First published as: Zhang, Y. J., Murray, M. H. and Ferreira. L. (2000). Modelling Rail Track Performance: An Integrated Approach. , Transport Journal, 187-194. Awarded the 2001 Webb Prize by the Institution of Engineers UK for the best paper in railway engineering and transportation. MODELLING RAIL TRACK PERFORMANCE: AN INTEGRATED APPROACH By Yu-Jiang Zhang1, Martin Murray2, Luis Ferreira3 Abstract: This paper reports on the work undertaken at the Queensland University of Technology to develop an integrated computer based tool for the prediction of track behaviour under changing traffic conditions. The track degradation model described here takes into account the degradation effects due to the interactions between track components. The model uses mechanistic relationships and embraces all the major factors which may influence service life of track components. The model was applied to a typical Australian railway track and the results are summarised here. The results showed that increasing axle load and train speed accelerates track degradation in general and rail wear in particular. Sub-grade stiffness and ballast depth are found to be the most important parameters for track roughness. Timber sleeper failure is sensitive to both traffic parameters and environmental factors, such as the decay index and track drainage condition. Increases in axle loads may not lead to higher sleeper damage failure rate at fixed traffic loads. Decay index, track drainage condition and average age of sleepers, all have a noticeable effect on sleeper failure. Key Words: modelling, track performance, computer model, track degradation, railway engineering
1
Research Fellow, School of Civil Engineering, Queensland University of Technology, P O Box 2434, Brisbane, Australia 4001.
2
Senior Lecturer, School of Civil Engineering Queensland University of Technology, P O Box 2434, Brisbane, Australia 4001. Tel: +61 7 3864 2513; Fax: +61 7 3864 1515;
[email protected] 3
Associate Professor, School of Civil Engineering, Queensland University of Technology, P O Box 2434, Brisbane, Australia 4001. Tel: +61 7 3864 1542 Fax: +61 7 38641515
[email protected]
1
INTRODUCTION The financial impact of railway track related decisions are felt mainly in train operating performance (transit times, arrival time reliability, and axle loading); and in on-going track maintenance costs required to keep track quality to a specified service level. There are strong linkages between track design standards, allowable train operating performance and track maintenance needs. A trade-off is often present between initial capital cost of construction and maintenance cost needs during the operating life of a track project. When planning for new track or upgrading of existing track, it is important to be able to predict the likely rate of asset degradation as a function of train and time related variables. An integrated set of computer based tools has been developed to address the interactions of the above train and track planning tasks facing a modern rail system. The main relationships between the various components of this integrated approach are shown in Figure 1. This paper describes an integrated track degradation model (ITDM) using mechanistic relationships to predict track behaviour. Unlike existing approaches, the modelling framework takes into account the degradation effects due to the interactions between track components, enabling prediction of either overall track condition or the condition of individual track components, starting from any initial track status. After describing the overall modelling approach, the paper deals with the main sub-models of rail and sleeper degradation. This is followed by a discussion of the way in which the degradation interaction between track components is modelled. The results of an application of the overall model, as well as of a comprehensive sensitivity analysis, are summarised and some oveerall conclusions are drawn. MODEL FRAMEWORK A comprehensive literature review has revealed that there are no track degradation models generally available which can serve as a single tool for analysis of deterioration of each railway track component (Zhang et al. 1997a). Adequate prediction of track degradation needs accurate quantification of in-track behaviour of each component and, more importantly, a good understanding of the interactions between degradation modes. Many factors can affect track degradation and suitable modelling techniques must be employed (Zhang et al. 1997b). A basic methodology in degradation analysis is the statistical approach which involves the analysis of large sample observations of actual track performance and the affecting parameters (Bing and Gross 1983). Correlation, variance, and regression analyses may then be used to develop track degradation models. However, variations in data recording and interpretation may invalidate the results. An alternative is the mechanistic approach which involves establishing, by theory or by testing, the mechanical properties of track components. Track structure analysis models based on these properties are then used to calculate the forces, stresses and likelihood of the development of defects in the components, for individual track components. The advantage of this approach is that the response of track to traffic parameters can be incorporated, though the response of some track components is difficult to quantify. The ITDM model uses mechanistic relationships wherever possible. It also endeavours to embrace all the major factors which may influence service life of track components. The framework of the model is shown in Figure 2. The model consists of inter-related deterioration sub-models for rail, sleepers, ballast and subgrade. At the starting point, the user needs to enter current track conditions, traffic parameters and the period for analysis. The model works through cycles determined by tonnage and time. RAIL SUB-MODEL Rail degradation has long been determined by wear and fatigue defects. Nowadays, however, the practice of rail grinding removes many defects before they become large enough to be predictable. The interactive relationships between grinding and rail fatigue has yet to be established. For these reasons, the current ITDM version carries out only an analysis for rail wear. In developing wear equations, it is assumed that the wear at rail/wheel contact area is of deformation wear type, following the methodology of Clayton and Steele (1987). The sliding between rail and
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wheels is considered proportional to the angle of attack of the wheel-set to the track, following the study of Ghonem and Kalousek (1984). The equations use in ITDM for calculating rail top and gauge face wear are: for top of the high rail:
whr _ top = 7.61 × 10 −6 k h k l _ hrtop Wh sin ψ
(1)
for top of the low rail: wlr _ top = 9.5 × 10 −6 k h k l _ lrtopWl sin ψ
(2)
and for high rail gauge face: ⎧⎪12.1 × 10 −6 k h k l _ hrgauge C1 Pl sin ψ R ≤ 500 whr _ gauge = ⎨ −6 12 . 1 × 10 sin ψ ( 17 . − 0 . 0014 ) 500 < R < 1200 k k C P R h l _ hrgauge 1 l ⎩⎪
(3)
The effect of material hardness on wear is non-linear (Clayton and Steele 1987, and Mutton et al. 1982). This effect has been taken in to account by extrapolating data from Facilities for Accelerated Service Track (FAST) (Clayton and Steele 1987):
k h = 5105 . e −0.0152 H
(4) Lubrication reduces the wear rate of rails by reducing the coefficient of friction. Laboratory studies by Tyfour et al. (1996) indicated that the coefficient of friction ranges from 0.115 for well lubricated conditions, to 0.497 for dry friction. These results agree well with the simulation results of 0.1-0.56 (Mutton et al. 1982). In practice, however, in Australian heavy haul lines, the friction coefficient is considered to vary from 0.15 (well lubricated) to 0.35 (poorly lubricated) representing a variation in wear performance of 1.6:1. To quantify this effect, a lubrication index is used in ITDM to represent lubrication conditions. As rail lubrication has a different effect on high and low rails, and on rail top and gauge face different relationship for lubrication correction factors have been developed. SLEEPER SUB-MODEL The sleeper sub-model draws from Lamson and Dowdall (1985). Stress conditions in a timber sleeper are correlated with sleeper life, based on the mechanistic analysis of timber sleepers. The asssumption is that each standardised wheel loading cycle causes an equal amount of sleeper damage. Hence, total sleeper replacement in a given section over a given time period is proportional to the total standardised wheel loading cycles, over the same track section and time period.
Since wheel loads are classified into categories, one pass of non-standard wheel load will not equal to one standard loading cycle. The equation used to standardise loading cycles is:
N ieqv
⎛ σ ⎞ = Ni ⎜ i ⎟ ⎝ σ std ⎠
kt
k ≥1
(5)
th
where Nieqv = the equivalent number of standardised loading cycles of i load category; Ni = number of loading th th cycles of i load category; σI = the stress level of i load category (cutting or splitting), kPa; σstd = the standard stress level (cutting or splitting), kPa; and kt = damage intensity factor accounting for sleeper age and the environmental effects. The damage intensity factor is calculated from:
kt
=e
( 0.019 ( Ag t − 1) + 0.087 ( I drn − 1) + 0.0007 I Decay )
= 0.99e
( 0.019 Ag t + 0.087 I drn + 0.0007 I Decay )
3
(6)
Timber sleepers are considered to fail mainly in three modes: spike killing, plate cutting and biological decay. The number of sleepers failed due to plate cutting is assessed against the calculated cutting stress at the edge of the plate. The number of splitting sleepers is assessed against the splitting stress at the spike holes, which is a combination of the compressive stress in the fibre direction due to spike pressure and sleeper bending. An age factor is incorporated to account for the weakening of timber sleepers due environmental decay. To quantify the effect of biological factors in Australia, a decay index map has been developed using the methodology developed by Russell (1986). Since the sleeper sub-model relates sleeper degradation with applied loads, analysis of sleeper degradation is independent of historical data and length of track section selected. Furthermore, effects of track quality on sleeper failure can be readily incorporated by varying the applied loads. This is particularly useful in an integrated track degradation model, where the interrelationships of failure modes of each component can be quantified. OTHER SUB-MODELS The ballast and sub-grade components is based on the work of Chrismer (1994). Track modulus is a key parameter in predicting track behaviour under passing traffic affecting calculations of track deflections, rail bending stresses, bearing stresses in track components, and the response of track to dynamic loading from trains. ITDM uses a method proposed by Cai et al. (1994) for estimating static modulus using elastic foundation models, taking into consideration sleeper bending rigidity and elastic properties of layered ballast/sub-grade foundation. The model has been reported in detail in Zhang et al. (1998). INTERACTION BETWEEN DEGRADATION OF COMPONENTS Track degradation is an integrated process in which degradation of one component affects that of the other, as discussed in some examples below. Gauge widening is affected by wear of rails and by failure of fasteners. Through increased dynamic forces, corrugations can accelerate the rail wear process and promote fatigue defects. These forces may also penetrate down to sub-layers of the track, exacerbating damage to sleepers and ballast.
Deterioration of timber sleepers will lead to loss of support and gauge. A single defective sleeper may not cause any noticeable effects. However, several adjacent defective sleepers will affect the degradation of other components and the track as a whole. Both concrete and steel sleepers provide very good gauge holding unless they crack or break due to inadequate support or rail irregularities. It is believed that deterioration of sleepers will also increase track roughness, though the relationship has not been established. The interactions between the various types of deterioration of track components are incorporated into ITDM. As illustrated by the feedback loop in Figure 2, the effect of deterioration of one component on that of the others is reflected by changes in dynamic forces on the rails. Furthermore, because the ITDM model simulates track deterioration in a cyclic manner, it is possible to incorporate the compounding effect of uncertainties in the track deterioration process. The uncertainties, which are expected to increase with time, are mainly due to modelling specification errors and to measurement errors in the input parameters. MODEL APPLICATION General The model has been applied to a section of Australian track on Queensland Rail’s heavy haul system which has concrete sleepers and 60 kg/m rails. Figure 3 shows that, under 18.25 tonne axle load, there is no risk of sleepers being cracked on this line. When the axle load is increased to 22.5 tonnes, a certain percentage of sleepers become prone to cracking. This percentage should not be interpreted as the percentage of sleepers that will crack, but rather the probability that an individual sleeper will receive a load which may cause it to crack. The saw-tooth shape of the curve for 22.5 tonne axle load reflects the probability of concrete sleeper damage due to steadily worsening track roughness, regularly improved by tamping.
As shown in Figure 4, increasing axle load from 18.25 to 22.5 tonnes is not predicted to affect tamping cycles, although the track deteriorates slightly faster under the higher load. However, if the axle load is increased to 30 tonnes, more frequent tamping is required to maintain the track to the same standard.
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Sensitivity Analysis of ITDM There are many uncertainties in track degradation and track design. Measurement errors in the input parameters may accumulate in the process of degradation modelling leading to uncertainties in model outputs. This section discusses the results of a sensitivity analysis undertaken by changing the main input parameters.
The following nominal conditions were used as the basis for model runs in a 200-MGT period, to obtain the output variances due to input errors: (a) 22 tone axle load; 80 km/hr. train speeds; unit train traffic (b) 40 MGT annual tonnage (c) 60 kg/m rails, standard carbon material; concrete sleepers (d) medium clay sub-grade; 250 mm crushed rock ballast (e) 1067 mm gauge; 500 curve radius; 70 mm superelevation (f) moderate track drainage condition Sensitivity of Rail Wear The model has identified that axle load and train speeds are the major parameters affecting rail wear. Other parameters such as sub-grade stiffness, ballast depth, ballast elastic modulus have very minor effect on rail wear. Therefore, the sensitivity of rail wear was examined for changes in axle load and train speed individually and in combination.
Figure 5 shows the sensitivity of rail wear to variations in nominal axle load. A ±10% variation in axle load will result in about ±4.5% change in high rail wear. There is no change observed in the low rail wear. This indicates that if the user specified axle load has a 10% error, there will be noticeable uncertainties in the predicted high rail wear. For the low rail, the wear is directly proportional to cumulative traffic tonnage if all the other input parameters remain unchanged. Using the same set of conditions with train speed varying ±10%, the predicted rail wear is shown in Figure 6. It can be seen that variation in train speed will result in changes in both high and low rail wear. A +10% variation in train speed leads to 6.7% increase in high rail total wear and -4.6% in low rail wear. A -10% variation in train speed results in -5.9% decrease in high rail wear and 4.1% increase in low rail wear. This indicates that high rail wear is more sensitive to train speed than the low rail wear component. The effects of the combination of errors in axle load and train speeds are examined for the following four scenarios: (1) upper limit of axle load and upper limit of train speed; (2) upper limit of axle load and lower limit of train speed; (3) lower limit of axle load and upper limit of train speed; and (4) lower limit of axle load and lower limit of train speed. The result for the four scenarios showed that the worst case for high rail is scenario 1 (the combination of upper limit of axle load and upper limit of train speed). In this case, a 10% increase in both axle load and train speed results in an average of 17% increase in high rail wear. The worst case for low rail is found to be scenario 2 (the combination of upper limit of axle load and lower limit of train speed). In this case a +10% error in axle load and 10% error in train speed will result in an average of +4.2% variation in the low rail wear. This analysis provides evidence of the compounded effect of errors in inputs on the variation in the outputs. Combination of up-shifting 10% of both axle load and train speed results in an increasing factor of 1.7 times in the high rail wear. The more significant variation in high rail wear than in low rail wear reflects the fact that gauge face wear (contained in high rail wear) is sensitive to errors in axle load and train speed. Sensitivity of Timber Sleeper Failure For timber sleeper failure, analyses were carried out for errors in axle loads, train speeds, sub-grade stiffness, and environmental factors including decay, age, and track drainage conditions. It was found that timber sleeper failure is insensitive to train speeds and sub-grade stiffness, although variations in these two parameters will result in slightly different outputs.
Sensitivity analysis of axle load produced results which seem to be controversial. A +10% increase in axle load will cause timber sleeper to fail at a slower rate and a 10% decrease in axle load will lead to timber sleeper to
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fail at a faster rate (see Figure 8). This can be explained by the fact that to accumulate the same amount of traffic tonnage the number of axle load passes will be smaller. There are two opposing effects: higher axle load causes more damage to sleepers per pass (loading cycle); but a smaller number of axle passes per MGT tends to reduce damage to sleepers. Furthermore, the net result depends not only on axle load but also on other factors. For example, if the track is located in a dry location with a very low decay index, with very good track drainage and with relatively new sleepers, increasing axle load may actually reduce sleeper failure rate at the same traffic tonnage (as in Figure 7). At the other extreme, if the track has a very high decay index, very poor drainage condition and relatively old sleepers, increasing axle load should lead to an increase in sleeper failure rate, for the same total tonnage. This is because the timber sleeper damage intensity factor will vary yielding different impacts of axle load on sleeper failure. Non-traffic related parameters are found to have significant effect on timber sleeper failure rates. These parameters include decay index, sleeper age and track drainage condition. For the model to predict sleeper failure with the influence of sleeper age an average sleeper age is required as the input. A more comprehensive way to account the effects of sleeper would be to use a “distributed” sleeper age. Because sleeper replacement policies varies from one railway system to another, it is impossible to develop a universal sleeper age distribution applicable to all railway systems. For example, the study of Worth et al. (1997) indicates some railway systems don not replace failed sleepers unless the cluster of failed sleepers is considered unacceptable. Track Roughness The analyses showed that small variations in axle load and train speed do not have significant effects on track roughness. The track modulus sub-model has revealed that sub-grade and ballast are primary factors affecting track modulus. Because the latter is a major factor influencing track settlement and hence track roughness, sub-grade and ballast are also significant in influencing track roughness.
The model showed that a possible variation of -20% in sub-grade stiffness will result in a faster track roughness deterioration without changing the tamping cycles. On the other hand, a +20% variation in sub-grade stiffness will lead to less frequent tamping. A 10% variation in ballast depth causes about 5% variation in track roughness within one tamping cycle. Figure 8 shows that either reducing or increasing ballast depth will cause changes in tamping frequency. CONCLUSIONS An integrated track degradation model for analysis of track degradation has been reported here. The model can deal with the entire track system or with individual components and has been designed to serve as a single tool for analysis of deterioration. It enables comprehensive and reliable prediction of track degradation through the following: account for a wide range of factors; accurate quantification of in-track behaviour of each track component; estimation of errors in the modelling process and input parameters; and especially incorporation of the interrelationships between degradation modes. Mechanistic relationships have been employed in the model, allowing new technology and new research results to be incorporated at later stages.
The model application revealed that increasing axle load and train speeds can result in concrete sleepers being exposed to the risk of cracking. Track roughness will also be affected significantly by heavier axle load and higher trains speed, with more frequent tamping being necessary to keep the track to the same standard. Sensitivity analysis of the model revealed that: (i)
Axle load and train speeds are the major factors in rail wear. They affect more high rail gauge face wear than rail top wear. The results reflect the reality that high rail gauge experiences the most wear. Axle load and train speeds do not have noticeable influence on sleeper damage and on ballast and sub-grade deterioration. They do not cause much variation to the degradation of rail track sub-layers.
(ii)
Sub-grade stiffness and ballast depth are found to be the most important parameters for track roughness. Measurement errors in sub-grade stiffness can lead to significant variations in the predicted track roughness.
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(iii)
Timber sleeper failure is sensitive to both traffic parameters and environmental factors, such as decay index and track drainage condition. Increases in axle loads may not lead to higher sleeper damage failure rate at fixed traffic tonnages because under heavier axle loads there will be less wheel passes to accumulate the same amount of traffic tonnage as under lighter axle loads. Decay index, track drainage condition and average age of sleeper all have a noticeable effect on sleeper failure. Sleeper damage is a result of interaction between traffic parameters and environmental factors.
(iv)
Concrete sleepers are safe under certain traffic conditions with track quality regularly being restored by tamping. However, if the traffic loading exceeds a certain limit a number of concrete sleepers will become prone to cracking.
ACKNOWLEDGEMENT The research work presented in this paper is funded by the Australian Research Council and Queensland Rail through the Queensland University of Technology. The views expressed in the paper remain those of the authors.
APPENDIX I. REFERENCES Bing, A.J., and Gross, A. (1983). “Development of railroad track degradation models.” Transportation Research Record 939, TRB, Washington D.C.. Cai, Z., Raymond, G. P., and Bathurst, R. J. (1994). “Estimate of static track modulus using elastic foundation models.” Transportation Research Record 1470, TRB, Washington D. C., 65-72. Chrismer, S. M. (1994). “Mechanistic-based model to predict ballast-related maintenance timing and cost.” PhD Thesis, Department of Civil and Environmental Engineering, University of Massachusetts, Amherst. Clayton, P., and Steele, R. K. (1987). “Wear processes at the wheel/rail interface.” AAR Report No. R-613, AAR Technical Center, Chicago, Illinois. Ghonem, H., and Kalousek, J. (1984). “A quantitative model to estimate rail surface failure.” Wear, 97, 65-81. Lamson, S. T., and Dowdall, B. “Tie life model.” CIGGT Report No. 84-12, Canadian Institute of Guided Transport, Queen’s University at Kingston, Ontario. Mutton, P.J., Epp, J., and Marich, S. (1982). “Rail assessment.” Proceedings of Second International Heavy Haul Railway Conference, Colorado Springs, Colorado, 330-338. Selig, E. T., and Li, D. (1994). “Track modulus: its meaning and factors influencing it.” Transportation Research Record 1470, TRB, Washington D. C., 47-54. Tyfour, W. R., Beynon, J. H., and Kapoor, A. (1996). “Deterioration of rolling fatigue life of pearlitic rail steel due to dry-wet rolling-sliding line contact.” Wear, 197, 255-265. Zhang, Y. J., Murray, M. H., and Ferreira, L. (1997a). “Track degradation prediction: criteria, methodology and models.” Proceedings, Australasian Transport Research Forum, University of South Australia, Adelaide, 21(1), 391-405. Zhang, Y. J., Murray, M. H. and Ferreira, L. (1997a). “Railway track performance models: degradation of track structure.” Road and Transport Research, Melbourne, Vic., 6(2), 4-19. Zhang, Y. J., Murray, M. H. and Ferreira, L. (1998). “A mechanistic approach for estimation of track modulus”. Conference on Railway Engineering (CORE98), 9-14. Central Queensland University, Rockhampton, Queensland.
APPENDIX II. NOTATION C1 = H = kh kl_hrtop kl_hrgauge kl_lrtop kt Nieqv =
constant accounting for the wheel profile and flange angle (3.5 - 4.4) hardness of rail material (BH) = rail material hardness correction factor; = lubrication correction factor for high rail top; = lubrication correction factor for high rail gauge; and = lubrication correction factor for low rail top = timber sleeper damage intensity factor the equivalent number of standardised loading cycles of ith load category;
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Ni Pl
= =
number of load cycles of ith load category; lateral wheel load on high rail, kN;
ψ R SL
= = = = = =
angle of attack, rad; curve radius, m. average track settlement resulting from sum of settlement of all sub-layers (mm). standard deviation (roughness of track profile in term of vertical offsets) (mm); standard deviation of track top line just after resurfacing (mm); and the stress level of ith load category (cutting or splitting), kPa;
σvo σ(vo)min
σi
σstd Wh Wl whr_top wlr_top
whr_gauge =
= the standard stress level (cutting or splitting), kPa; and = wheel load on high rail, kN; = wheel load on low rail, kN; = high rail gauge wear, mm2 per wheel pass; = high rail gauge wear, mm2 per wheel pass; high rail gauge wear, mm2 per wheel pass.
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Figure 1. Track Degradation & Maintenance Planning Tool
Manag. Info. & Control Systems Rail Customer Needs
Service Plans
Operating Plans •service standards •resource utilisation
Train Operations Track •design •maintenance
•speeds •axle loads •line capacity •train reliability •train control •loco/crew scheduling •train scheduling
Completed
•customer info. •Asset management •Costing/Pricing Systems
Rollingstock •standards •utilisation
Terminals
Proposed Project
Vertical separation track/ops.
Rail Research at QUT
Track Degradation & Maintenance Planning Tool
Train scheduling Track degradation model Track Maintenance scheduling model
Sleeper replacement model 1
Maint. planning model
current values of: track & track condition
Yes
START Update input
track deterioration warrants change in track conditions ?
maintenance happened ?
no
user inputs: initial track data, track condition data, traffic data, and period of analysis (MGT)
yes
no Structure Analysis Module: Dynamic forces & Stresses etc.
dynamic force due to defects & corrugations
dynamic force due to track roughness
Next increment in traffic MGT
Sleeper Submodel
Rail Submodel Rail Wear Rate; Fatigue Defects
Corrugations Probability
Timber Sleeper Con. Sleeper Failure Rate Cracking Rate
Ballast & Subgrade Submodel Ballast Settlement
Subgrade Settlement
change in track roughness due to settlement and sleeper deterioration
Yes
Total MGT increments < MGT period required ?
no
2
Display of outputs
1 22.5 tonne axle load
Sleepers prone to crack, %
0.8 0.6 0.4 0.2 0
18.25 tonne axle load
-0.2 -0.4 0
100
200
300
Accumulating Trffic, MGT
3
400
500
14
Track roughness, mm
12 10 8 6 4 18.25 tonne axle load 22.5 tonne axle load 30 tonne axle load
2 0 0
50
100
150
200 250 300 Accumulating traffic, MGT
4
350
400
450
500
600 22+10% tonne, HR 22 tonne, HR
500
22-10% tonne, HR 22, 22+10% and 22-10% tonne, LR
Rail Wear, mm
2
400
300
200
100
0 0
50
100 Cumulative MGT
5
150
200
600
80+10% kph, HR 500
80 kph 80-10% kph, HR 80-10% kph, LR
Rail Wear, mm2
400
80 kph, LR 80+10% kph, LR
300
200
100
0 0
50
100 Cum ulative MGT
6
150
200
12.00
10.00
Nominal axle load
Sleeper Failure, %
Axle load -10% Axle load +10%
8.00
6.00
4.00
2.00
0.00 0
50
100
Cum ulating MGT
7
150
200
14
12
Track Roughness, mm
10
8
6
4
Ballast Depth -10% Nominal Ballast Depth Ballast Depth +10%
2
0 0
50
100 Cum ulative MGT
8
150
200
Figure 1. Track Degradation and Maintenance Planning Tools Figure 2. Framework of Integrated Track Degradation Model (ITDM) Figure 3. Effect of Increased Axle Load on Probability of Sleeper Cracking Figure 4. Effect of Axle Load on Tamping Demands Figure 5. Sensitivity of Rail Wear to Axle Load Figure 6. Sensitivity of Rail Wear to Train Speed Figure 7. Sensitivity of Timber Sleeper Failure to Axle Load Figure 8. Sensitivity of Track Roughness to Ballast Depth
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