LCC-based approach for design and requirement specification for railway track system

doi:10.21203/rs.3.rs-3737541/v1

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LCC-based approach for design and requirement specification for railway track system

https://doi.org/10.21203/rs.3.rs-3737541/v1

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published 30 Jul, 2024

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Life cycle cost (LCC) analysis is an important tool for effective infrastructure management. It is an essential decision support methodology for selection, design, development, construction, and maintenance of railway infrastructure system. Effective implementation of LCC analysis will assure cost-effective operation of railways from both investment and life-cycle perspectives. A major setback in the successful implementation of LCC by infrastructure managers is the availability of relevant, reliable, and structured data. Different cost estimation methods and prediction models have been developed to deal with this challenge. However, there is a need to integrate prediction models as an integral part of LCC methodology to account for possible changes in the model variables.

This article presents an LCC based approach for requirement specification. It integrates degradation models with an LCC model to study the impact of change in design speed on key decision criteria such as track possession time, service life of track system, and LCC. The methodology is applied to an ongoing railway investment project in Sweden to investigate and quantify the impact of design speed change from 250 km/h to 320 km/h.

The results from the studied degradation models show that the correction factor for a change in speed varies between 0,79 and 0,96. Using this correction factor to compensate for changes in design speed, the service life of ballasted track system is estimated to decrease by an average of 15%. Further, the LCC of the route under consideration will increase by an expected value of 30%.

LCC

track system

requirement specification

degradation models

correction factor

Life cycle cost analysis is a method for evaluating the total cost of ownership for a component, system, or complex system over its entire lifespan. The goal of LCC analysis is to identify the most cost-effective option by comparing the total costs of different alternatives over the life of the system. It is a decision-making tool commonly used in engineering, construction, and procurement to evaluate the cost-effectiveness of different design and technical solutions. It provides a comprehensive and objective way to evaluate and analyse the total cost of a system and support engineers or other asset managers in making effective decision from a cradle-to-grave viewpoint.

By considering the total cost which includes the costs of operating, maintaining, and disposing a system and not only the initial acquisition cost, LCC analysis allows engineers to make more informed decisions about the long-term economic feasibility of different design options. Additionally, this systematic approach to decision making can help to identify trade-offs between different designs such as between a more expensive, high-performance product and a less expensive, low-performance product.

The application areas of LCC analysis in railway infrastructure management include: ranking or selection of railway project, requirement specification, design and construction of railway system. It is also applied in selection of technical system, procurement of new system or component, selection of maintenance methods, maintenance intervention cycles, decision between maintenance and renewal, optimal timing of technological shift, decision between improvement and modification, as well as system upgrade or replacement.

Life cycle cost analysis has been used in the railway industry to evaluate the total cost of ownership for various rail infrastructure, rolling stock, and other railway-related systems. For example, LCC analysis has been used to evaluate the cost-effectiveness of different track maintenance and renewal options, for example slab or traditional ballasted tracks. Further, it has been used to evaluate the cost-effectiveness of different rolling stock options, such as diesel or electric trains, and to compare the costs of different signaling and train control systems. Additionally, LCC analysis can also be used to evaluate the sustainability of different railway-related systems and components. This is done by considering the environmental, social, and economic impacts of different alternatives over the entire service life. The method can help to identify options that are more sustainable in terms of their long-term impacts.

A vital advancement in the application of LCC in railway industry is the use of probabilistic and statistical methods such as Monte Carlo simulation which allows systematic incorporation of uncertainty and risk into the LCC analysis (1). This enables decision-makers to evaluate the potential impacts and cost implication of variations in design, operational parameters, or market conditions. The contribution of this paper is the application of deterioration models to estimate correction factor that is included in the presented LCC model. This is aimed at studying the impact of increased design speed on track system. The result of this study is used to support the design specification at an early lifecycle phase of an ongoing railway project in Sweden.

The structure of the paper is as follows, section 1 gives an introduction to the implementation of LCC in the railway industry and vital advancement in the field. Section 2 provides an overview of the three levels of LCC analyses i.e., network, asset, and component levels followed by description of the methodology used in section 3. A case study is presented in section 4 and the result is discussed in section 5 with the concluding section presented afterwards.

There are various LCC-based approaches for assessing economic performance of infrastructure asset considering various factors including complexity, cost bearing object, cost estimate, RAMS modeling, and decision-making level. Various mathematical formulations have been developed and utilized to (see Table 1). Each of these indicators can be classified based on cost aggregation structure consist of LCC at network, asset, and component levels.

Table 1

Modelling levels of LCC analyses.
	Cost Aggregate	Cost atom	Cost element
Complexity	Simple	Moderate	Detailed
Cost bearing object	Asset	Activity	Resource
Cost estimate	Relative	Moderate	Absolute
RAMS modelling	Rarely included	Often and partially included	Always included
Decision	Netowrk/Route design	Asset type design	Component design and maintenance
Formulation	$\sum _{t=1}^{T}\sum _{n=1}^{N}{C}_{t,n}$	$\sum _{t=1}^{T}\sum _{n=1}^{N}{\sum _{a=1}^{A}{C}_{t,n,a}}_{}$ $\sum _{t=1}^{T}\sum _{n=1}^{N}{\sum _{a=1}^{A}{(C.f)}_{t,n,a}}_{}$	$\sum _{t=1}^{T}\sum _{n=1}^{N}\sum _{a=1}^{A}\sum _{r=1}^{R}{\left(C.f.t\right)}_{t,n,a,r}$

2.1 LCC at Network level

In railway structure, the network level encompasses the whole or a large part of a country’s railway infrastructure assets. Various researchers estimate the LCC at the network level. For instance, Patra et al (2008) have assessed the various LCC formulation for the iron ore line (Malmbanan) case study that runs from Luleå in Sweden to Narvik in Norway. They proposed a methodology for estimation of uncertainty in LCC by introducing a hybrid model including a combination of design of experiment and Monte Carlo simulation. Stenström et al. (2016) presented preventive maintenance (PM) and corrective maintenance (CM) actions costs through the analysis of historical data, with the aim of finding the shares of PM and CM actions cost for a specific line of Sweden railway network. Furthermore, a novel framework was presented to assess railway infrastructure LCC in a part of railway line that involves constructing a performance model of the railway (Rama and Andrews, (2016)). The findings of the study indicate that considering the interdependencies among intervention activities has a significant impact on the performance of the infrastructure and its LCC. Kasraei and Ali Zakeri, (2022) in their research, assessed estimated LCC associated with RAMS framework. They employed a stochastic linear degradation model, considering different thresholds to optimize the cost functions including inspection, PM and CM actions cost, and penalty for exceeding the specific threshold. In addition, according to EN 13848-5, they used the alert limit (AL) as an indicator to plan maintenance operations and differentiated between the PM and CM actions to enable better estimation of PM and CM actions and reducing the level of uncertainty.

2.2 LCC at Asset level

The railway track is a heterogeneous system including interdependent assets, i.e., steel rail, sleepers, joints and fasteners, ballast, subgrade, overhead powerlines, and etc. At this level, economic dependency indicates the increase or decrease in maintenance cost when multiple components are maintained simultaneously (Kobbacy et al., (2008)). For instance, Nissen, (2009) proposed a LCC framework which comparing various types of S&Cs. The model can be utilized to identify the main cost drivers and perform sensitivity analysis to find effective parameters that have a significant influence on the result. Furthermore, a study has proposed a new multiple integer linear programing that uses predictive maintenance operation to optimize maintenance activities scheduling of ballast, rail and sleeper renewal (Caetano and Teixeira, (2015)). This model established a connection between the decision-making process for the renewal of these components and their respective conditions. By leveraging the integrated planning of renewal activities, the model aims to minimize the LCC of the railway track. In other research, the different factors, both on a global and local scale that play a role in the decision on the type of track system have been evaluated by Sárik, (2018). Through LCC calculations, he provided recommendations regarding the preferred track system among three alternatives: ballasted track, ballast less track, and an alternating system option. Said Rajab, (2019) conducted an integrated RAMS analysis combined with an LCC analysis to enhance the maintenance optimization of the signaling system. Formulas were developed to allocate costs associated with repairing or restoring asset failures, addressing service losses, and managing overall LCC during both the operation and maintenance phases.

2.3 LCC at Component Level

It is essential for infrastructure manager to have a deep insight regarding the performance of components to handle the demand for spare parts management and procurements, and maintenance planning (Galar et al., (2017)). Degradation of each component is influenced by traffic and operational factors, e.g., train speed, axle loads, accumulative tonnage and performed maintenance. The cost that is consequence of degradation of component level can help decision maker to planed budget and financial program for each component (Galar et al., (2017)).

Calle-Cordón et al., (2018) presented a methodology for combining RAMS in the LCC analysis of linear transport infrastructures. The method has been demonstrated in a railway use case at the component level of S&C. They showed how to obtain cost estimates and cost driver’s dependencies using historical maintenance records, together with individual cost figures, in LCC formulas. Shang et al., (2020) proposed a reliability-LCC model for the embedded rail system at the level crossing to optimize the replacement time and decrease associated uncertainties. Senaratne et al., (2020) investigated the imperative to evaluate the LCC associated with adopting alternative materials for railway transoms. Their research introduces a cost-effective and sustainable material specific to the Australian railway industry, offering an alternative to the widely used traditional timber in railway bridges. Alabdullah et al., (2020) investigated the reduction in railway track cost due to the use of geogrid reinforcement. The result show that the reduced cost of reinforced railway track compared to the unreinforced track is the following: the sub-ballast reinforced model (4.1%), the ballast reinforced model was (6.5%), and the double reinforced model was (3.73%). Sasidharan et al., (2020) introduced a comprehensive LCC approach under different uncertainty to determine the economically optimal maintenance strategy for the substructure of traditional ballasted railway tracks. The results showed that to maximize economic benefits for each route, different maintenance strategies are required. Senaratne et al., (2020) investigated the imperative to evaluate the LCC associated with adopting alternative materials for railway transoms. Their research introduces a cost-effective and sustainable material specific to the Australian railway industry, offering an alternative to the widely used traditional timber in railway bridges Thompson et al., (2022) conducted life cycle analyses of emissions and costs associated with timber, concrete, short-fiber, and long-fiber composite railway sleepers. Their objective was to identify the sleepers that exhibit superior environmental friendliness and cost competitiveness. The findings unmistakably underscore the environmental benefits of short-fiber plastic composites. Camille et al., (2022) examined the economic viability of incorporating macro synthetic fiber reinforced concrete (MSFRC) as an alternative material for railway sleeper applications. Utilizing a LCC approach, the research assesses the influence of various asset phases acquisition, operation and maintenance, and end-of-life on a comprehensive cost comparison between MSFRC and conventional sleeper materials.

This study entails the development of a data-based decision model to support the selection of track system during the design phase of a new railway line. The method used in the study includes literature survey, degradation modelling, Monte-Carlo simulation and LCC modelling.

The first step in the study is to collect available cost data for the track system under consideration. Construction and maintenance costs for ballast system with speed up to 250km/hr was collected from literature and available database. Similar costs for slab track with operating speed up to 320 km/hr were obtained from benchmarking. Thus, these operating conditions became the baseline for both systems.

Furthermore, literature study was conducted to study how change in operating speed can affect the degradation, maintenance need and cost of ballasted track system. Several models were studied to capture the behaviour of ballasted track system under varying operating conditions. Based on relevance, robustness, model formulations, result accuracy as presented in the literature studied, 10 models were selected for detailed study. At the final stage, 1 model was exempted due to very high discrepancy in comparison to other models and poor performance with variation in speed, which is the parameter of interest. A list of the degradation model with simplified model formulation, remarks and literature source is given in Table 2. More information about these models can be found in the references provided.

Table 3

Degradation models [see references 2–20]
Model nr	Degradation Models (References)	Formulation	Comments
1	Sato, (1995); sveld and Esveld, (2001); Lichtberger, (2011)	${N}_{S}=\text{2,04}{10}^{-3}{T}^{\text{0,31}}{V}^{\text{0,98}}{M}^{\text{1,1}}{L}^{\text{0,21}}{P}^{\text{0,26}}$	V-velocity T- Annual Tonnage M-Structural factor L-CWR factor P-Subgrade factor
2	ORE : Doyle, (1980); Esveld and Esveld, (2001); Larsson, (2004); Lichtberger, (2011)	${N}_{ORE}=1+ {\alpha }+\beta +\gamma$	${\alpha }, \beta , \gamma$are dimensionless speed coefficients that depend on track and vehicle parameters
3	Bing and Gross, (1983)	${N}_{BG}=\text{1,25}{\left(\frac{TQI1}{TQI2}\right)}^{-\text{0,58}}{\left(\frac{V1}{V2}\right)}^{-0.18}{\left(\frac{RA1}{RA2}\right)}^{-\text{0,11}}{\left(\frac{BI1}{BI2}\right)}^{\text{1,04}}*{\left(1+FS\right)}^{-\text{0,44}}$	TQI- Track quality indices at time 1&2 V- Velocity at time 1&2 RA- Track age at time 1 and 2 BI- Ballast index at time 1 and 2 FS- Substructure factor
4	Indian formula: Doyle, (1980)	${N}_{IN}=1+\text{0,5}\left(\frac{V}{\text{58,14}k}\right)$	k- Track stiffness
5	South African formula: Doyle, (1980)	${N}_{SA }= 1+\text{4,92}*\left(\frac{V}{D}\right)$	D-Wheel diameter
6	Clarke: Doyle, (1980)	${N}_{CL}=1+\text{0,5}\left(\frac{\text{19,5}V}{D*k}\right)$	k- Track stiffness
7	WMATA formula: Doyle, (1980)	${N}_{WHATA}={\left(1+\text{3,86}{10}^{-5} {V}^{2}\right)}^{\text{0,67}}$	V-Velocity
8	British railway formula: Doyle, (1980)	${N}_{BR}=1+\left(\frac{\text{8,784}\left(a1+a2\right)V}{Ps}\right){\left(\frac{DjPu}{g}\right)}^{\text{0,5}}$	g- Gravity Dj-sleeper pressure Pu Ps- axle load (a1 + a2)-total rail joint dop angle
9	AREA formula: Doyle, (1980)	${N}_{AREA}=1+\text{5,21}\frac{V}{D}$	D- wheel diameter

It should be noted that the main aim of using these models is not to predict the exact evolution of the track system but to estimate a correction factor which will be used to adjust the maintenance and cost of the known baseline scenario.

For the data collection, different sources of data were used to obtain the input values for the LCC model. The default values of the degradation model parameters presented in Table 2 are collected from literature and the technical specification document for the construction of the route. Maintenance and cost data are collected from benchmarking and best practices. Line specification and system description are collected from the ongoing investment project within the Swedish Transport Administration, Trafikverket.

Correction factor is a common method used in engineering design and construction to compensate for changes or uncertainties in design parameters and operating conditions. Correction factor makes it possible to use the results of the different degradation models even though there are uncertainties in predicting the exact degradation value. The correction factor is estimated using the ratio of the predicted deterioration of the track system under 2 different operating speeds i.e., 250 and 320 km/h, see Eq. 1. It should be noted that other parameters remain unchanged in the study. The correction factor is then used to adjust the possible change in maintenance need, service life of track components and costs due to speed change.

$${CF}_{mod}=\frac{Degradation value 250 km/h}{Degradation value 320 km/h} \left(1\right)$$

CF_mod is the correction factor for a given degradation model m.

A range of correction factors was obtained using the degradation models in Table 2. A normal distribution was assumed for the factors and then used for a probabilistic estimation of the values of the unknown LCC parameters under varying operational condition. These parameters include service life and intervention interval of track systems.

In the design of track system and selection of optimal operating parameters, the amount of possession time that will be required for the purpose of maintenance and renewal is a very vital criterion. This is a good indication of the availability performance that a track system can deliver over a given period. A benchmarking of existing maintenance plan of ballasted track with speed up to 250 km/h was carried out to create a realistic maintenance plan for the base scenario. The correction factors were used to adjust the intervention intervals for the different activities for ballasted track system with speed up to 320 km/h. A simplified formulation used to estimate the possession time requirement for the two systems is given in Equations 2 and 3. Note that this does not include the time required for the design, projection, and initial installation of the systems.

$$Possession time=Maintenance Possession Time+Renewal Possession Time \left(2\right)$$

$$Possession time=\sum _{m=1}^{M}\frac{120}{{Interval}_{m}}*{MPT}_{m}+\sum _{r=1}^{R}\frac{120}{{Interval}_{r}}*{RPT}_{r} \left(3\right)$$

Where MPTm and RPTr are maintenance possession time for activity m and renewal possession time for activity r respectively.

For the life cycle cost of the two track designs, the costs of acquisition, operation and maintenance, renewal and disposal of replaced items are summed up over the required period of 120 years. Only significant, distinctive, and future cost elements are included in the model since the purpose of the study is to compare the proposed alternatives and to provide input for decision making. A simplified formulation of the LCC model is given in Eq. 4.

$$LCC=Cost acquisition + Cost operation \& maintenance$$

$$+ Cost renewal and disposal \left(4\right)$$

For calculation without discount rate, LCCs for the alternative designs under consideration are estimated using the formulation below.

$$LCC=CA+\sum _{m=1}^{M}\frac{120}{{Interval}_{m}}*{CM}_{m}+\sum _{r=1}^{R}\frac{120}{{Interval}_{r}}*{CR}_{r} \left(5\right)$$

LCC estimation with discount can be estimated using the formulation in Eq. 6.

$$LCC=CA+\sum _{m=1}^{M}\sum _{n=1}^{\frac{120}{{Interval}_{m}}}\frac{{CM}_{m}}{{(1+i)}^{\left(n*{Interval}_{m}\right)}}+\sum _{r=1}^{R}\sum _{n=1}^{\frac{120}{{Interval}_{r}}}\frac{{CR}_{r}}{{(1+i)}^{\left(n*{Interval}_{r}\right)}} \left(6\right)$$

For ballast system with traffic speed equal to 320 km/h, the interval used in the LCC formulation is adjusted for speed change using the formular below:

$${Interval}_{m\_320}=\frac{{Interval}_{m}}{{CF}_{mod}} \left(7\right)$$

Where CA is the cost of acquisition of the different systems. CM_m and CR_r are the cost of maintenance for intervention m and cost of renewal for activity r respectively. Interval_m and Interval_r are the intervention intervals for maintenance and renewal activities m and r respectively. Interval_{m_320} is the maintenance interval for activity m under an operating speed condition of 320 km/h.

The variation in the intervals of track intervention due to the impact of operational parameters was modelled using Monte Carlo Simulation assuming a normal distribution of the correction factor. 10 000 simulations were carried out to have a reliable estimate of the decision criteria.

Note that a discount rate of zero is used in the LCC calculation due to the long calculation period of 120 years required for technical specifications within the project.

The main aim of the study was to support the specification of track design requirement for an ongoing investment project in Sweden. An overall description of the route is given in Table 3 below. Only the features that are needed for modeling the impact of speed change is given in the table. The description of the route presented below was as it were in June 2022 when this study was carried out.

Table 4

Description of line section
Line section	Alternative 1- Design Track system	Alternative 2- Design Track system	Speed (km/h)	Estimated Trains per day	Estimated annual load (MGT)	Track length (km)
Line 1	Ballasted	Ballasted	250	179	16	177
Line 2	Slab	Ballasted	320	160	15	122
Line 3	Slab	Ballasted	320	100	9	83
Line 4	Ballasted	Ballasted	250	210	15	82,5
Line 5	Slab	Ballasted	320	100	9	207
Line 6	Slab	Ballasted	320	130	11	57

For analysis and comparison, a third option called alternative 3 with ballasted track and speed of 250 km /h on all the lines was drawn up.

Using all the degradation models presented in Table 2, a range of values was obtained for the correction factors. These values were used to adjust parameters in the LCC models whose values were expected to vary with change in speed. Figure 1 presents the values obtained for the different models. The variation in the factor is assumed to be due to several factors including: assumption of the models, parameter values used in the model, number of parameters in the model, application area of the model (heavy haul, mixed traffic), approximation and simplification in the model. However, using several models gives a robust and practical range of the possible variation in the LCC parameters. In a nutshell, the values presented in the figures is a quantification of the impact of track force on maintenance and renewal actions as caused by speed increase in an uneven track. Technically, an increase in track force leads to the following: rail head wear, fatigue of the rail, increased bending stress of the rail foot, increased pressure at sleeper-ballast interface & rail-sleeper interface, higher bearing stress in ballast & subgrade, increased force on fasteners and higher track deformation. Using the distribution of the correction values to compensate for increase in design speed from 250 to 320 km/h, the service life of ballasted track system is expected to decrease by an average of 15%. This means a reduction from 30 years to a range between 23 and 28 years is expected. Similar level of decrease is expected in the service life of some other track components and their maintenance intervals. Note that there is no change in the design speed of any line section with slab track thus a design life of 60 years is used in the analysis. In other words, correction factors were not applied to slab track scenarios. Using the procedure described in section 2 and in Eq. 3, the amount of possession time that will be required for the purpose of maintenance and renewal is estimated and presented in Fig. 2 . This is an indication of the change in availability performance of ballasted track system when the speed is increased from 250 to 320 km/h. An increase of about 18% is expected for the required track possession time. In reality, the outcome can be far above the estimated value in this study if an effective maintenance programme is not adopted. Similarly, the procedure described in section 2 was used to estimate the LCC for the different design alternatives. It should be noted that intervention intervals in the models are treated as probabilistic variables due to uncertainties in the degradation models used to predict the impacts of speed. As a result, a probabilistic approach was used in the LCC estimations, and the results are presented in Fig. 3. The results show that there is a clear difference between the estimated LCC for alternatives 1 and 2. Alternative 1 which has about 67% of the entire route as slab track has a lower LCC than the competing alternative 2 (100% ballast) even though the initial investment is relatively higher. Alternative 1 is estimated to be 30% less expensive than alternative 2 from a lifecycle viewpoint. Also, the variation in the estimated LCC for alternative 1 is lower than the second alternative based on the variations in the input parameters. The results also show that increasing the speed from 250km/h to 320 km/h on a ballasted section would lead to an average of 10% increase in the LCC. The magnitude of the difference in LCC for alternative 2 and comparative alternative 3 should be interpreted carefully due to the overlap of the box plot. This is due to the uncertainty of the impact of speed on LCC parameters such as intervention intervals and service life of track components. More data collection or expert opinion would be needed to confirm if the difference in LCC of these two alternatives is significant. However, there is an indication that increase in speed would lead to higher maintenance needs and if not well handled with appropriate preventive maintenance, cost of ownership could grow exponentially at an undesirable rate. To further investigate the impact of speed on LCC, the speed of lines 2, 3 and 5 were varied for the second design alternative with ballast design. Note that the speed of lines 1 and 4 were unchanged since 250 km/h is the design requirement for these lines and it’s not expected to change. The result is presented in Fig. 4 . The result shows a near-linear logarithmic relationship between speed and LCC, for every 10 km/h rise in speed an increase of about 1 MSEK is expected over the entire life length of the railway infrastructure (120 years). This translates to about 10 000 Sek per km and year. For the case-study under analysis, 10% increase in LCC is expected for a change in speed from 250 to 320km/h. It is worth mentioning that this study assumed relatively good ground formation along the route thus no need for extensive ground preparation and reinforcement for slab track in some areas. In future study, location-specific geotechnical works that is required for construction of both slab and ballast track system along the route will be considered in detail. Other important design and maintenance aspects that can reduce the impact of speed increase on LCC are ballast quality and profile, ground treatment, (differential) settlement requirements, use of under-sleeper pad (USP) and maintenance strategy.

This study has contributed to the advancement of LCC application in the design phase of railway infrastructure where there is inadequacy of data. It addresses the integration of deterioration model and LCC model to study the impact of increasing design speed on important selection criteria for track system. A probabilistic approach has been used to capture the uncertainty in predicting the effect of speed change on LCC parameters such as maintenance interval and service life of track system. In summary, an increase in the operational speed of a ballasted section from 250 to 320 km/h will lead to an average of 15% decrease in service and 18% increase in possession time. From a route perspective, about 30% increase in LCC is expected with a change of the design from alternative 1 to alternative 2. On the other hand, an average of 10% increase in LCC is expected with a change in speed from 250 to 320 km/h on a ballasted track. In conclusion, this study is used by track engineer to motivate the selection of track system and to specify an optimal design requirement for an ongoing investment project in Sweden.

Conflict of interest There are no conflicts of interest declared by any of the authors.

Human participants and/or animals Human Participants and/or Animals are not involved in this research.

Informed consent was obtained from all individual participants included in the study.

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LCC-based approach for design and requirement specification for railway track system

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Abstract

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1. INTRODUCTION

2. STATE OF THE ART

2.1 LCC at Network level

2.2 LCC at Asset level

2.3 LCC at Component Level

3. RESEARCH METHODOLOGY

4. CASE STUDY

5. RESULTS AND DISCUSSIONS

6. CONCLUSIONS

Declarations

References

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Model nr	Degradation Models (References)	Formulation	Comments
1	Sato, (1995); sveld and Esveld, (2001); Lichtberger, (2011)	\({N}_{S}=\text{2,04}{10}^{-3}{T}^{\text{0,31}}{V}^{\text{0,98}}{M}^{\text{1,1}}{L}^{\text{0,21}}{P}^{\text{0,26}}\)	V-velocity T- Annual Tonnage M-Structural factor L-CWR factor P-Subgrade factor
2	ORE : Doyle, (1980); Esveld and Esveld, (2001); Larsson, (2004); Lichtberger, (2011)	\({N}_{ORE}=1+ {\alpha }+\beta +\gamma\)	\({\alpha }, \beta , \gamma\)are dimensionless speed coefficients that depend on track and vehicle parameters
3	Bing and Gross, (1983)	\({N}_{BG}=\text{1,25}{\left(\frac{TQI1}{TQI2}\right)}^{-\text{0,58}}{\left(\frac{V1}{V2}\right)}^{-0.18}{\left(\frac{RA1}{RA2}\right)}^{-\text{0,11}}{\left(\frac{BI1}{BI2}\right)}^{\text{1,04}}*{\left(1+FS\right)}^{-\text{0,44}}\)	TQI- Track quality indices at time 1&2 V- Velocity at time 1&2 RA- Track age at time 1 and 2 BI- Ballast index at time 1 and 2 FS- Substructure factor
4	Indian formula: Doyle, (1980)	\({N}_{IN}=1+\text{0,5}\left(\frac{V}{\text{58,14}k}\right)\)	k- Track stiffness
5	South African formula: Doyle, (1980)	\({N}_{SA }= 1+\text{4,92}*\left(\frac{V}{D}\right)\)	D-Wheel diameter
6	Clarke: Doyle, (1980)	\({N}_{CL}=1+\text{0,5}\left(\frac{\text{19,5}V}{D*k}\right)\)	k- Track stiffness
7	WMATA formula: Doyle, (1980)	\({N}_{WHATA}={\left(1+\text{3,86}{10}^{-5} {V}^{2}\right)}^{\text{0,67}}\)	V-Velocity
8	British railway formula: Doyle, (1980)	\({N}_{BR}=1+\left(\frac{\text{8,784}\left(a1+a2\right)V}{Ps}\right){\left(\frac{DjPu}{g}\right)}^{\text{0,5}}\)	g- Gravity Dj-sleeper pressure Pu Ps- axle load (a1 + a2)-total rail joint dop angle
9	AREA formula: Doyle, (1980)	\({N}_{AREA}=1+\text{5,21}\frac{V}{D}\)	D- wheel diameter