**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.

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.

*Traffic Load and Its Impact on Track Maintenance DOI: http://dx.doi.org/10.5772/intechopen.110800*

**Figure 1.** *Input categories for the track deterioration model.*

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 (Dn for damage) and cost calibration factors (cn 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.

$$\mathbf{C}\_{\text{Veh}\_{\text{S},\text{R}}} = \sum\_{\mathbf{n}=1}^{7} \mathbf{c}\_{\text{n}} \times \mathbf{D}\_{\text{n}} \tag{1}$$

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

\**Unit of the damage term: kN<sup>3</sup> (D1, D6 and D7), kN1.2 (D2), kW/mm<sup>2</sup> (D3), kN (D5) and D4.1 and D4.2 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 (kN1.2). 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 Dn 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).

$$\begin{aligned} \mathbf{C}\_{\text{Vech},\text{R}} &= \mathbf{c}\_{1} \times \mathbf{P}\_{2,\text{S}}^{2} + \mathbf{c}\_{2} \times \mathbf{P}\_{2,\text{S}}^{1,2} + \mathbf{c}\_{3} \times \mathbf{T} \mathbf{P} \mathbf{V} + \mathbf{c}\_{4,1} \times \mathbf{D}\_{4,1} + \mathbf{c}\_{4,1} \times \mathbf{D}\_{4,2} + \mathbf{c}\_{5} \\ &\times \sqrt{\left(\mathbf{0}.5 \times \mathbf{P}\_{2,\text{S}}^{2} + \mathbf{0}.5 \times \mathbf{Y}\_{\text{R}}^{2}\right)} + \mathbf{c}\_{6} \times \mathbf{P}\_{1,\text{S}}^{3} + \mathbf{c}\_{7} \\ &\times \sqrt{\left(\mathbf{f}\_{7\_{1\text{R}}} \times \mathbf{P}\_{2,\text{S}}^{2} + \mathbf{f}\_{7\_{2\text{R}}} \times \mathbf{Y}\_{\text{R}}\right)^{3}} \end{aligned} \tag{2}$$

CVeh,S,R—costs per vehicle kilometer depending on speed and radius (costs/km); cn—cost calibration factors (n = 1, 2, 3, 4.1, 4.2, 5, 6, 7) (costs/(unit km\*); P2,S dynamic vertical wheel force (long-waved) depending on speed (kN); P1,S—dynamic vertical wheel force (short-waved) depending on speed (kN); YR—lateral force of the guiding wheel on the outer rail within radius R 190 m (kN); TPV—traction power value (kW/mm<sup>2</sup> ); D4.1—damage index for rolling contact fatigue (RCF); D4.2—damage index for plastic deformation/rail abrasion; f71,R—weighting factor for the vertical dynamic wheel force depending on radius R; f72,R—weighting factor for the lateral wheel force depending on radius R.

\**Unit of the damage term: kN3 (D1, D6 and D7), kN1.2 (D2), kW/mm<sup>2</sup> (D3), kN (D5) and D4.1 and D4.2 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).

$$\mathbf{D}\_1 = \mathbf{P}\_{2,\mathbf{S}}^{\;\;\;\!\!\mathbf{S}} \tag{3}$$

D1—damage term 1 (kN<sup>3</sup> ); P2,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.

$$\mathbf{D}\_2 = \mathbf{P}\_{2,\mathbf{S}} \mathbf{s}^{1,2} \tag{4}$$

D2—damage term 2 (kN1.2); P2,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.

$$\mathbf{D}\_3 = \mathbf{T} \mathbf{P} \mathbf{V} \tag{5}$$

D3—damage term 3 (kW/mm<sup>2</sup> ); TPV—traction power value (kW/mm<sup>2</sup> ).

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 D4.1 and D4.2 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

#### **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).*

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).

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 D4.1 as depicted in Eq. (6) and damage term D4.2 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 D4.1. Scenarios (d) and (e) are linked to damage term D4.2 that represents rail exchange in curved tracks.

$$\begin{array}{ll} \text{(a)} \; \text{D}\_{4.1} = \text{0} & \text{for } \text{T}\_{\text{\text{\textdegree}}} < 15 \text{Nm}/\text{m} \text{and } \text{T}\_{\text{\textdegree}} \ge 175 \text{Nm}/\text{m} \\\\ \text{(b)} \; \text{D}\_{4.1} = \text{n} \times \left( \mathbf{0}.02 \times \text{T}\_{\text{\textdegree}} - \mathbf{0}.3 \right) & \text{for } 15 \le \text{T}\_{\text{\textdegree}} < 65 \text{Nm}/\text{m} \\\\ \text{(c)} \; \text{D}\_{4.1} = \text{n} \times \left( \frac{-\text{T}\_{\text{\textdegree}} + 175}{110} \right) & \text{for } 65 \le \text{T}\_{\text{\textdegree}} < 175 \text{Nm}/\text{m} \\\\ \text{(d)} \; \text{D}\_{4.2} = \text{0} & \text{for } \text{T}\_{\text{\textdegree}} < 65 \text{Nm}/\text{m} \\\\ \text{(e)} \; \text{D}\_{4.2} = \text{n} \times \left( \frac{\text{T}\_{\text{\textdegree}} - 65}{110} \right) & \text{for } \text{T}\_{\text{\textdegree}} \ge 65 \text{Nm}/\text{m} \end{array}$$

D4.1—damage term 4.1; D4.2—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 D4.1 and D4.2 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 D4.1 and D4.2 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 (YR) forces. P2 is thereby analogous to the use in D1. Whereas YR is estimated by a multiplebody 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%.

$$\mathbf{D}\_{\mathbf{5}} = \sqrt{\left(\mathbf{0.5} \times \mathbf{P\_{2,S}}^{2} + \mathbf{0.5} \times \mathbf{Y\_{R}}^{2}\right)}\tag{8}$$

D5—damage term 5 (kN); P2,S—dynamic vertical wheel force (long-waved) depending on speed (kN); YR—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).

$$\mathbf{D}\_{\mathsf{6}} = \mathbf{P}\_{\mathbf{1},\mathsf{S}}{}^{\mathsf{3}} \tag{9}$$

D6—damage term 6 (kN3 ); P1,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 shortwaved 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.

$$\mathbf{D}\_{7} = \sqrt{\left(\mathbf{f}\_{7\_{1,\mathbb{R}}} \times \mathbf{P}\_{2,\mathbb{S}}\right)^{2} + \mathbf{f}\_{7\_{2,\mathbb{R}}} \times \mathbf{Y}\_{\mathbb{R}}^{2}}^{3} \tag{10}$$

D7—damage term 7 (kN<sup>3</sup> ); P2,S—dynamic vertical wheel force (long-waved) depending on speed (kN); YR—lateral force of the guiding wheel on the outer rail within radius R (kN); f71,R—weighting factor for the vertical dynamic wheel force; f72,R—weighting factor for the lateral wheel force.

The speed-dependent vertical force P2 and the speed and radius-dependent lateral force YR 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 (f7.1 and f7.2) 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 YR. **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.

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


**Table 1.**

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

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.
