction.

2.2. Sleeper

The sleepers play important roles in railwaytrack system. The primary function of thesleepers is to transfer the vertical, lateral andlongitudinal rail seat loads to the ballast, sub-ballast and subgrade layers. They also serve tomaintain the track gauge and alignment byproviding a stable support for the rail fasteners.

The vertical loads induce bending moments tothe sleeper which are dependent upon the degreeand quality of ballast layer compactionunderneath the sleeper. Besides, the performanceof a sleeper to withstand lateral and longitudinalloading is relied on the sleeper’s size, shape,surface geometry, weight, and spacing [17].

Current practices regarding the analysis anddesign of sleepers in vertical direction comprisethree steps. These are [4]: 1) estimation ofvertical rail seat load, 2) assuming a stressdistribution pattern under the sleeper, and 3)applying vertical static equilibrium to a structuralmodel of the sleeper.

Vertical wheel load is transferred through therail and distributed on certain numbers ofsleepers due to rail continuity. This is usually

referred to as vertical rail seat load. The exactmagnitude of the load applied to each rail seatdepends upon several parameters including therail weight, the sleeper spacing, the trackmodulus per rail, the amount of play between therail and sleeper, and the amount of play betweenthe sleeper and ballast [4]. Based on theseconsiderations, various relations are proposedand summarized in Table 3.

However, for the purpose of simplification, itwould be more practical to consider only theeffect of some of the above mentionedparameters and define, for example, the value ofvertical rail seat load as a function of sleepers’type and spacing. This approach is widelyaccepted by many railway authorities. For an

Table 3. Relations for the calculation of rail seat load

Fig. 6. Diagrams for the estimation of rail seat load [12]

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instance and as illustrated in Figure 6, AREMA[12] recommends diagrams in which thepercentage of wheel load transferred to thesleeper are drawn against the sleeper spacing.The effect of sleeper type and track modulus isalso included in these diagrams.

The exact contact pressure distributionbetween the sleeper and the ballast and itsvariation with time, will be of importance in thestructural design of sleepers. When track isfreshly tamped the contact area between thesleeper and the ballast occurs below each railseat. After the tracks have been in service thecontact pressure distribution between the sleeperand the ballast tends towards a uniform pressuredistribution [4]. This condition is associated witha gap between the sleeper and the ballast surfacebelow the rail seat. The most accepted contactpressure distribution patterns between sleeperand ballast are presented in Table 4.

Although some researches have beenconducted to determine the in-track distribution

of the pressure under the sleeper for designpurposes such as those recommended by Sadeghi[16] (see Table 5), but the results are still not aspractical as it is needed.

As indicated in Table 6 and to take intoaccount the sleeper support condition as real as itis possible, it is usually presumed that theuniform pressure under the sleeper distribute incertain portion of the sleeper length (area). Thislength (area) is referred to as “Effective Length(Area)” and commonly shown with “L (Ae)” inthe literature. This assumption is made tofacilitate the procedure of design calculations.The static equilibrium in vertical direction is thenapplied to acquire the magnitude of contactpressure under the sleeper. A factor of safety isalso included to account for variations in thesleeper support. Therefore, the average contactpressure between the sleeper and the ballast Pa

(kPa) can be acquired by:

(15)3.FB.Lrq

aP ��

��

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Table 5. In-track sleeper loading pattern [16]

Table 4. Some contact pressure distribution patterns [4]

Developer Description

AREMA [12]Ae: Two third of sleeper area at its bottom surface

UIC [13]

Ae = 6000 cm2 for l = 2.5 m Ae = 7000 cm2 for l = 2.6 m

Australia [14, 15])( glL ��

(1)

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l

g

L L

Table 6. Effective length (area) of sleeper support at railseat

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Having determined the sleeper loading pattern,its structural model could be defined for thecalculation of sleeper bending moments. Thissubject has drawn the special attention of manyrailway researchers and consequently, severalformulations have been suggested in theliterature. All of these proposed formulas andmethods could be classified based on sleeper typeas shown in Figure 7. Formulations of thesemethods are also summarized in Table 7.

It should be noted that there are somedifferences between the sleeper structural models(i.e. sleeper loading patterns) used for thecalculation of bending moments in each type ofthe sleepers indicated in Table 7. The completedescription of these differences along with otheradditional criteria could be found in references13, 14, 15, and 17.

Having calculated sleeper bending moments,bending strength of the sleeper should then beexamined and controlled. AREMA [18] hasdeveloped a practical method for the estimationof ultimate bending capacities of sleepers. In thismethod, based on sleeper length, type of bendingmoment (i.e. positive or negative), and locationof sleeper bending moment calculation (i.e. at railseat or at sleeper mid-span), limit values aredetermined.

2.3. Rail Fastener

Rail fasteners also known as fastening systemsare used in railway track structure to fasten therails on the sleepers and to protect the rail frominadmissible vertical, lateral, and longitudinal

movements as well as rail overturning. Moreover,these components serve as tools for gaugerestraining, wheel load impact attenuation,increasing track elasticity, etc. There areenormous different types of rail fasteners whichmainly are classified based on two importantaspects as presented in Figure 8.

Despite the important roles rail fasteners playin track system, current practices in analysis anddesign of these track components are notextensively explained in the literature. Designcriteria mentioned in many railway related designcodes are restricted to those dealt with theacceptance criteria of laboratory qualificationtests. AREMA manual [19], Australian Standard[20], and European Standards [21-25] are themost important design codes which include suchcriteria. A comparison between these standards ispresented in Table 8.

2.4. Ballast and Sub-ballast Layers

Ballast and sub-ballast layers are composed ofgranular materials and used in track structuremainly to sustain the loads transferred from thesleeper. Other important functions of these layersinclude: 1) to reduce the stress intensity to thelevel to be tolerable for subgrade layer, 2) toabsorb impact, noise and vibration induced fromthe wheels, 3) to restrict the track excessivesettlement, 4) to facilitate track maintenanceoperations, especially those related to thecorrection of track geometry defects, and 5) toprovide adequate drainage for the track structure,thereby track settlement as well as vegetation

Sleeper bending moment calculation

TimberSleepers

Steel Sleepers

Concrete Sleepers

Battelle

Schramm

Raymond

Australian Standard

UIC

Australian Standard

Fig. 7. Recommended methods for sleeper bendingmoment calculation

Sleeper bending moment calculation

TimberSleepers

Steel Sleepers

Concrete Sleepers

Battelle

Schramm

Raymond

Australian Standard

UIC

Australian Standard

Fig. 8. Classification of rail fasteners

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growth will be limited. In addition to the abovementioned functions, the sub-ballast layer is usedto act as a filter layer which prevents ballast andsubgrade materials to be mixed together.

Current practices in structural analysis anddesign of ballast and sub-ballast are dealt with thedetermination of minimum required depth fortheses granular layers. Theoretical, semi-empirical, and empirical methods have beendeveloped and are being used in order to satisfythis design criterion.

Theoretical determination of minimumrequired depth based on the results of Boussinesqelastic theory applied to a uniform rectangularloaded area (see Figure 9) is performed using the

numerical solution of the following equation:

(16)

It should be noted that, ballast and sub-ballastlayers are assumed as a single homogenous andisotropic layer in Boussinesq elastic theory.Although such an assumption seems not to besufficiently accurate, however, based on thecomparison to the available field tests results, theORE [26] investigations have proved the validityof Boussinesq elastic theory. This comparison ispresented in Figure 10. It is apparent from this

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Developer Rail Seat Moment Center Moment Mr

+ (kN.m) Mr- (kN.m) Mc

+ (kN.m) Mc- (kN.m)

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Steel Australian Standard [15]

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Australian Standard [14]

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Table 7. Comparison of different recommended methods for calculation of sleepers bending moments

Qualification Test

Design Code AREMA AS

1085.19EN

13146Uplift restraint � � �Longitudinal restraint

� � �

Repeated load � � �Torsional restraint

� � �

Lateral restraint � � �Clip spring rate � � �Fatigue strength � � �Impact attenuation

� � �

Table 8. Comparison of different railway design codeswith the consideration of rail fastener qualification tests

Fig. 9. Application of Boussinesq’s elementary singlevertical concentrated load over a uniformly loaded

rectangular bearing area [17]

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figure that the results obtained from Boussinesqmethod, reasonably remain within the envelopeacquired from experimental investigations.

Simplified semi-empirical methods are alsoemployed which assume that the load isdistributed vertically with a load spread slope of1 vertical to 1 horizontal or a slope of 2 verticalto 2 horizontal. Moreover, in this method, it isassumed that the stress distribution to be uniformat any given depth below the surface. Thismethod calculates only the average verticalpressure at depth while the Boussinesq methodcalculates the maximum vertical pressure at adepth below the loaded area [17]. A comparisonof the vertical stress distribution calculated forboth 1:1 and 2:1 load spread assumptions withthe theoretical Boussinesq solution is presentedin Figure 11. It is clearly apparent that theassumed 2:1 load spread distribution of verticalpressure more closely approximates theBoussinesq pressure distribution than that of 1:1load spread distribution.

Considering the 2:1 load spread distribution asindicated in Figure 12, the minimum requiredballast and sub-ballast depth can be calculatedfrom the following formula:

(17)

Having determined the allowable subgradeload carrying capacity and substituting it in theequations (16) or (17) the minimum requiredballast and sub-ballast depth can be calculated.

3. Discussion and Conclusions

Current practices of anlysis and design ofrailway track systems are entirely based on loadbearing approach in which every trackcomponent is evaluated separately for itssufficient strength to sustain the environmentaland traffic loads. Various analysis methods anddesign criteria have been reviewed in this paper.Based on comparisons of the available railwaydesign methods, some shortcomings arediscussed in this section.

Results of the main proposed dynamic impactfactors presented in Table 1 are drawn againsttrain speed in Figure 13. It is apparent from thisfigure that train speed can considerably influence

))((.

zLzBLB

aPz ��

Fig 10. Comparison of experimental vertical stressdistribution with depth and the Boussinesq solution [26]

Fig. 11. Comparison of vertical stress distribution under auniformly loaded circular area based on Boussinesq

equations and 1:1 and 2:1 distributions [17]

Fig. 12. Recommended semi-empirical pressuredistribution in track supporting granular layers [17]

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the values of dynamic impact factor obtainedfrom different formulas. This result could be usedto make suggestions on proper utilization ofdifferent methods used for the dynamic impactfactor estimation. Since AREMArecommendations are basically in accordance toheavy haul operations in which trains usuallypass over the track with lower speeds, it wouldnot be practical to use the AREMA formula forthe railway transit operations. Therefore, the useof the AREMA recommendation for dynamicimpact factor is suggested for heavy haul railwaywith the train speed less that 80 km/h. On theother hand, the use of equation proposed byEisenmann is more justifiable for the calculationof dynamic impact factor in high speed railwaytracks. Although the consideration of dynamicimpact factor somehow compensate for theeffects of dynamic impact of the loads, thecurrent railway design approach still needsfurther improvement by the consideration oftransient characteristics of the wheel loads.Furthermore, the above review of the currentapproach indicates that the effect of higher trainspeeds is not included in the estimation of railwear life. This urges the modification of currentrail design criteria in accordance to high speedtrain operations.

The main proposed contact pressure betweensleepers and ballast are presented in Table 4. Asindicated in the previous section, a uniformcontact pressure distribution is assumed betweenballast and sleeper. This means that the currentdesign criteria do not clearly include the effect ofballast material degradation on the ballast-sleeperpressure distribution pattern. In other words, thesleeper design approach awaits furtherimprovements by the incorporation of long-termeffects of the loads and in turn, consideration ofthe ballast degradation.

Current practices in analysis and design ofrailway tracks do not lead to a clear approach forthe analysis and design of rail fasteners. Theresults of this study indicate that except thelimited controlling criteria suggested byAustralian Standard, there is no other obviouscriteria proposed regarding rail fasteners’ fatiguestrength. Therefore, the procedure of the analysisand design of rail fasteners needs to be further

developed. This importance would be morerealized when considering the great variety ofdifferent rail fasteners currently used in theconstruction of railway track systems.

The only criterion determined for the structuralanalysis and design of granular ballast and sub-ballast supporting layers is the estimation ofminimum required depth of these trackcomponents. This criterion is based on theamount of reduction in vertical pressure intensityto be tolerable for the underlying subgrade layer.However, there are no specific suggestionsregarding the effect of the material gradation onoptimum ballast and sub-ballast layers thicknessdetermination. The influence of trackmaintenance operations as well as newconstruction techniques and facilities such asgeosynthetics application, is not clearly included.The effect of accumulative loading and theconsequent track plastic strain behavior is notincorporated clearly in track components analysisand design especially with respect to the sleepersas well as ballast and sub-ballast layers.

The results obtained in this research clearlyindicate the need for further developments ofcurrent railway track design methods by moreinvestigations of railway track short and longterm behaviors with particular attention to theresults obtained from laboratory and field tests.The aim is to define new criteria which includetrack parameters and loading conditions omittedin the current design approaches.

List of Symbols

B Sleeper breadth, mC height of the rail neutral axis from rail base,

Fig. 13. Comparison of some recommended methods forthe estimation of dynamic coefficient factor

0.9

1.2

1.5

1.8

2.1

2.4

0 20 40 60 80 100 120 140 160 180 200

Speed (Km/hr)

Dyn

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coe

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acto

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AR E MA (D= 965mm)

S outh A fric a(D= 965mm)

O R E

E is enman G ood track

C urent S tudy(D= 965mm)Sadeghi

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mmD Wheel diameter, mmDA Annual gross tonnage, MGT/yearDc Degree of curve, degreesDi Sub-tonnage, MGT/yearDj Track stiffness at the joint, kN/mmD.F. Load distribution factore Width of the rail seat load distribution

along sleeper thicknessE Rail modulus of elasticity, N/mm2

F1 Track support variation safety factorF2 Factor accounting for adjacent wheels

interactionsF3 Factor depending on the sleeper type and

the standard of track maintenanceG Gravitational constant, m/s2

I Rail moment of inertia, mm4

Ic Horizontal moment of inertia of the centerof the sleeper cross section, mm4

Ir Horizontal moment of inertia of the sleepercross section at rail seat position, mm4

j Load amplification factor

Wheel load class factorKC Track curvature and lubrication factorKG Track gradient factorKR Rail factor

Service type factor

Speed class factorKw Wear factor varying with the degree of

curvel Sleeper length, mLp Distance between rail seat axles and the end

of the sleepern Length of steel rail plateP Wheel load, kNPs Static wheel load, kNPu Unsprung weight of the wheel, kNRw Wheel radius, mmRwt Rail weight per unit length, kg/mS Sleeper spacing, mmT Rail wear life, MGTTy Rail wear life, yearu Track modulus, N/mm2

V Train speed, km/hWt Average rail head area wear term, mm2/MGTGreek symbols

Coefficient of expansionSpeed coefficientSpeed coefficientFactor related to track conditionTemperature variationSpeed coefficientLoad distribution factorDynamic increment of bending momentfactor due to sleeper support irregularitiesImpact attenuation factor of rail fastenersSleeper support condition factor Speed related amplification factorSpeed factorEffective lever arm, mPoisson’s ratio

References

[1] Kerr, A.D., “Fundamentals of Railway TrackEngineering”, Simmons-Boardman Books, Inc,2003.

[2] Esveld, C., “Modern Railway Track”, MRT-Productions, the Netherlands, 2001.

[3] Boresi, A.P., Schmidt, R.J., 2003, “AdvancedMechanics of Materials”, 6th Edition. JohnWiley & Sons, New York, NY: Chap. 5, 10.

[4] Sadeghi, J., and Yoldashkhan, M., 2005,“Investigation on the accuracy of currentpractices in analysis of railway track sleepers”,International Journal of Civil Engineering, Vol.3, No. 1, pp. 9-15.

[5] American Railway Engineering andMaintenance of way Association, Manual forRailway Engineering, Volume 4, Chapter 16,Part 10, “Economics of Railway Engineeringand Operations – Construction andMaintenance Operations”, 2006.

[6] Robnett, Q.L., Thompson, M.R., Hay, W.W.,Tayabji, S.D., Peterson, H.C., Knutson, R.M.,and Baugher, R.W., “Technical Data BasesReport”, Ballast and Foundation MaterialsResearch Program, FRA Report No. FRA-OR&D-76-138, National Technical InformationService, Springfield, Virginia, USA, 1975.

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[7] Eisenmann, J., “Stress Distribution in thePermanent way due to Heavy Axle Loads andHigh Speeds”, AREMA Proceedings, Vol. 71,pp. 24-59, 1970.

[8] Smith, J.O., and Liu, C.K., 1953, “Stresses Dueto Tangential and Normal Loads on an ElasticSolid with Application to some Contact StressProblems”, Transactions of ASME, Journal ofApplied Mechanics, pp. 157-166.

[9] Lundgren, J.R., Martin, G.C., and Hay, W.W.,“A Simulation Model of Ballast Support and theModulus of Track Elasticity”, (Masters Thesis),Civil Engineering Studies, TransportationSeries, No. 4, University of Illinios, 1970.

[10] Hay, W.W., Schuch, P.M., Frank, M.W., andMilskelsen, M.J., “Evaluation of Rail Sections”,2nd Progress Report, Civil Eng., TransportationSeries No. 9, Univ. Illinois Project No. 44-22-20-332, Univ. Illinois, Urbana, 1973.

[11] Danzing J. C., Hay W., and Reinschmidt A.,“Procedures for analyzing the economic costsof railway roadway for pricing purposes”,Volume 1 and 2, Report No. RPD-11-CM-R,Tops on-line Service Inc, 1976.

[12] American Railway Engineering andMaintenance of Way Association, Manual forRailway Engineering, Volume 1, Chapter 30,Part 1, “Ties – General Considerations”, 2006.

[13] International Union of Railways, “Design ofMonoblock Concrete Sleepers”, UIC CODE,713 R, 1st Edition, 2004.

[14] Australian Standard, AS 1085.14, Railway trackmaterials, Part 14: “Prestressed ConcreteSleepers”, 2002.

[15] Australian Standard, AS 1085.17, RailwayTrack Materials, Part 17: “Steel Sleepers”,2002.

[16] Sadeghi, J., 2008, “Experimental Evaluation ofAccuracy of Current Practices in Analysis andDesign of Railway Track Sleepers”, CanadianJournal of Civil Engineering. Vol. 35, pp. 881-893.

[17] Doyle, N.F., “Railway Track Design: A Reviewof Current Practice”, Occasional Paper No. 35,Bureau of Transport Economics,Commonwealth of Australia, Canberra, 1980.

[18] American Railway Engineering andMaintenance of Way Association, Manual forRailway Engineering, Volume 1, Chapter 30,Part 4, “Ties – Concrete Ties”. 2006.

[19] American Railway Engineering andMaintenance of Way Association, Manual forRailway Engineering, Volume 1, Chapter 5, Part9, “Track – Design Qualification Specificationsfor Elastic Fasteners on Timber Cross Ties”,2006.

[20] Australian Standard, AS 1085.19, Railway trackmaterials, Part 19: “Resilient fasteningassemblies”, 2002.

[21] European Committee for Standardization, EN13146-1, Railway Applications – Track, TestMethods for Fastening Systems, Part 1:“Determination of Longitudinal Rail Restraint”,2002.

[22] European Committee for Standardization, EN13146-2, Railway Applications – Track, TestMethods for Fastening Systems, Part 2:“Determination of Torsional Rail Resistance”,2002.

[23] European Committee for Standardization, EN13146-3, Railway Applications – Track, TestMethods for Fastening Systems, Part 3:“Determination of Attenuation of ImpactLoads”, 2002.

[24] European Committee for Standardization, EN13146-4, Railway Applications – Track, TestMethods for Fastening Systems, Part 4: “Effectof Repeated Loading”, 2002.

[25] European Committee for Standardization, EN13146-7, Railway Applications – Track, TestMethods for Fastening Systems, Part 7:“Determination of Clamping Force”, 2002.

[26] Office of Research and Experiments (ORE),Stresses in the Formation, Question D17,“Stresses in the Rails, the Ballast and the

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Wheel Load Contact Area