Traffic Load and Its Impact on Track Maintenance

Abstract

Transport volume is often addressed as gross-tonnage when it comes to track degradation, maintenance needs and maintenance costs. Tonnage and thus weight are insufficient to address track loading: The vehicle properties, mainly axle load, unsprung masses and bogie stiffness, as well as train speed have a major impact on track maintenance demand. This chapter gives an overview on vehicle-specific track deterioration models and describes the maintenance needs of different track as well as the influence on track maintenance cost of different vehicles and trains. The approach proposed is both simple enough to be used and sufficiently detailed to address the main technical aspects. The differences of track maintenance necessary for mixed traffic lines, high-speed lines and heavy haul freight lines can be derived from the vehicles used and the existing train speeds. In passenger transport, the difference between push-pull loco hauled trains (with wagons) and multiple units are a major aspect.

1. Introduction

Transport volume is one major trigger of track maintenance [1]. For all structures, the loading drives damage, wear and overall system deterioration. In the case of railway track, the loading can be described in different ways. The number of trains gives information on the track utilization. This indicator is important in terms of capacity and linked issues like timetabling, but does not directly influence the maintenance as trains range from heavy freight trains to fast long-distance passenger trains to regional trains. The number of trains does have influence on the maintenance costs as most maintenance tasks need closed-down tracks. Line utilization specifies that the duration of track closures which in turn defines the possible length maintenance tasks can be carried out and thus impacts the cost per meter [2]. Additionally, so-called cost of operational hindrances emerges as trains are delayed, re-routed or simply do not run [3, 4]. Alternatively, the accumulated weight of trains can be applied which describes the intensity of track usage. The calculated gross-tonnage is widely used as an indicator for track loading and used for both classifying tracks [5] and defining maintenance frequencies [6, 7, 8]. Consequently, track maintenance cost is allocated to gross-ton-kilometers [9, 10]. Also charging is based on this unit [11, 12]. On the other hand, tracks are designed, constructed and classified for a maximum permissible static axle load and a maximum allowed speed [13, 14, 15]. Forces are generally used as loading for the design of structures, and the static axle load at least approximates the vertical loading. Summarizing, the gross-tonnage is a feasible indication for the vertical loading and thus works as an approximation for ballast maintenance and possible rail fatigue. However, the gross-tonnage is definitely not sufficient for all damage mechanisms that are additionally triggered by lateral forces, slip and/or applied traction forces and thus for determining maintenance requirements for rails and turnouts. Moreover, gross-tonnage does not cover axle loads and speed.

There are some approaches covering those—and even more—aspects in calculating so-called equivalent gross-tons. In [16, 17] speed and high axle loads act as increasing factors, so does the aggressiveness of powered axles. [18, 19] additionally address the unsprung mass. Those models improve the description of vertical impacts, but still do not address lateral aspects. The approach of Burstow adds these in providing a damage index for the rails [20].

Finally, there are existing track deterioration models combining all the mentioned aspects [18, 21]. These models have been the inspiration for the proposed model in this paper which is close to the SBB-model. We specified the loading for crossings in turnouts and also added an additional damage term for the entire track structure being restored by a track renewal only. In the result section, we give several examples using this model for both the prediction of track maintenance and the effects of different train service and vehicles. These examples leave out the damage mechanisms for turnouts and the entire track and turnout structures so that one can follow the examples well.

2. Methodology

2.1 Track deterioration model

A track deterioration model (TDM) enables damage assessment by using one mathematical formula. The TDM presented is therefore an analytic approach. The model is based on three input categories: track characteristic, rolling stock and track maintenance. Figure 1 gives an overview of the input categories that have an impact on the TDM.

Figure 1.

Input categories for the track deterioration model.

The first category track characteristics provides components of the track’s superstructure such as rail type (profile and steel grade), sleeper type and ballast bed condition. Furthermore, the category track characteristic includes information about the condition of substructure and functionality of drainage. Radius and maximum permissible speed (alignment caused) can be found as well in this category. In Section 2.3.1, the track characteristic is discussed in more details.

In the second category Maintenance track works are documented by their type, frequency and costs. This category is strongly linked to the track characteristic. Types of maintenance work considered in the model can be found in the subchapters of Section 2.2.

Vehicle parameters that are necessary for the TDM and therefore those which can cause damage to the track can be allocated to Rolling Stock. Parameters like maximum vehicle speed, number of powered/unpowered axles, static and dynamic vertical wheel load, etc., are collected in the third category. This is discussed in Section 2.3.2.

Linking all parameters of those three input categories together enables setting up a track deterioration model. The core of the TDM is a developed mathematical wear formula that is composed of damage terms (D_n for damage) and cost calibration factors (c_n for costs). The wear formula consists of seven damage terms which range from D₁ to D₇. Each of the seven damage terms describes a partial damage that occurs due to a vehicle’s run through the track and the arising vehicle track interaction. As those damage terms do differ in their units and do cause different maintenance costs, the damage terms are multiplied by cost calibration factors. Eq. (1) depicts the calculation method and general composition of the wear formula, in which the products of damage terms and cost calibration factors are added up to give total costs per vehicle kilometer.

$${\mathrm{C}}_{{\mathrm{Veh}}_{\mathrm{S},\mathrm{R}}}=\sum _{\mathrm{n}=1}^7{\mathrm{c}}_{\mathrm{n}}\times {\mathrm{D}}_{\mathrm{n}}$$E1

C_Veh,S,R—costs per vehicle kilometer depending on speed and radius (costs/km); c_n—cost calibration factors (n = 1, 2, 3, 4.1, 4.2, 5, 6, 7) (costs/(unit km*)); D_n—damage term (n = 1, 2, 3, 4.1, 4.2, 5, 6, 7)(unit*).

*Unit of the damage term: kN³ (D₁, D₆ and D₇), kN^1.2 (D₂), kW/mm² (D₃), kN (D₅) and D_4.1 and D_4.2 are unitless

Damage terms D₁, D₂, D₅, D₆ and D₇ represent physical forces that are expressed in kN. However, D₁, D₆ and D₇ are weighted by the exponent 3 (kN³), while damage term D₂ is weighted by the exponent 1.2 (kN^1.2). D₃ describes the rail deterioration caused by the physical power per wheel in kW/mm². The contact patch frictional energy described in D₄ is expressed in Nm/m. By the fact that the damage terms differ in their physical units, the cost calibration factors correlate to the track damage terms in costs/unit-kilometer.

Precising the general Eq. (1) by means of its damage terms D_n leads to the track deterioration formula set up by Graz University of Technology based on the existing SBB-formula [21]. The track deterioration model and its detailed approach for each damage term are depicted in Eq. (2).

$$\begin{array}{l}{\mathrm{C}}_{{\mathrm{Veh}}_{\mathrm{S},\mathrm{R}}}={\mathrm{c}}_1\times {{\mathrm{P}}_{2,\mathrm{S}}}^3+{\mathrm{c}}_2\times {{\mathrm{P}}_{2,\mathrm{S}}}^{1.2}+{\mathrm{c}}_3\times \mathrm{TPV}+{\mathrm{c}}_{4.1}\times {\mathrm{D}}_{4.1}+{\mathrm{c}}_{4.1}\times {\mathrm{D}}_{4.2}+{\mathrm{c}}_5\\ \times \sqrt{\left(0.5\times {{\mathrm{P}}_{2,\mathrm{S}}}^2+0.5\times {{\mathrm{Y}}_{\mathrm{R}}}^2\right)}+{\mathrm{c}}_6\times {{\mathrm{P}}_{1,\mathrm{S}}}^3+{\mathrm{c}}_7\\ \times {\sqrt{\left({\mathrm{f}}_{7_{1,\mathrm{R}}}\times {{\mathrm{P}}_{2,\mathrm{S}}}^2+{\mathrm{f}}_{7_{2,\mathrm{R}}}\times {{\mathrm{Y}}_{\mathrm{R}}}^2\right)}}^3\end{array}$$E2

C_Veh,S,R—costs per vehicle kilometer depending on speed and radius (costs/km); c_n—cost calibration factors (n = 1, 2, 3, 4.1, 4.2, 5, 6, 7) (costs/(unit km*); P_2,S—dynamic vertical wheel force (long-waved) depending on speed (kN); P_1,S—dynamic vertical wheel force (short-waved) depending on speed (kN); Y_R—lateral force of the guiding wheel on the outer rail within radius R 190 m (kN); TPV—traction power value (kW/mm²); D_4.1—damage index for rolling contact fatigue (RCF); D_4.2—damage index for plastic deformation/rail abrasion; f7_1,R—weighting factor for the vertical dynamic wheel force depending on radius R; f7_2,R—weighting factor for the lateral wheel force depending on radius R.

*Unit of the damage term: kN³ (D₁, D₆ and D₇), kN^1.2 (D₂), kW/mm² (D₃), kN (D₅) and D_4.1 and D_4.2 are unitless

The track deterioration model consists of seven damage terms, whose calculation is based on one of the four physical parameters vertical or lateral force, power or friction work. These four parameters lead to wear mechanism in the track and therefore to maintenance work. A more detailed explanation of each damage term can be found in the following Section 2.2, while Section 2.3.2 describes the four physical parameters.

2.2 Damage terms

2.2.1 D1: Track geometry deterioration

Damage term D₁ describes track geometry deterioration and ballast destruction on the basis of dynamic vertical wheel contact force P₂. [22] and [23] are the foundation for the approach of the P₂ force (see Section 2.3.2 for more details). This force is not only vehicle-dependent but also a function of speed and represents the long-waved force influence that is caused by track joints/isolated defects. As shown in Eq. (3), the dynamic vertical wheel contact force P₂ is weighted by the exponent 3. The approach of the over linear influence (exponent 3) of the representative axle load bases on [24]. In 1987, the dynamic effects due to increasing axle load from 20 to 22.5 metric tons were investigated on the railway test circuit in Velim (Czechia).

$${\mathrm{D}}_1={{\mathrm{P}}_{2,\mathrm{S}}}^3$$E3

D₁—damage term 1 (kN³); P_2,S—dynamic vertical wheel force (long-waved) depending on speed (kN).

In the calculation of damage term D₁, each vehicle’s wheelset is classified as damage-relevant. Ballast bed cleaning, line and spot tamping are types of maintenance work that are related to this damage term.

2.2.2 D2 and D3: Rail surface damage (straight tracks)

The damage terms D₂ and D₃ are discussed together in this chapter because both describe rail surface failures in straight tracks, however, due to different impacts. Damage term D₂ describes rail surface failures due to vehicle’s dynamic vertical wheel force. The wheel force P₂ used in D₂ corresponds to the force applied in damage term D₁. As depicted in Eq. (4), the P₂ force is weighted by the exponent 1.2. The approach of the power 1.2 does also form the basis of [24, 25], in which the influence of increasing axle load from 20 to 22 metric tons was investigated. In damage term D₂, every wheelset is relevant for determining the surface failure due to vertical force impact.

$${\mathrm{D}}_2={{\mathrm{P}}_{2,\mathrm{S}}}^{1.2}$$E4

D₂—damage term 2 (kN^1.2); P_2,S—dynamic vertical wheel force (long-waved) depending on speed (kN).

As not only dynamic vertical forces affect the rail surface, the model does also consider longitudinal forces on the rail. Longitudinal forces are induced by the traction power of the rolling stock [26]. Damage term D₃ describes the influence of traction power on the rail surface by means of the traction power value (TPV). This value is based on the vehicle’s power density. The power density is related to the cumulated contact area between rail and wheel of powered axles. Hence, a multiplication with the number of axles (like it is done, e.g., in D₁ and D₂) is not necessary. Furthermore, only powered axles are considered in damage term D₃. Eq. (5) describes the content of damage term D₃ with its unit. A more detailed description of the traction power value can be found in Section 2.3.2.

$${\mathrm{D}}_3=\mathrm{TPV}$$E5

D₃—damage term 3 (kW/mm²); TPV—traction power value (kW/mm²).

Both damage terms D₂ and D₃ describe rail surface fatigue in straight tracks, where head checks, squats and corrugation occur. The vertical (D₂) and lateral (D₃) force impact cause maintenance work in form of rail surface treatment, such as grinding and milling.

2.2.3 D4.1 and D4.2: rail surface damage and wear (curved tracks)

As damage term D₂ and D₃ describe the rail surface damage in straight tracks, damage terms D_4.1 and D_4.2 do so for curved tracks. In curved tracks, a distinction between three damage characteristics can be drawn: rolling contact fatigue (RCF), rail abrasion/plastic deformation and a mixture of both effects. The three damage characteristics are evaluated due to the contact patch frictional energy Tγ (T-Gamma) that is calculated by a multiple-body simulation. The evaluating function that describes the relationship between the contact patch frictional energy (Tγ) and the fatigue damage is based on findings by Burstow [18]. This function with its different wear areas (A to D) is depicted in Figure 2. In the function’s area A (Tγ < 15 Nm/m), no rail surface wear occurs, whereas in area B RCF takes place (15 ≤ Tγ < 65 Nm/m). RCF as well as abrasion/deformation appears in function area C (65 ≤ Tγ < 175 Nm/m). Isolated rail abrasion/deformation is described in area D (Tγ > 175 Nm/m).

Figure 2.

Burstow’s evaluating damage function [18] that distinguishes in no wear (A), RCF (B), RCF and abrasion/deformation (C) and abrasion/deformation (D).

As the TDM distinguishes between rail surface treatments and rail exchange, Burstow’s evaluating damage function is split up into descriptive functions for damage term D_4.1 as depicted in Eq. (6) and damage term D_4.2 as shown in Eq. (7). These two equations reflect different scenarios (a to e). Scenarios (a)-(c) describe rail surface wear that is connected to rail surface treatments in curved tracks as it is meant by damage term D_4.1. Scenarios (d) and (e) are linked to damage term D_4.2 that represents rail exchange in curved tracks.

$$\left(\mathrm{a}\right){\mathrm{D}}_{4.1}=0\text{for}{\mathrm{T}}_{\mathrm{\gamma }}<15\mathrm{Nm}/\text{mand}{\mathrm{T}}_{\mathrm{\gamma }}\ge 175\mathrm{Nm}/\mathrm{m}$$

$$\left(\mathrm{b}\right){\mathrm{D}}_{4.1}=\mathrm{n}\times \left(0.02\times {\mathrm{T}}_{\mathrm{\gamma }}-0.3\right)\text{for}15\le {\mathrm{T}}_{\mathrm{\gamma }}<65\mathrm{Nm}/\mathrm{m}$$

$$\left(\mathrm{c}\right){\mathrm{D}}_{4.1}=\mathrm{n}\times \left(\frac{-{\mathrm{T}}_{\mathrm{\gamma }}+175}{110}\right)\text{for}65\le {\mathrm{T}}_{\mathrm{\gamma }}<175\mathrm{Nm}/\mathrm{m}$$E6

$$\left(\mathrm{d}\right){\mathrm{D}}_{4.2}=0\text{for}{\mathrm{T}}_{\mathrm{\gamma }}<65\mathrm{Nm}/\mathrm{m}$$

$$\left(\mathrm{e}\right){\mathrm{D}}_{4.2}=\mathrm{n}\times \left(\frac{{\mathrm{T}}_{\mathrm{\gamma }}-65}{110}\right)\text{for}{\mathrm{T}}_{\mathrm{\gamma }}\ge 65\mathrm{Nm}/\mathrm{m}$$E7

D_4.1—damage term 4.1; D_4.2—damage term 4.1; n—number of leading wheelsets of a bogie; Tγ—contact patch energy (Nm/m).

The total damage potential of a vehicle is obtained by multiplying each term D_4.1 and D_4.2 by the number of all leading wheelsets in a bogie, since only these are considered as relevant for damage. The contact patch friction that appears in curved tracks not only leads to rail surface treatments but also to rail exchanges of the outer rail. Damage terms D_4.1 and D_4.2 therefore enable an evaluation of curve-friendly vehicles. Nerlich and Holzfeind [27] shows that actively controlled wheelsets that enable a better radial position in curved track sections lower the contact patch friction by over 60%. A more detailed description of the contact patch energy Tγ is discussed in Section 2.3.2.

2.2.4 D5: Wear of turnout components

Damage term D₅ describes wear of turnout components with the exception of the crossing nose. The wear of the crossing nose is specified in an extra damage term (D₆). As can be seen in Eq. (8), term D₅ is described by the vertical (P₂) and lateral (Y_R) forces. P₂ is thereby analogous to the use in D₁. Whereas Y_R is estimated by a multiple-body simulation and represents the lateral force of the guiding wheel on the outer rail that occurs while driving through a s-curve. As a reference for the simulation, a turnout deviation at a radius of 190 m and speed of 40 kmph was chosen as representative for turnouts in the Austrian railway network. This turnout geometry not only represents the vehicle’s characteristic significantly, and this geometry also occurs frequently in the network of the Austrian Federal Railways (appearance of still 15% on the mainlines). With regard to [24] and [25], a linear damage approach is selected for D₅, in which the vertical and lateral forces are weighted in each case by 50%.

$${\mathrm{D}}_5=\sqrt{\left(0.5\times {{\mathrm{P}}_{2,\mathrm{S}}}^2+0.5\times {{\mathrm{Y}}_{\mathrm{R}}}^2\right)}$$E8

D₅—damage term 5 (kN); P_2,S—dynamic vertical wheel force (long-waved) depending on speed (kN); Y_R—lateral force of the guiding wheel on the outer rail within radius R 190 m (kN).

Calculating damage term D₅, each wheelset is concerned for the vertical force. For the lateral force the number of leading axles in a bogie is considered. Damage in D₅ correlates to maintenance work in turnouts, such as exchange of turnout parts (switch and guard rail), grinding, welding, singular sleeper exchange and deburring. As mentioned in the beginning of this chapter, D₅ does not deal with crossing noses. This is done in the following chapter.

2.2.5 D6: Wear of crossing nose

Damage term D₆ describes wear of the turnout’s crossing nose. The short-waved dynamic vertical wheel force P₁ is used to estimate the wear, as depicted in Eq. (9).

$${\mathrm{D}}_6={{\mathrm{P}}_{1,\mathrm{S}}}^3$$E9

D₆—damage term 6 (kN³); P_1,S—dynamic vertical wheel force (short-waved) depending on speed (kN).

The approach of P₁ is based on [22] and is justified due to the fact that a wheel running through a crossing nose results in an immediate impact stress. This short-waved force impact leads to wear in the crossing nose. The damage influence of the P₁ force on the crossing nose is evaluated super linearly with the exponent 3 and is therefore comparable with the damage term D₁. The power of 3 is also based on [24, 25]. As every wheelset has its impact on the crossing nose during a drive through the turnout, each wheelset is classified as damage relevant in the calculation of damage term D₆. The described wear in D₆ results in maintenance work, such as exchange of the crossing nose and surface-layer welding/build-up welding. Section 2.3.2 gives a more detailed description of the vertical wheel force P₁.

2.2.6 D7: track renewal (reinvestment)

Damage term D₇ was invented to describe the damage to ballast, sleeper and rail components and implies the renewal of tracks and turnouts. Eq. (10) shows the composition of damage term D₇ and its parameters.

$${\mathrm{D}}_7={\sqrt{\left({\mathrm{f}}_{7_{1,\mathrm{R}}}\times {{\mathrm{P}}_{2,\mathrm{S}}}^2+{\mathrm{f}}_{7_{2,\mathrm{R}}}\times {{\mathrm{Y}}_{\mathrm{R}}}^2\right)}}^3$$E10

D₇—damage term 7 (kN³); P_2,S—dynamic vertical wheel force (long-waved) depending on speed (kN); Y_R—lateral force of the guiding wheel on the outer rail within radius R (kN); f_71,R—weighting factor for the vertical dynamic wheel force; f_72,R—weighting factor for the lateral wheel force.

The speed-dependent vertical force P₂ and the speed and radius-dependent lateral force Y_R are used in D₇ similarly to damage term D₅. Again, the ORE report for question D161 [24] is the basis for the over-linear approach (3rd power) to the component damage in this term. The weighting factors (f_7.1 and f_7.2) represent the damage allocation of the vertical and lateral forces depending on the radius. In straight track sections, the lateral force between wheel and rail is negligible, which is why only the vertical force P₂ contributes to the component damage in straight sections. Furthermore, the smaller the radius, the higher the share of lateral force Y_R. Table 1 includes the radius-dependent weighting factors for the vertical and lateral force. The classification of radii is explained in Section 2.3.1 in more detail.

Radii class	Track alignment; in m	f7.1,R; unitless	f7.2,R; unitless
R5	R > 1000	1.0	0.0
R4	600 < R ≤ 1000	0.8	0.2
R3	400 < R ≤ 600	0.7	0.3
R2	250 < R ≤ 400	0.6	0.4
R1	R ≤ 250	0.5	0.5

Table 1.

Weighting factors for vertical and lateral forces in damage term D7.

For calculating the component renewal of tracks and turnouts, each vehicle’s wheelset is considered in the calculation for the vertical stress. The leading wheelsets of bogies are used for the lateral stress in curved tracks.

2.3 Input data

Applying the TDM needs input data. On the one hand these input data refer to the characteristics of rolling stock. The vehicle’s information about their characteristics is the foundation for calculations and simulations regarding stresses on the track. On the other hand, the input data refer to the track alignment radii and speeds. Classifying the track in radii and speed classes allows distinguishing between wear in straight and curved tracks. One and the same vehicle running once through a straight and once through a curved track section causes therefore wear to a different extent. Furthermore, maintenance work that appears due to vehicle–track interaction needs to be defined and its strategic costs determined. In this TDM, the three categories, namely rolling stock, track characteristics and maintenance are strongly connected to each other (see Figure 1). The following chapters expand on these impact categories.

2.3.1 Track characteristics: Radii and speed classes

Based on the differences between alignment of track sections, a classification of radii and speeds was done. For the TDM 5 radii classes (R) were defined. These 5 radii classes correspond to those used in the Standard Elements [1]. Further, defining different radii classes comprises not only the law of superelevation of the outer rail in curved tracks, but also the wear of this outer rail.

The fifth radius class (R₅) represents track sections that show a radius of more than 1000 m. These sections are classified as straight track sections. Radius class R5 is again divided into eight speed classes (S₁ to S₈). For each radius and speed class one reference speed is allocated with which the vehicle parameters are calculated. The reference speeds of radius class R₁ to R₄ each represent the rounded median value of the line speeds of the Austrian core network. The reference speed of 60 kmph of radius class R₁ thus indicates that on the Austrian main lines in curves with radii less than 250 m a median speed of 60 kmph is driven. The calculation of the reference radii is done analogously. Table 2 lists the radius and speed classes with their reference radius and speed.

Classification of radius; in m	Reference radius; in m	Classification of speed; in kmph	Reference speed; in kmph
R1 R ≤ 250	215	SR1 -	60
R2 250 < R ≤ 400	315	SR2 -	70
R3 400 < R ≤ 600	500	SR3 -	90
R4 600 < R ≤ 1000	800	SR4 -	110*
R5 R > 1000	—	S1 S ≤ 80	75
		S2 80 < S ≤ 100	90
		S3 100 < S ≤ 120	110*
		S4 120 < S ≤ 140	130*
		S5 140 < S ≤ 160	150*
		S6 160 < S ≤ 200	190*
		S7 200 < S ≤ 230	220*
		V8 230 < S ≤ 250	240*

Table 2.

Speed and radii classification.

^*

For fright rolling stock 90 kmph is assumed as a reference speed.

In the Austrian network vehicles used in freight traffic are always treated with a maximum speed of 100 kmph, in the model 90 kmph is therefore assumed as a reference speed for freight traffic. This speed limitation applies to both freight wagons and also freight locomotives. For speed classes S₃ to S₈ of radius class R₅ (straight tracks) and radius class R₄, the reference speed is applied in the model, as the reference speed exceeds 90 kmph in these classes. The speed classes S₂ to S₈ are thus identical and further lead to equal calculated parameters for freight rolling stock. For universal locomotives, it therefore makes sense to differentiate according to type of traffic (passenger or freight traffic).

Additionally, it should be mentioned here that the reference speed is a reference value that is based on the maximum permissible speed of track sections due to their track alignment (S_Track). If the maximum permissible speed of the vehicle/train (S_max,Veh) exceeds the speed due to track alignment (S_max,Veh > S_Track), in the calculation of the vehicle parameters (Section 2.3.2), the reference speed (S_Track) of Table 2 is applied. This also applies analogously vice versa, as shown in Eq. (11).

$$\mathrm{S}=min\left[{\mathrm{S}}_{max,\mathrm{Veh}},{\mathrm{S}}_{\text{Track}}\right]$$E11

S—relevant speed (kmph); S_max,Veh—maximum permissible speed due to rolling stock (kmph); S_Track—maximum permissible speed due to track alignment (kmph).

Using the minimum of both speeds S_max,Veh and S_Track reflects the operational reality. Regarding the speed the determined damage that is speed dependent (D₁, D₂, D₅, D₆ and D₇) is thus also close to reality.

2.3.2 Rolling stock parameters

The TDM and its damage terms described in Section 2.2 are based on the following four parameters:

• Dynamic vertical force: P₁ and P₂

• Lateral force: Y_R

• Contact patch energy: Tγ

• Traction power value: TPV.

The parameters P₁ and P₂ stand for the dynamic vertical force, while Y describes the lateral force that occurs in a track curve on the outer wheel of a bogie in the leading wheelset. The traction power value (TPV) indicates the physical power per wheel due to traction. The fourth physical input parameter for the deterioration formula is Tγ that stands for the friction work due to longitudinal and lateral slip. The following sections include a discussion of the four vehicle parameters in more details.

2.3.2.1 Dynamic vertical force (P₁ and P₂)

Due to [23] and [22], the P₁ and P₂ force is applied in the TDM. Eqs. (12)–(16) depict the composition and calculation scheme of the vertical impact forces P₁ and P₂.

$$\text{Iteration}:{\mathrm{P}}_{1,\mathrm{S}}={\mathrm{P}}_0+\mathrm{S}\times 2\mathrm{\alpha }\times \sqrt{\frac{{\mathrm{K}}_{\mathrm{H}}\times {\mathrm{m}}_{\mathrm{e}}}{1+\raisebox{1ex}{${\mathrm{m}}_{\mathrm{e}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}_{\mathrm{u}}$}\right.}}$$E12

$${\mathrm{m}}_{\mathrm{e}}\approx 0.4\times \left({\mathrm{m}}_{\mathrm{r}}+\frac{{\mathrm{m}}_{\mathrm{s}}}{\mathrm{l}}\right)$$E13

$${\mathrm{K}}_{\mathrm{H}}=\frac{{\mathrm{P}}_{1,\mathrm{est}}-{\mathrm{P}}_0}{\mathrm{G}\times \left({\mathrm{P}}_{1,\mathrm{est}}^{2/3}-{\mathrm{P}}_0^{2/3}\right)}$$E14

$$\mathrm{G}=\frac{3.86}{{{\mathrm{R}}_{\text{Wheel}}}^{0.115}}\times {10}^{-8}$$E15

$${\mathrm{P}}_{2,\mathrm{S}}={\mathrm{P}}_0+\mathrm{S}\times 2\mathrm{\alpha }\times \sqrt{\frac{{\mathrm{m}}_{\mathrm{u}}}{{\mathrm{m}}_{\mathrm{u}}+{\mathrm{m}}_{\mathrm{t}}}}\times \left(1-\frac{{\mathrm{c}}_{\mathrm{t}}\times \mathrm{\pi }}{4\times \sqrt{{\mathrm{K}}_{\mathrm{t}}\times \left({\mathrm{m}}_{\mathrm{u}}+{\mathrm{m}}_{\mathrm{t}}\right)}}\right)\times \sqrt{{\mathrm{K}}_{\mathrm{t}}\times {\mathrm{m}}_{\mathrm{u}}}$$E16

P_1,S,P_2,S—vehicle dynamic wheel forces (N); P₀—vehicle static wheel force (N); S—relevant speed (limited by vehicle or track alignment) (m/s); 2α—total joint angle (rad); m_u—unsprung mass per vehicle wheel (kg); m_t—effective vertical track mass per vehicle wheel (kg); c_t—effective track damping per vehicle wheel (Ns/m); K_t—effective vertical track stiffness per vehicle wheel (N/m); m_e—effective track mass per vehicle wheel (kg); m_r—rail mass per unit length (kg/m); m_s—mass of half a sleeper (kg); l—sleeper spacing (m); P_1,est—estimated vertical dynamic wheel force (N); G—Hertzian flexibility constant (for worn tyre profiles) (m/N^2/3); R_Wheel—wheel radius (m); K_H—linearized Hertzian contact stiffness per vehicle wheel (N/m).

The total vertical force that arises due to the interaction between rail and wheel comprises not only the static gravitational loading (P₀) but also the dynamic forces activated by speed (S), unsprung mass (m_u) and the rail’s alignment (2α). Both calculation schemes of the vertical wheel forces P₁ (Eq. (12)) and P2 (Eq. (16)) are based on this approach. While P₂ does also depend on track parameters (track stiffness K_t, track damping c_t and track mass m_t per wheel), P₁ is additionally conditioned by the effective track mass (m_e) and the linearized Hertzian contact stiffness (K_H) per wheel. The Hertzian contact stiffness K_H is subject to an iterative calculation between Eqs. (12) and (14) and therefore depending on the P₁ force. P₁ can be summarized to be the high-frequency portion of the impact force that is primarily responsible for surface wear and local stress peaks of the rail material. P₂ is the low-frequency and long-waved force component that mainly stresses the sleepers and ballast bed [22, 28].

In the TDM the included constants m_t, K_t, c_t and 2α are applied according to [23]. Further, rail mass (m_r) belongs to UIC60 rail, the mass of half a sleeper (m_s) to concrete sleepers and the sleeper spacing (l) to the standard spacing in Austria. The included constants in the TDM have the following values:

• 2α = 0.02 rad

• m_t = 245 kg

• c_t = 55,400 Ns/m

• K_t = 62,000,000 N/m

• m_r = 60 kg

• m_s = 150 kg

• l = 0.6 m

As the approach for P₁ and P₂ depends on the unsprung mass (m_u), the model differs between the wheels of powered and unpowered axles. Wheels of powered and unpowered axles therefore show varying values in both, P₁ and P₂. Section 2.4 gives an example calculation of the P₁ and P₂ force for a universal locomotive at a speed of 90 kmph.

2.3.2.2 Lateral force (Y_R)

The lateral force Y_R describes the horizontal force component that arises in a track curve on the outer wheel of a bogie in the leading wheelset. On the basis of a multiple-body simulation (MBS) Y_R can be determined. Representing the case of a typical turnout (for Austria EW190-1:9: single turnout at an inclination 1:9 and radius of the junction of 190 m) and calculating Y_R values for damage term D₅ at 40 kmph, a s-curve with radii 190 m was used in the MBS. In this s-curve no cant or transition curves, but an intermediate straight line of 6 m were used to ensure a representative simulation of a vehicle’s run through a turnout. Due to the s-curve, both the left and right wheel run on the outside of the curve. Two Y_R values (left and right) are therefore produced within one simulation. The higher value of both is considered for the TDM [21].

As not only damage term D₅ (turnout wear) depends on the lateral force but also D₇ (track renewal), an MBS is done also for tracks representing radius class R₁ to R₄. In this MBS a cant was implemented, so that the lateral acceleration is constantly 0.85 m/s² in each radius class. As a reference for the classification (R₁ to R₄) of the track curves, radii in the amount of 215, 315, 500 and 800 m were chosen. The Y_R was simulated at the reference speed of each radii class and for both, powered and unpowered wheelsets. Table 3 gives an overview of the input parameters that were chosen for the MBS and calculating Y_R.

Radii class	Reference Radius; in m	Reference Speed; in kmph	Superelevation; in mm	Lateral acceleration; in m/s^2
R1	215	60	67.6	0.85
R2	315	70	53.6	0.85
R3	500	90	61.2	0.85
R4	800	110*	48.5	0.85

Table 3.

Input parameters for MBS of Y_R.

*

For fright rolling stock 90 kmph is assumed as a reference speed.

2.3.2.3 Contact patch energy (Tγ)

Not only the lateral force is determined by a multiple-body simulation (MBS), but also the contact patch energy Tγ. Tγ represents the input parameter for Burstow’s [20] evaluating function for rail surface wear in curved tracks. For determining Tγ on the basis of MBS, again a s-curve has been used. The applied curves in the simulation belong to the radii classes R₁ to R₄ and their reference radius (215 m for R₁, 315 m for R₂, 500 m for R₃ and 800 m for R₄). Further, the simulation includes superelevation and transition curves. Table 3 in Section 2.3.2.3 depicts the input parameters for the MBS of Tγ. As the simulation is not only done for powered but also unpowered wheelsets up to 8 Tγ values can be assigned to one specific vehicle due to 4 radii classes. The contact patch energy Tγ is given in Nm/m for the TDM. A detailed specification for the MBS is described in the SBB guidance for vehicle pricing [21].

2.3.2.4 Traction power value (TPV)

As damage term D₃ describes rail surface wear due to the impact of vehicles power, the traction power value (TPV) is calculated. TPV describes the power of a vehicle (P_Wheel in kW) related to the contact area between rail and wheel (A_eff in mm²) as shown in Eq. (17). The TPV therefore only exists for powered wheelsets.

$$\mathrm{TPV}=\frac{{\mathrm{P}}_{\text{Wheel}}}{{\mathrm{A}}_{\mathrm{eff}}}$$E17

TPV—traction power value (kW/mm²); P_Wheel—power per wheel (kW); A_eff—effective contact area between rail and vehicle wheel (mm²).

The vehicle’s power is a value that should be available from the data sheet of the vehicle, whereas the contact area between rail and wheel can be calculated. Basis for this is the methodology due to the Hertzian contact area for arbitrarily curved surfaces as it is described in [29]. However, the Hertzian contact area is downsized by a factor of 2/3 to include system-related uncertainties. Eq. (18) depicts the general approach of the downsized Hertzian contact area due to the major (a) and minor (b) radius. In Eq. (19) the major and minor radius of the ellipse are specified as well as the auxiliary angle (ϑ). Due to the formula of Hertz, the major and minor radius are functions dependent on static wheel force (P₀), Poisson’s ratio (ν), modulus of elasticity (E) and the wheel and rail radius (R_Wheel, R_Rail). The coefficients η and ξ are functions of ϑ. Summarizing the general Eq. (18) of Hertz with the parameters in Eq. (19) leads to a simplified Hertzian formula, as depicted in Eq. (20).

$${\mathrm{A}}_{\mathrm{eff}}=\frac{2}{3}\mathrm{\pi }\mathrm{a}\mathrm{b}$$E18

$$\mathrm{a}=\sqrt[3]{\frac{3{\mathrm{\xi }}^3\left(1-{\mathrm{\nu }}^2\right){\mathrm{P}}_0}{\mathrm{E}\left(\frac{1}{{\mathrm{R}}_{\text{Wheel}}}+\frac{1}{{\mathrm{R}}_{\text{Rail}}}\right)}}\times {10}^6\text{and}\mathrm{b}=\sqrt[3]{\frac{3{\mathrm{\eta }}^3\left(1-{\mathrm{\nu }}^2\right){\mathrm{P}}_0}{\mathrm{E}\left(\frac{1}{{\mathrm{R}}_{\text{Wheel}}}+\frac{1}{{\mathrm{R}}_{\text{Rail}}}\right)}}\times {10}^6$$

$$\mathrm{\vartheta }=arccos\left(\frac{{\mathrm{R}}_{\text{Wheel}}-{\mathrm{R}}_{\text{Rail}}}{{\mathrm{R}}_{\text{Wheel}}+{\mathrm{R}}_{\text{Rail}}}\right)$$E19

$${\mathrm{A}}_{\mathrm{eff}}=\frac{2}{3}\mathrm{\pi }\mathrm{\xi }\mathrm{\eta }{\left(\frac{3\left(1-{\mathrm{\nu }}^2\right)}{\mathrm{E}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\times {\left(\frac{{\mathrm{P}}_0}{\left(\frac{1}{{\mathrm{R}}_{\text{Wheel}}}+\frac{1}{{\mathrm{R}}_{\text{Rail}}}\right)}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\times {10}^6$$E20

A_eff—effective contact area between rail and vehicle wheel (mm²); a—major radius of ellipse (mm); b—minor radius of ellipse (mm); P_0—vehicle static wheel force (kN); R_Wheel—wheel radius (m); R_Rail—rail radius (m); ϑ—auxiliary angle (rad); ξ, η—coefficients; ν—Poisson’s ratio; E—modulus of elasticity (kN/m²).

For the analytic calculation of the Hertzian contact area, the vehicle is considered to be standing on a straight track. The rail head radius (R_Rail) is therefore expected to be 0.3 m. Further, the Poisson’s ratio (ν) is uniformly set to 0.3 and the modulus of elasticity (E) to 2.1 × 10⁸ kN/m². These parameters are constant values in the TDM and do not get changed. The coefficients ξ and η can be summarized (approximation) in a function for each, as shown in Eq. (21) due to [21] and [29].

$$\mathrm{\xi }\left(\mathrm{\vartheta }\right)=1.5281739\times {\mathrm{\vartheta }}^{-0.8571601}$$

$$\mathrm{\eta }\left(\mathrm{\vartheta }\right)=0.4724037\times \mathrm{\vartheta }+0.2366389$$E21

ϑ—auxiliary angle (rad); ξ, η—coefficients.

Implementing the functions of the coefficients ξ and η into Eq. (20) gives the final description of the effective contact area due to the Hertzian methodology, as shown in Eq. (22). This formula is valid for the Austrian TDM due to constant values for R_Wheel, E and ν.

$${\mathrm{A}}_{\mathrm{eff}}=\frac{8.3593707\times \mathrm{\vartheta }+4.1874191}{{\mathrm{\vartheta }}^{0.8571601}}\times {\left(\frac{{\mathrm{P}}_0}{\frac{1}{{\mathrm{R}}_{\text{Wheel}}}+\frac{1}{0.3}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

$$\text{with}\mathrm{\vartheta }=arccos\left(\frac{{\mathrm{R}}_{\text{Wheel}}-0.3}{{\mathrm{R}}_{\text{Wheel}}+0.3}\right)$$E22

A_eff —effective contact area between rail and vehicle wheel (mm²); P_0—vehicle static wheel force (kN); R_Wheel—wheel radius (m); ϑ—auxiliary angle (rad).

The determined TPV in kW/mm² corresponds directly to the damage term D₃. A multiplication by the number of driven axles is not necessary, since the vehicle’s power per wheel is related to the Hertzian contact area between wheel and rail.

2.4 Exemplary calculations

In the following subchapters, exemplary calculations are done for a sample vehicle. A universal locomotive is chosen as a sample vehicle that goes through a track segment of radius class R₃. The maximum permissible speed of this vehicle is 230 kmph at an axle load of 22 t. In Table 4, all input parameters are given that are necessary to calculate the rolling stock parameters and the damage increments for the sample vehicle.

Universal locomotive: axle load = 22 t | Smax,Veh = 230 kmph; R3: 400 < R ≤ 600 m | Sreference = 90 kmph | Rreference = 500 m
Parameter		Value (per wheel)	Unit
P0	Vehicle static wheel force	22 × 9.81 × 1000/2 = 107,910	N
S	Vehicle speed (limited by vehicle or track alignment)	90 × 3.6 = 25	m/s
S40	Vehicle speed for D5	40 × 3.6 = 11.11	m/s
2α	Total joint angle	0.02	rad
mu	Unsprung mass per vehicle wheel	1247.5	kg
mt	Effective vertical track mass per vehicle wheel	245	kg
ct	Effective track mass per vehicle wheel	55,400	Ns/m
Kt	Effective vertical track stiffness per vehicle wheel	62,000,000	N/m
RWheel	Wheel radius	1.15	m
mr	Rail mass per unit length (rail UIC60/60E1)	60	kg/m
mS	Mass of half a sleeper	300/2 = 150	kg/m
l	Sleeper spacing	0.6	m
P1,est	Estimated dynamic vertical wheel force	Iteration (367,283.51)	N
np	Number of powered wheelsets	4	—
nup	Number of unpowered wheelsets	0	—
np‘	Number of leading powered wheelsets in a bogie	2	—
nup‘	Number of leading unpowered wheelsets in a bogie	0	—
PWheel	Power per wheel	6400/4/2 = 800	kW
Smax,Veh	Maximum permissible speed of vehicle	230	kmph

Table 4.

Input parameter of the sample vehicle.

2.4.1 Rolling stock parameters

In this chapter, the rolling stock parameters of the sample vehicle are calculated due to the described formula in Section 2.3.2. The vertical wheel forces P₁ and P₂ and the TPV are determined for powered axles as the sample vehicle has four powered axles and no unpowered axles. Furthermore, Tγ and Y_R are not treated in this chapter as they are estimated by MBS. Determining the dynamic P₂ force for radius class R₃ at 90 kmph as shown in Eq. (23) and for 40 kmph (damage term D₅) in Eq. (24):

$$\begin{array}{ccc}{\mathrm{P}}_{2,90}& =& (\mathrm{107,910}+25\times 0.02\times \sqrt{\frac{\mathrm{1,247.5}}{\mathrm{1,247.5}+245}}\times \left(1-\frac{\mathrm{55,400}\times \mathrm{\pi }}{4\times \sqrt{\mathrm{62,000,000}\times \left(\mathrm{1,247.5}+245\right)}}\right)\\ & & \times \sqrt{\mathrm{62,000,000}\times \mathrm{1,247.5}})\times {10}^{-3}\\ & =& \underset{\_}{216.8\mathrm{kN}}\end{array}$$E23

$$\begin{array}{ccc}{\mathrm{P}}_{2,40}& =& (\mathrm{107,910}+11\times 0.02\times \sqrt{\frac{\mathrm{1,247.5}}{\mathrm{1,247.5}+245}}\times \left(1-\frac{\mathrm{55,400}\times \mathrm{\pi }}{4\times \sqrt{\mathrm{62,000,000}\times \left(\mathrm{1,247.5}+245\right)}}\right)\\ & & \times \sqrt{\mathrm{62,000,000}\times \mathrm{1,247.5}})\times {10}^{-3}\\ & =& \underset{\_}{156.3\mathrm{kN}}\end{array}$$E24

Determining the dynamic P₁ force for radius class R₃ as shown in Eq. (25).

to Eq. (28):

$${\mathrm{m}}_{\mathrm{e}}\approx 0.4\times \left(60+\frac{150}{0.6}\right)=124\mathrm{kg}/\mathrm{m}$$E25

$$\mathrm{G}=\frac{3.86}{{1.15}^{0.115}}\times {10}^{-8}=3.798\times {10}^{-8}\mathrm{m}/{\mathrm{N}}^{2/3}$$E26

$${\mathrm{K}}_{\mathrm{H}}=\frac{{{\mathrm{P}}_{1,\mathrm{est}}}^{\ast }-\mathrm{107,910}}{3.798\times {10}^{-8}\times \left({{\mathrm{P}}_{1,\mathrm{est}}}^{2/3}-{\mathrm{107,910}}^{2/3}\right)}=2.386\times {10}^9\mathrm{N}/\mathrm{m}$$E27

$${\mathrm{P}}_{1,90}=\left(\mathrm{107,910}+25\times 0.02\times \sqrt{\frac{2.386\times {10}^9\times 124}{1+\raisebox{1ex}{$124$}\!\left/ \!\raisebox{-1ex}{$\mathrm{1,247.5}$}\right.}}\right)\times {10}^{-3}=367.3\mathrm{kN}$$E28

*At the end of the iteration process P_1,est is expected to be ∼367,283.5 N.

Determining the TPV as shown in Eq. (29):

$${\mathrm{A}}_{\mathrm{eff}}=\frac{8.3593707\times \mathrm{\vartheta }+4.1874191}{{\mathrm{\vartheta }}^{0.8571601}}\times {\left(\frac{\mathrm{107,910}\times {10}^{-3}}{\frac{1}{1.15}+\frac{1}{0.3}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=110.4{\mathrm{mm}}^2$$

$$\text{with}\mathrm{\vartheta }=arccos\left(\frac{1.15-0.3}{1.15+0.3}\right)=0.94442737$$E29

$$\mathrm{TPV}=\frac{{\mathrm{P}}_{\text{Wheel}}}{{\mathrm{A}}_{\mathrm{eff}}}=\frac{800}{110.4}=7.243\mathrm{kW}/{\mathrm{mm}}^2$$

Table 5 summarizes the parameter values for the powered wheels of a universal locomotive in a track segment of radius class R₃. This locomotive does not have unpowered wheelsets. In the case of vehicles with both powered and unpowered wheelsets there exist another table with vehicle parameters for unpowered wheels. In the following table, the parameters Y_R and Tγ that belong to MBS are added too for further calculations. P₂ and Y_R values at 40 kmph are needed later on for damage term D₅.

Universal locomotive: axle load = 22 t | Smax,Veh = 230 kmph; R3: 400 < R ≤ 600 m | Sreference = 90 kmph | Rreference = 500 m
Parameters are related to one wheel
P1,90; in kN	P2,40; in kN	P2,90; in kN	TPV; in kW/mm^2	YR,90; in kN	YR,190; in kN	Tγ; in Nm/m
367.3	156.3	216.8	7.243	40.0	69.0	291.0

Table 5.

Vehicle parameters for powered wheels of a sample vehicle (universal locomotive).

For every vehicle type (such as locomotives, freight wagons, passenger wagons or multiple units) that appears in a network these five physical parameters shown in Table 5 need to be estimated. As not every parameter occurs in every radius and/or speed class in the following their appearance is summarized:

• P₁: R₁ to R₄ and values for R₅ S₁ to S₈ (12 input values)

• P₂: R₁ to R₄ and values for R₅ S₁ to S₈, one value at 40 kmph (13 input values)

• Y_R: R₁ to R₄ and one value at 40 kmph (5 input values)

• TPV: one input value

• Tγ: R₁ to R₄ (4 input values).

If those input parameters are available for all vehicles in the network, the seven damage increments of the TDM can be estimated for each vehicle. This is illustrated for the sample vehicle in the following Section 2.4.2.

2.4.2 Damage increment (D_n)

In this chapter exemplary calculations for the seven damage increments of the TDM are given. Required data and information to do so are.

• the estimated vehicle parameters (Table 5),

• number of powered and unpowered wheelsets and the number of leading wheelsets of bogies (Table 4) and

• the maximum permissible vehicle speed (Table 4).

The basic formula of the vehicle-specific damage increments D_n can be seen in Eq. (30). Since the vehicle parameters are given per wheel, they must be multiplied by the number of wheelsets (powered and unpowered).

$${\mathrm{Dn}}_{\mathrm{Veh},\mathrm{Si}|\mathrm{Rj}}=\begin{array}{l}\text{formula for damage termn}\left[{\text{VehPn}}_{\mathrm{Si}|{\mathrm{Rj}}_{\mathrm{p}}\left({\mathrm{S}}_{\min }\right)}\right]\times {\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{p}}}\\ +\text{formula for damage termn}\left[{\text{VehPn}}_{\mathrm{Si}|{\mathrm{Rj}}_{\mathrm{up}}\left({\mathrm{S}}_{\min }\right)}\right]\times {\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{up}}}\end{array}$$

$$\text{with}:\mathrm{S}=min\left[{\mathrm{S}}_{max,\mathrm{Veh}},{\mathrm{S}}_{\text{Track}}\right]$$E30

${\mathrm{D}}_{\mathrm{Veh},\mathrm{Si}\mid \mathrm{Rj}}$—damage increment Dn of the sample vehicle at speed (i) and radius class (j) (with i=1-8 und j=1-4); ${\mathrm{n}}_{\mathrm{Veh}}$—number of powered (p) or unpowered (up) wheelsets of the sample vehicle; ${\text{VehP}}_{\mathrm{Si}\mid \mathrm{Rj}\left({\mathrm{S}}_{\min }\right)}$—vehicle parameter of powered/unpowered wheelsets at R/S-class, depending on relevant speed; ${\mathrm{S}}_{max,\mathrm{Veh}}$—maximum permissible vehicle speed in kmph; ${\mathrm{S}}_{\text{Track}}$—reference speed at a certain speed and radius class in kmph; $\mathrm{S}$—relevant speed for determining the vehicle parameter in kmph.

Calculating the vehicle-specific damage increments, it is important to distinguish between powered and unpowered wheelsets. On the other hand, attention must be paid to the relevant speed due to the vehicle and the radius/speed-class. The minimum of both is considered in the calculation.

Exemplary calculation for damage increment D₁ is given in Eq. (31).

$$\text{Formula for}{\mathrm{D}}_1:{{\mathrm{P}}_{2,\mathrm{S}}}^3$$

$$\mathrm{S}=min\left[230\text{kmph},90\text{kmph}\right]=90\text{kmph}$$

$${\mathrm{D}}_{1,\mathrm{R}3}={216.8}^3\times 4+0=\mathrm{40,793,916}{\mathrm{kN}}^3$$E31

${\mathrm{D}}_{1,\mathrm{R}3}$—damage increment D1 of the sample vehicle in radius class R3 (kN³/vehicle); ${\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{p}}}$—4; ${\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{up}}}$—0; ${\mathrm{P}}_{2,\mathrm{R}{3}_{\mathrm{p}}}$—216.8 (kN/wheel); ${\mathrm{P}}_{2,\mathrm{R}{3}_{\mathrm{up}}}$—0 (kN/wheel).

Exemplary calculation for damage increment D₂ is given in Eq. (32).

$$\text{Formula for}{\mathrm{D}}_2:{{\mathrm{P}}_{2,\mathrm{S}}}^{1,2}$$

$$\mathrm{S}=min\left[230\text{kmph},90\text{kmph}\right]=90\text{kmph}$$

$${\mathrm{D}}_{2,\mathrm{R}3}={216.8}^{1.2}\times 4+0=\mathrm{2,543.7}{\mathrm{kN}}^{1.2}$$E32

${\mathrm{D}}_{2,\mathrm{R}3}$—damage increment D2 of the sample vehicle in radius class R3 (kN^1.2/vehicle); ${\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{p}}}$—4; ${\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{up}}}$—0; ${\mathrm{P}}_{2,\mathrm{R}{3}_{\mathrm{p}}}$—216.8 (kN/wheel); ${\mathrm{P}}_{2,\mathrm{R}{3}_{\mathrm{up}}}$—0 (kN/wheel).

Exemplary calculation for damage increment D₃ is given in Eq. (33).

$$\text{Formula for}{\mathrm{D}}_3:\mathrm{TPV}=7.243\mathrm{kW}/{\mathrm{mm}}^2$$E33

${\mathrm{D}}_3$—damage increment D₃ of the sample vehicle (kW/mm²); ${\mathrm{TPV}}_{\mathrm{Veh}}$—7.243 (kW/mm²)

Exemplary calculation for damage increment D_4.1 and D_4.2 is given in Eq. (34).

$$\text{Formula for}{\mathrm{D}}_{4.1}\left(\text{area}\mathrm{a}\right):{\mathrm{D}}_{4.1}=0\text{for}{\mathrm{T}}_{\mathrm{\gamma }}\ge 175\mathrm{Nm}/\mathrm{m}$$

$${\mathrm{D}}_{4.1,\mathrm{R}3}=\underset{}{0.00}$$

$$\text{Formula for}{\mathrm{D}}_{4.2}\left(\text{area}\mathrm{e}\right):{\mathrm{D}}_{4.2}=\left(\frac{{\mathrm{T}}_{\mathrm{\gamma }}-65}{110}\right)\text{for}{\mathrm{T}}_{\mathrm{\gamma }}\ge 65\mathrm{Nm}/\mathrm{m}$$

$${\mathrm{D}}_{4.2,\mathrm{R}3}=2\ast \left(\frac{291-65}{110}\right)+0=4.11$$E34

$\mathrm{D}{4.1}_{\mathrm{R}3}$—damage increment D4.1 of the sample vehicle in radius class R₃ (-/vehicle); $\mathrm{D}{4.2}_{\mathrm{R}3}$—damage increment D4.2 of the sample vehicle in radius class R₃ (-/vehicle); ${\mathrm{n}}_{\mathrm{p}}^{\prime }$—2; ${\mathrm{n}}_{\mathrm{up}}^{\prime }$—0; ${\mathrm{T}}_{\mathrm{\gamma },\mathrm{p}}$—291 (Nm/m); ${\mathrm{T}}_{\mathrm{\gamma },\mathrm{up}}$—0 (Nm/m).

Exemplary calculation for damage increment D₅ is given in Eq. (35).

$$\text{Formula for}{\mathrm{D}}_5:\sqrt{\left(0.5\times {{\mathrm{P}}_{2,\mathrm{S}=40}}^2+0.5\times {{\mathrm{Y}}_{\mathrm{R}=190}}^2\right)}$$

$${\mathrm{D}}_{5,\mathrm{R}3}=\sqrt{\left(0.5\times {156.3}^2+0.5\times {69.0}^2\right)}\ast 2+0=241.6\mathrm{kN}$$E35

${\mathrm{D}}_{5,\mathrm{R}3}$—damage increment D5 of the sample vehicle in radius class R₃ (kN/vehicle); ${\mathrm{n}}_{\mathrm{p}}^{\prime }$—2; ${\mathrm{n}}_{\mathrm{up}}^{\prime }$—0; ${\mathrm{P}}_{2,40,\mathrm{p}}$—156.3 (at 40 kmph) (kN/wheel); ${\mathrm{P}}_{2,40,\mathrm{up}}$—0 (at 40 kmph) (kN/wheel); ${\mathrm{Y}}_{\mathrm{R}=190,\mathrm{p}}$—69.0 (at 40 kmph and radius 190 m) (kN/wheel); ${\mathrm{Y}}_{\mathrm{R}=190,\mathrm{up}}$—0 (at 40 kmph and radius 190 m) (kN/wheel).

Exemplary calculation for damage increment D₆ is given in Eq. (36).

$$\text{Formula for}{\mathrm{D}}_6:{{\mathrm{P}}_{1,\mathrm{S}}}^3$$

$$\mathrm{S}=min\left[230\text{kmph},90\text{kmph}\right]=90\text{kmph}$$

$${\mathrm{D}}_{6,\mathrm{R}3}={367.3}^3\times 4+0=\mathrm{198,208,728}{\mathrm{kN}}^3$$E36

${\mathrm{D}}_{6,\mathrm{R}3}$—damage increment D6 of the sample vehicle in radius class R₃ (kN³/vehicle); ${\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{p}}}$—4; ${\mathrm{n}}_{{\mathrm{Veh}}_{\mathrm{up}}}$—0; ${\mathrm{P}}_{1,\mathrm{R}{3}_{\mathrm{p}}}$—367.3 (kN/wheel); ${\mathrm{P}}_{1,\mathrm{R}{3}_{\mathrm{up}}}$—0 (kN/wheel).

Exemplary calculation for damage increment D₇ is given in Eq. (37).

$$\text{Formula for}{\mathrm{D}}_7:{\sqrt{\left({\mathrm{f}}_{7_{1,\mathrm{R}}}\times {{\mathrm{P}}_{2,\mathrm{S}}}^2+{\mathrm{f}}_{7_{2,\mathrm{R}}}\times {{\mathrm{Y}}_{\mathrm{R}}}^2\right)}}^3$$

$$\mathrm{S}=min\left[230\text{kmph},90\text{kmph}\right]=90\text{kmph}$$

$${\mathrm{D}}_{7,\mathrm{R}3}=2\ast {\sqrt{\left(0.7\times {216.8}^2+0.3\times {40.0}^2\right)}}^3+0=\mathrm{12,207,944}{\mathrm{kN}}^3$$E37

${\mathrm{D}}_{7,\mathrm{R}3}$—damage increment D7 of the sample vehicle in radius class R₃ (kN³/vehicle); ${\mathrm{n}}_{\mathrm{p}}^{\prime }$—2; ${\mathrm{n}}_{\mathrm{up}}^{\prime }$—0; ${\mathrm{P}}_{2,\mathrm{R}{3}_{\mathrm{p}}}$—216.8 (kN/wheel); ${\mathrm{P}}_{2,\mathrm{R}{3}_{\mathrm{up}}}$—0 (kN/wheel); ${\mathrm{Y}}_{\mathrm{R}3,\mathrm{p}}$—40.0 (kN/wheel); ${\mathrm{Y}}_{\mathrm{R}3,\mathrm{up}}$—0 (kN/wheel); ${\mathrm{f}}_{7_{1,\mathrm{R}}}$—0.7; ${\mathrm{f}}_{7_{2,\mathrm{R}}}$—0.3.

2.4.3 Damage per vehicle-kilometer (D_n,km)

In a further step the damage increments of every vehicle are related to length by means of unit-kilometer. The absolute value of the damage increment stays the same; however, the unit changes (see Eq. (38)).

$${\mathrm{D}}_{\mathrm{n},\mathrm{km}}={\mathrm{D}}_{\mathrm{n}}\times \mathrm{km}$$E38

D_n,km—damage increment per vehicle of one unit kilometer (unit*km/vehicle); D_n—damage increment (n = 1, 2, 3, 4.1, 4.2, 5, 6, 7) (unit*/vehicle); km—length (km).

*Unit of the damage term: kN³ (D₁, D₆ and D₇), kN^1.2 (D₂), kW/mm² (D₃), kN (D₅) and D_4.1 and D_4.2 are unitless

Due to Eq. (38), a length reference is established in the TDM and vehicle-kilometer are therefore included. This approach is similar to the conventional vehicle gross-tonnes that turn into vehicle gross-ton-kilometer when length is considered.

3. Results

In the first part of the result section, we assess the influence of traffic on track maintenance. We start with a given situation in an artificially configurated network with a mixed traffic load of some 30,000 gross-tons per day (∼10 mio. Gross-tons per year). We analyze the influence of changes in the traffic mix preliminary with focus on the different traffic segments, long-distance passenger traffic, regional passenger traffic, and freight transport. We also include by way of example the influence of different superstructure and substructure conditions (see more details in [1]).

In the second part, the influence of different vehicles is highlighted (see Section 3.2). This influence is much smaller at the network level of course, but might be significant in some parts, line segments or whenever entire vehicle fleets are replaced. We compare different vehicles at different speed levels and also powered and running axles. Estimating future track maintenance needs the assessment of future vehicles on the network. New vehicles might be track-friendly or track-harming and train operators focus on many other aspects than track maintenance consequences. Again, in these evaluations, we see the strong impact of the technical configuration of track, as different tracks indicate a higher or lower importance of the one or the other vehicle characteristics.

3.1 The impact of loading on track maintenance

In order to highlight the main difference between treating track loading as gross-tonnage or as specific damage for different track components, this paper delivers results based on an artificial network. Note, this network is also used in [1] dealing with the impact of different track-related boundary conditions on overall track maintenance and renewal. We add the line speed as a parameter and use the modulation done in [1] concerning superstructure components.

The network covers 10,000 km with a given radii distribution and line speed distribution for the straight sections given in Table 6. To keep the focus on the loading, we model the maintenance for a given, constant track consisting of heavy superstructure (concrete sleepers, wooden sleepers in the sharp curves R ≤ 250 m 60E1-R260 rails) on medium ballast quality, good subsoil and good drainage condition. The tracks are loaded with 30,000 gross-tons per day everywhere in the network. Both assumptions are never true for any existing network, but were chosen for illustration purposes.

Radii Class	Track Alignment; in m	Radii Distribution; in %	Line Speed*; in kmph	Line Speed Distribution; in %
R5	R > 1000	76.0	140 < S ≤ 160	12
			120 < S ≤ 140	31
			100 < S ≤ 120	30
			80 < S ≤ 100	19
			S ≤ 80	8
R4	600 < R ≤ 1000	12.5	140	—
R3	400 < R ≤ 600	6.5	110	—
R2	250 < R ≤ 400	4.5	90	—
R1	R ≤ 250	0.5	80	—

Table 6.

Reference network.

*

Line Speed is not reached by all trains/vehicles.

According to the track maintenance assessment provided by [1], this network characteristics deliver a certain track maintenance and renewal demand depicted in Table 7.

Track work	Amount; in km
Leveling-Lining-Tamping	1747
Ballast Cleaning	0
Rail Grinding	504
Rail Exchange	5.5
Renewal	283

Table 7.

Maintenance and renewal demand for the reference network.

For applying the alternative description of track loading proposed in Section 2, we need to have a closer look at the transport load. The traffic mix shows long-distance passenger trains (LDP), regional passenger trains (RP) and freight trains (F) consisting of different vehicles (Table 8).

	No. of Trains; unitless	[relative]; in %	Loading; in gross- tons	[relative]; in %
Long-distance Passenger	13	[20]	6284	[21]
Regional Passenger	35	[53]	5072	[17]
Freight	18	[27]	18,814	[62]
Sum	66	[100]	30,170	[100]

Table 8.

Traffic mix.

The train configuration of the different market segments is depicted in Table 9. The long-distance train is a loco-wagon train with a maximum speed of 160 kmph (and thus reaching the maximum line speed). For the regional passenger traffic, the traffic mix consists of two different types of electric multiple units (EMU), one (EMU1) with a total weight of 160 tons consists of four powered bogies and two trailer bogies (axle scheme Bo’Bo’2’2’Bo’Bo’), the other one (EMU2) is a lighter and shorter trainset with two powered bogies and three trailer bogies (axle scheme Bo’2’2’2’Bo’) and a total weight of 127 tons. The freight trains are modeled as some 1000 ton-trains with one four-axle loco and a mix of empty, medium-loaded and full-loaded freight wagons (detailed loads see Table 9), all with two Y25-bogies.

	No. of vehicles; in Vehicle/Train	Vehicle weight; in tons	Train weight; in tons
Long-distance Passenger
4ax-Universal-Loco	1	88	483.36
4ax-2Bogie-Passenger Car	7	56.48
Regional Passenger
EMU1	19	160	160.00
EMU2	16	127	127.00
Sum/Average	35		144.91
Freight
4ax-Universal-Loco	1	88	1045.23
4ax-2Bogie-Y25 Freight Car “empty” (Axle-load 6.5 t)	7	25.56
4ax-2Bogie-Y25 Freight Car “medium” (Axle-load 14 t)	8	56.65
4ax-2Bogie-Y25 Freight Car “full” (Axle-load 20 t)	4	81.28

Table 9.

Train configuration.

3.1.1 Ballast maintenance

The transport volume for the entire network can be addressed as gross-ton-kilometers or as damage-kilometer following Section 2. For the latter, for the ballast maintenance we need to calculate the P₂-forces, specifically for the different line speed sections. Thereby it is to be considered that the train speed may be limited by the allowed maximum vehicle speed and not by the given line speed. For freight trains, the maximum speed assessed for this calculation is 100 kmph. Thus, for all sections with higher line speeds the damage increments D₁ (P₂-force) are calculated with 100 kmph only. What we can see directly from Table 10 is that the alternative description of “loading” moves relative damage shares toward long-distance passenger traffic. This effect is mainly driven by the higher speed as axle loads for these two traffic segments are about the same on average. The share of regional passenger traffic remains almost the same as speeds are higher than in freight transportation, but axle loads lower.

	Gross-ton-kilometers per year		D1: kN^3-km per year
Long-distance Passenger	2,29E+10	21%	9,83E+15	28%
Regional Passenger	1,85E+10	17%	5,57E+15	16%
Freight	6,87E+10	62%	1,94E+16	56%
Sum	1,10E+11	100%	3,48E+16	100%

Table 10.

Track loading.

We know that this loading leads to 1747 kilometers of necessary tamping in the network or a tamping interval of 5.7 years (6 years in the straight sections). If we allocate this tamping demand to the loading, we can calculate an incremental tamping demand for the unit gross-ton-km and the alternative unit kN³km.

As both tamping demand and the damage according to D₁ vary over the track radius, we split this calculation into radii classes. All values are presented in Table 11. The incremental tamping demand in Table 11 can also be seen as the calibration of the damage model D₁.

	R5	R4	R3	R2	R1
LDP gross-ton-km	1,75E+10	2,86E+09	1,48E+09	1,03E+09	1,13E+08
RP gross-ton-km	1,41E+10	2,30E+09	1,19E+09	8,28E+08	9,15E+07
F gross-ton-km	5,23E+10	8,55E+09	4,43E+09	3,07E+09	3,39E+08
Sum gross-ton-km	8,38E+10	1,37E+10	7,10E+09	4,93E+09	5,44E+08
Tamping demand [km]	1269	208	133	112	26
incremental tamping demand ton-km	1,51E-08	1,51E-08	1,87E-08	2,27E-08	4,72E-08
LDP kN^3-km	7,99E+15	1,18E+15	4,35E+14	2,06E+14	1,85E+13
RP kN^3-km	4,52E+15	6,74E+14	2,48E+14	1,17E+14	1,05E+13
F kN^3-km	1,49E+16	2,49E+15	1,29E+15	6,10E+14	5,46E+13
Sum kN^3-km	2,74E+16	4,34E+15	1,97E+15	9,34E+14	8,36E+13
Tamping demand [km]	1269	208	133	112	26
incremental tamping demand kN^3-km	4,62E-14	4,78E-14	6,73E-14	1,20E-13	3,07E-13

Table 11.

Incremental tamping demand.

If we now change the transport volume, the differences between those two approaches become visible, as shown in Table 12. In the gross-ton-approach, it does not matter which trains generate the ton-km. Therefore, the tamping demand calculated with the incremental tamping demand based on ton-km according to Table 11 stays constant for constant gross-ton-kilometers. In the case of the specified loading derived from the damage function D₁, the results change significantly: In the straight sections, the higher speeds of long-distance trains lead to 40% higher taping demands, whereas the lower speeds decrease the demand for the “freight trains only” scenario (note: on average, freight transport delivers lower axle loads than long-distance passenger traffic).

		R5	R4	R3	R2	R1
Sum gross-ton-km		8,38E+10	1,37E+10	7,10E+09	4,93E+09	5,44E+08
Tamping demand	in km	1269	208	133	112	26
LDP kN^3-km		3,87E+16	5,70E+15	2,11E+15	1,00E+15	8,95E+13
Tamping demand; LDP only	in km	1791	273	142	120	28
	in %	141	131	107	107	107
RP kN^3-km		2,69E+16	4,01E+15	1,48E+15	6,98E+14	6,23E+13
Tamping demand; RP only	in km	1244	192	99	84	19
	in %	98	92	75	75	75
F kN^3-km		2,41E+16	4,01E+15	2,08E+15	9,83E+14	8,80E+13
Tamping demand; F only	in km	1112	192	140	118	27
	in %	88	92	105	105	105

Table 12.

Tamping demand for different track loading scenarios.

These results are well in line with common experience and still count, even if we consider the extremes on both ends of railway operation: slow freight traffic needs less ballast maintenance, even in case of heavy haul operation, while high-speed train operation leads to very frequent interventions in order to keep track geometry on the necessary level [30].

In mixed traffic though, these effects occur at a much lower level. To show the limited, but nevertheless existing effects, we double the transport volume in our example. For the gross-ton-approach, this simply gives double amount of ballast related maintenance. Increasing the transport volume to some 60,000 gross-tons per day by adding trains of one market segment only delivers different maintenance needs using the specific damage function D₁ (Table 13).

		R5	R4	R3	R2	R1
Doubled gross-ton-km		1,68E+11	2,74E+10	1,42E+10	9,85E+09	1,09E+09
Tamping demand	in km	2538	415	266	224	51
	in %	200	200	200	200	200
	SUM in km	3493
Doubling with LDP only kN^3-km		6,56E+16	9,95E+15	4,05E+15	1,92E+15	1,72E+14
Tamping demand; LDP only	in km	3032	476	272	230	53
	SUM in km	4062
Doubling with RP only kN^3-km		5,46E+16	8,38E+15	3,46E+15	1,64E+15	1,46E+14
Tamping demand; RP only	in km	2524	401	233	196	45
	SUM in km	3399
Doubling with F only kN^3-km		2,41E+16	4,01E+15	2,08E+15	9,83E+14	8,80E+13
Tamping demand; F only	in km	2381	399	273	230	53
	SUM in km	3335

Table 13.

Tamping demand with increasing track loading.

Again, we learn that maintenance demands for increasing transport volumes can be estimated sufficiently well by using gross-tonnage as long as this increasing volume consists mainly of regional passenger trains (the difference to specific damage function is less than 3%) or freight trains (the difference is some 5%). Adding faster long-distance trains add much more tamping needs than estimated by the simplified gross-ton-approach with a resulting 16% higher total tamping demand.

In order to deepen these findings, we performed another variation: instead of simply increasing the amount of tonnage of long-distance passenger trains, we introduced an additional long-distance passenger service with a speed up to 200 kmph in the sections with maximum line speed of 160 kmph (12% of the straight line-section according to Table 6) covering half of the trains (scenario LDP+ only, see Table 14). Of course, this assumes that technically the maximum line speed can be increased to this level. In this case, the tamping demand in the radii class R₅ rises by another 250 kilometers.

Doubling with LDP only kN^3-km		6,56E+16	9,95E+15	4,05E+15	1,92E+15	1,72E+14
Tamping demand; LDP only	in km	3032	476	272	230	53
	SUM in km	4062
Doubling with LDP+ only kN^3-km		7,10E+16	9,95E+15	4,05E+15	1,92E+15	1,72E+14
Tamping demand; LDP+ only	in km	3282	476	272	230	53
	SUM in km	4312
Doubling with LDP+ only kN^3-km		7,10E+16	9,95E+15	4,05E+15	1,92E+15	1,72E+14
Tamping demand; LDP+ only; improved Track Quality	in km	2894	476	272	230	53
	SUM in km	3925

Table 14.

Tamping demand with increasing track loading—Scenario LDP+ only.

Usually this effect is not significantly recognized even though higher speeds in passenger long-distance services are introduced quite intensively in European mixed traffic networks. The reason for this is that introducing faster passenger services is often accompanied by establishing new lines or the total rehabilitation of existing lines. This comes along with tracks on perfect substructure and robust components such as padded concrete sleeper. Such tracks generally perform better, and ballast-related maintenance is reduced by 50%. Considering this for our example, the scenario “LDP+ only, improved Track Quality” even delivers a lower tamping demand (Table 14).

Summarizing, we can state that the influence of the specific loading is considerably high even though it is hard to extract in top-down figures for extended mixed traffic networks.

3.1.2 Rail maintenance

Similar to the ballast maintenance, we can analyze the rail maintenance in more detail using the damage function according to Section 2, the damage mechanism D₄ for both rail surface damage in wider curves and rail side wear in tighter curves. Note: Rail grinding in tangent track is not assessed, as this is driven by the damage functions D₂ and D₃.

Differently to ballast maintenance, regional passenger trains contribute significantly to both rail surface damage and rail side wear (Table 15). Again, doubling the transport volume (and thus the rail maintenance following the gross-ton-approach) by increasing one market segment only, we see distinct differences.

		Rail Surface Damage1	Rail Wear2
Doubled gross-ton-km		1,37E+11	1,37E+11
Grinding1/Rail Exchange2demand	in km	584	11.0
	in %	200	200
Doubling with LDP only D4-km		1,07E+09	8,76E+08
Grinding1/Rail Exchange2demand; LDP only	in km	653	14,6
	in %	224	265
Doubling with RP only D4-km		1,16E+09	7,81E+08
Grinding1/Rail Exchange2demand; RP only	in km	707	13,0
	in %	242	236
Doubling with F only D4-km		9,13E+08	5,48E+08
Grinding1/Rail Exchange2demand; F only	in km	558	9,1
	in %	191	165

Table 15.

Rail maintenance demand with increasing track loading.

¹

According to damage term D_4.1.

²

According to damage term D_4.2.

This result is not linked to train speeds, but mainly to the vehicles in use. Rail maintenance is mainly driven by the type of bogie and the longitudinal stiffness of the car body. Therefore, we need to look deeper into different vehicles.

3.2 The influence of vehicle properties on track maintenance

As shown in Section 2, the alternative description of track loading as specific damage goes along with vehicle properties such as axle load, unsprung mass, traction power or longitudinal stiffness and also for some damage mechanisms with vehicle speed. Since the total damage is calculated as a sum of those impacts, it is in turn also possible to allocate track maintenance to single vehicles. This chapter gives examples for the influence of different vehicle properties in combination with operational scenarios. In order to sum up different track maintenance works, for this task, we use the maintenance costs. Thus, the following results are track maintenance cost per vehicle-kilometer, sometimes re-calculated to track maintenance cost per gross-ton-kilometer. As the absolute cost level varies from infrastructure manager to infrastructure manager and are moreover not of importance for this investigation, the costs are normalized. The 100% base level is a fully laden 4-axle freight car with Y25-bogies (axle load 20.5 tons) running at 90 kmph on a straight track. The track is according to the network configuration in Section 3.1 a ballasted concrete sleeper track on good subsoil with 60E1-R260 rails. Note: In order to keep results comprehensible, we do not add any costs of turnouts so that the damage mechanisms D₅ and D₆ are not added. Also, D₇ is not addressed as we model track maintenance only, without incorporating renewals.

To better understand the results, it is essential to know the track maintenance expenses allocated to the damage mechanisms. According to the network configuration and the superstructure parameters, track maintenance costs split into the percentages displayed in Figure 3. We see that the overwhelming part of the costs (85%) is triggered by the damage mechanism D₁, the dynamic vertical forces. These costs contain mainly ballast-related maintenance and small reactive maintenance. In this network, some 12% of track maintenance costs are due to rail surface damage (D₂, D₃ and D_4.1) and thus rail grinding or milling costs. Only 2% of the costs are the consequence of rail side wear in curves.

Figure 3.

Track maintenance expenses.

Looking at the reference vehicle (Figure 4), the heavy freight car, we learn that ballast-related maintenance costs increase with decreasing track radius. Moreover, rail surface maintenance costs in straight tracks are very low and increase with decreasing track radius to be finally replaced by rail exchange costs (rail wear) in the smallest radius (250 m).

Figure 4.

Track maintenance costs per vehicle-kilometer—4ax-Y25 freight wagon_20 t.

Locomotives and generally powered axles come along with higher unsprung masses and in addition with traction forces. The latter deliver a significant contribution to the rail surface damage in straight sections (damage D₃). Figure 5 shows both effects directly compared to the unpowered axles of the freight wagon. Both vehicles have the same tonnage but contribute very differently to the track maintenance needs. This is in accordance with [16]. The third column shows the influence of speed: Due to the high additional dynamic forces, the allocated track maintenance costs double when operating the same loco at 160 kmph instead of 90 kmph.

Figure 5.

Track maintenance costs per vehicle-kilometer—Powered and unpowered axles.

The high differences between the allocated maintenance costs in straight and curved track originate mainly in the maintenance needs themselves. Track maintenance in tight curves reaches much higher levels than in straight sections (see [1]). In a network, curves form only a small part. Looking at the network used in this paper as reference (Table 6), only 24% of the track show radii below 1000 m. The wear formula allows for calculating associated track maintenance costs as a consequence of track radius and speed level. For this example, we use nine different operational situations (four radii classes and five speed levels according to Table 6). In our simplified example, all vehicles run along all lines exactly in the distribution of these operational situations. Thus, we can calculate the average track maintenance cost per average vehicle-kilometer (vehicles according to Table 9) and sum up how the single vehicles to trains deliver results as shown in Figure 6.

Figure 6.

Track maintenance costs per train-kilometer for different trains.

These trains have different weights and cannot be compared directly. This counts especially for the light electrical multiple units and the heavy freight trains. We thus re-calculate the train costs to gross-ton-kilometers, simply by dividing by the train weight (see Table 9). Figure 7 shows the damage impact of one gross-ton-kilometer on the track: Again, the long-distance passenger train is the 100% level. The regional trainsets deliver 20 to 40% less track damage. This is due to the lighter axles, less unsprung masses and the lower speeds compared to the fast intercity-train. The freight train has similar axle loads on average compared to the long-distance passenger train and the same loco, but runs at 100 kmph maximum so that the impact per gross-ton-km is 31% lower. Figure 7 also shows the gross-ton-kilometer approach with the dashed light gray bars: In this approach, all these trains are meant to cause the same track damage.

Figure 7.

Track maintenance costs per gross-ton-kilometer for different trains.

These first examples compare very different trains and/or vehicles. It is not surprising that the track damage caused by these vehicles differ widely. But these differences occur for similar vehicles in the same way. We looked at locos with similar total weights (which is always close to the maximum allowed weight of 90 tons), but very differing unsprung masses, and bogie stiffnesses. In Figure 8, we see differences up to 65% for operating the locos on passenger trains with speeds up to 160 kmph in the artificial network. In freight operation, the differences are much lower, 20% maximum.

Figure 8.

Track maintenance costs per gross-ton-kilometer for different locos.

From a track maintenance point of view, the use of track-friendly vehicles is of course preferable. However, the selection of vehicles or even train concepts focuses on many other aspects. We highlight the use of different train concepts as our next example: As an alternative choice for the long-distance passenger train consisting of the loco and seven wagons, an electrical multiple unit enabling higher speeds is analyzed. This trainset comes along with distributed powered axles, comparably high axles loads as the trainset is a double-decked unit and a total weight of some 400 tons. In Figure 9, we see three different options for performing a comparison. The red line demonstrates the allocated track maintenance costs of the loco hauled passenger train for the different radii and speed ranges (100% is the cost at speed level S₅, 150 kmph). This red line is the same for all three comparisons. Initially, we look at the track maintenance cost per vehicle-kilometer (black solid line). The EMU causes some 20% lower damage in curves, only slightly lower damage in straight sections with moderate speeds, and only at the highest speed (velocity range 5) higher track maintenance cost (plus 9%). As the trainset is lighter than the push/pull train, we again re-calculate the cost to gross-ton-level (dashed black line). We learn that the impact of these two trains is very similar on gross-ton-level; in the highest speed range the inserted damage of the EMU is 30% higher. The third possibility for comparing different passenger train configurations is to refer the track maintenance cost caused to seat-kilometers. In this option, the double-decked trainset option performs best: As the dotted line in Figure 9 indicates, the difference is remarkable and reaches values up to 35% less track maintenance costs caused.

Figure 9.

Track maintenance costs per gross-ton-kilometer for different long-distance passenger trains.

Concluding, we want to underline that not only loading triggers track maintenance. Improving track structure and installing innovative, robust components helps to drop the maintenance demands—much more than track-friendly vehicle concepts. Figure 10 shows, for example, the impact of using concrete sleepers with under sleeper pads [31, 32] on the ballast-related maintenance costs (dashed yellowish bars) and on top, for example, a rail steel grade of R350HT in curves (dashed purple bars—rail grinding and rail exchange). The track maintenance cost decreases by more than 40% using these components in curves and still by one third in straight sections where higher rail steel grades do not reduce maintenance demands.

Figure 10.

Track maintenance costs per vehicle-kilometer—4ax-Y25 freight wagon_20 t with varying track components.

4. Conclusions

Track loading is a main trigger for track maintenance needs. Gross-tonnage is widely used, but cannot cover the different damage mechanisms properly. Damage processes on rails and in ballast are the most relevant causes for track maintenance. Dynamic vertical forces created by unevenness of track can be seen as loading for the ballast-related deterioration process, while rail surface damage and rail wear needs a more holistic description of the loading. Existing track deterioration models still need improvement, especially for new rail surface failures and for turnouts, but deliver a much more detailed insight on both track maintenance demand for changing traffic compositions and vehicle properties triggering track maintenance. Following this approach of a track deterioration model based on vehicle properties will help to forecast necessary track maintenance and the associated budgets much more precisely than the simplified gross-ton approach. The possibility to calculate track maintenance costs on vehicle basis enables of course also to allocate these costs much better in track access charge schemes [21, 33, 34]. This would support the construction and use of track-friendly vehicles and thus to reduce the costs of railway operation.

Acknowledgments

The research was funded by ÖBB-Infrastruktur AG and SBB.