The effect of rolling stock characteristics on differential railway track settlement: An engineering-economic model

Transportation Geotechnics 37 (2022) 100845 Available online 24 August 2022 2214-3912/Crown Copyright © 2022 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). The effect of rolling stock characteristics on differential railway track settlement: An engineering-economic model C. Charoenwong a , D.P. Connolly a , K. Odolinski b , f , P. Alves Costa c , P. Galvín d , e , A. Smith b a Institute for High Speed Rail and Systems Integration, School of Civil Engineering, University of Leeds, UK b Institute for Transport Studies, University of Leeds, UK c Faculty of Engineering, University of Porto, Portugal d Escuela T´ ecnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain e Laboratory of Engineering for Energy and Environmental Sustainability, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain f The Swedish National Road and Transport Research Institute (VTI), Malvinas v¨ ag 6, Box 55685, SE-102 15, Stockholm, Sweden ARTICLE INFO Keywords: Railway track geometry Railroad track tamping Marginal railway cost Freight heavy haul Rolling stock linespeed Railway operation economics ABSTRACT Railway track geometry deteriorates under repeated train loading. When linespeed is increased or new rollingstock is introduced this can alter the future rate of change of differential settlement and track geometry. Therefore this paper presents a novel combined engineering-economic approach to investigate the effect of increasing train speeds, adding additional passenger movements, and adding additional freight movements to an existing line. Firstly a numerical algorithm is presented to compute differential track settlement. An important novelty of the model is its use of the wavenumber finite element method coupled with settlement relationships in a manner that allows for track irregularities to evolve after every load passage (i.e. taking into account the evolution of the track unevenness profile before applying each subsequent train passage). Unlike traditional approaches this allows the model to faithfully simulate mixed traffic conditions, including the coupled interactions between different rolling stock types and track geometry. The engineering model is used to predict tamping intervals, and then coupled with an economic model capable of calculating deterioration elasticities and marginal costs. It is shown that higher speeds result in higher dynamic forces and cause a faster rate of deterioration of track geometry, thus increasing marginal cost. The model is then used to investigate the effect of adding additional train movements to a passenger line. It is shown that additional movements increase the rate of track degradation and marginal costs, particularly if the additional traffic is freight. This is because freight vehicles typically have one only layer of (stiff) suspension, thus generating elevated dynamic forces compared to passenger vehicles. Introduction Under repeated train passages, railway tracks settle differentially along the line due to dynamic train-track interaction and changing track support conditions [1]. Deterioration in vertical track geometry results in a cycle of increased train-track dynamic interaction forces and further track deterioration. These track irregularities evolve with each load passage, meaning the characteristics of the train-track dynamic interaction forces, track stress distributions and the settlements also evolve over time. If each track section is subject to relatively constant operational conditions (i.e. no significant changes to rolling stock types, speeds and timetables), and no heavy maintenance/renewal is performed, then future track geometry changes can be approximated by extrapolating from historical data, with minimal regard for the underlying physics of track degradation [2–7]. This means variables such as changes in traintrack interaction and track stress distributions can be ignored. Although useful for routine cases, if a railway operator wishes to increase linespeed, or increase the annual number of rolling stock movements (either freight or passenger), maintenance frequency prediction using statistical extrapolation of historical records becomes more difficult because experimental data to perform forecasting analysis is not available. This then introduces uncertainties into marginal cost calculations, and thus the operator’s economic appraisal. An important driver behind operational cost is track maintenance frequency. To predict future maintenance intervals requires modelling of operational conditions that are different than the past. Rather than a statistical approach, this task is well suited to numerical modelling from first principles, where arbitrary future operational conditions can be simulated without relying on past track geometry records. This approach has been studied by [8–11], where an iterative train-passage simulation approach combined with settlement models is used to compute differential settlement, considering changes Contents lists available at ScienceDirect Transportation Geotechnics journal homepage: www.elsevier.com/locate/trgeo https://doi.org/10.1016/j.trgeo.2022.100845 Received 23 May 2022; Received in revised form 16 August 2022; Accepted 18 August 2022 Transportation Geotechnics 37 (2022) 100845 2 in track geometry. However, modelling cyclic loading in the time domain demands significant computational effort, and is thus challenging when calculating the 3D dynamic stress fields in the track and ground over a large number of loading cycles [12–16]. It is also important that the distribution of track-ground stresses is modelled explicitly because the deviatoric stress is one of the most influential parameters when computing settlement [17]. Two fundamental parts of a settlement prediction model are first calculating the stress/strain response of the structure, and secondly calculating the corresponding settlement. Regarding settlement calculation, two common modelling approaches for calculation are constitutive [18–23] and empirical [17,24–32]. The constitutive approach often requires higher computational effort and a non-trivial number of material input properties that are difficult to quantify, thus making reallife application challenging [33,34]. An alternative modelling approach is to use empirical equations which have fewer input parameters and require minimal computation, yet if used with care, can provide similar accuracy to constitutive models [35]. Most railway administrations use a track quality index (TQI) to measure track quality. Although many TQI variants exist [36], the standard deviation (SD) of vertical track geometry over a given track length is the most commonly used [37]. When the SD value, for wavelengths within a given range, exceeds a threshold limit (which is linked to linespeed), maintenance action is required. Tamping is then the most common corrective maintenance technique to restore the track geometry to an acceptable level of quality. The settlement of railway trackbed is non-linear [35], and thus changes in TQI are also often non-linear. Thus, to analyse changes in different settlement and TQI requires a numerical model that is capable of simulating daily rolling stock patterns, accounting for interactions between the different traffic types that operate in the railway infrastructure. For example, if a solely passenger line was upgraded to carry multiple freight train movements, the track resulting geometry would likely be different depending upon whether these freight trains were all run consecutively, or interspersed between passenger movements. Therefore, to most faithfully replicate changes in TQI, the traffic timetable should closely replicate true operation conditions, and vertical track profile should be updated after every axle passage [38]. To determine the relationship between TQI and tamping frequency, and thus economic cost, marginal cost analysis [39] is a useful tool. It is an important area academically and in practice given the vertical separation between infrastructure management and train operations, requiring track access charges to be set. For example, EU legislation (Directive 2012/34/EU, and the European Commission’s Implementing Regulation EU 2015/909) stipulates that those charges should be based on direct costs of a train service, which can contribute to an efficient use of the infrastructure. To address the above issues, this paper uses an engineering-economic approach to calculate marginal costs considering the evolution of railway track geometry. To do so, first a novel numerical algorithm is presented to compute differential track settlement. It uses an equivalentlinear wavenumber finite element method coupled with empirical settlement relationships in a manner that allows for the track irregularities to evolve after every load passage, before applying the next load. Compared to some traditional approaches this allows the model to faithfully simulate mixed traffic conditions, and the coupled interactions between different rolling stock types and track geometry. The model is used to calculate marginal costs of two scenarios: 1) increasing the linespeed on a passenger line, and 2) increasing the number of passenger or freight movements on a predominantly passenger line. Engineering-economic modelling Model architecture A combination of engineering and economic modelling is used to predict tamping intervals and marginal costs as outlined in Fig. 1. To do so, first the engineering model is used to calculate tamping frequencies, which are then passed to the economic model which computes the related marginal costs. The engineering model is divided into two coupled steps: Step A, pre-calculation and Step B, iterative process, solved across both frequency-wavenumber and time–space domains. Step B updates the track geometry profile after every train passage, considering the evolution of deviatoric stresses in the track and ground, thus allowing for the simulation of bespoke train timetables that accommodate both passenger and freight traffic. Therefore it is capable of simulating changes to linespeed and passenger-freight traffic ratios. Based on an initial track geometry profile and a threshold intervention limit, the time-to-maintenance is calculated, as an input for marginal cost calculation. Engineering model Track and settlement model The engineering sub-model uses a FEM-PML approach developed in Matlab, where the 3D stress fields are coupled with empirical settlement laws to estimate track settlement and, subsequently, future tamping intervals. The model is solved in a hybrid manner across frequencywavenumber and time–space domains. This approach makes it wellsuited for modelling repeated dynamic loads, because the track geometry profile can be updated after every axle passage with minimal computational effort. This is particularly important for operational scenarios where different types of rolling stock are operated in sequence on a daily timetable. As shown in Fig. 1, Step A is a pre-calculation step which involves computing the 3D elastodynamic response and the geo-static stresses in the track and ground. The moving load transfer function considering non-linear track-ground stiffness is computed in the frequencywavenumber domain. An equivalent linear approach is used to assess non-linear behaviour in the track and the ground in the frequency domain [40–43]. The effective octahedral shear strain is used to update the properties of each track-ground element until reaching convergence between the material properties and strain-adjusted properties. This non-linear calculation is performed during Step A and the strainadjusted material properties are passed to Step B for settlement calculation. For low/medium track geometry SD’s, the total stress field is dominated by quasi-static stresses, meaning the majority of nonlinearity can be assumed to occur in Step A. The moving load transfer functions are then used to calculate the 3D stress transfer functions due to quasi-static and dynamic loading. Several matrices required for computing the train-track dynamic interaction are also prepared in advance. Step A is only computed once for each moving speed of vehicle and track design being considered. All variables from this part can be pre-calculated and used repeatedly in Step B. In Step B, an iterative solver is implemented using a combination of wavenumber-frequency and space–time domains. Based on the track irregularity profile, track compliance and rolling stock, the train-track dynamic interaction force is calculated using a multi-body model. Considering the stress field in 3D, the quasi-static, geostatic, and dynamic stresses are all calculated at the track centreline and used to compute the deviatoric stress. The total deviatoric stress (combined quasi-static, dynamic and geo-static) is then used to calculate settlements in the track and the ground over the entire model track length (typically ≈200 m), in the direction of train passage. The vertical track geometry profile is updated after every axle passage and thus the traintrack dynamic force and deviatoric stress are also recalculated at each iteration. These steps are repeated until a threshold limit value is reached, based upon the SD of the vertical rail profile, as commonly defined in standards. It should be noted that due to the pre-calculations performed in Step A, each Step B iteration requires minimal computational effort, thus allowing for the rapid simulation of many axle passages. To repeatedly update the track profile and account for the C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 3 settlement of previous axle passages, an incremental form of settlement equation is required. Therefore the ballast [25] and subgrade [17] settlements are calculated using Eqs. (1) and (2). Additional information regarding the empirical settlement laws can be found in [38]. Δ ε p s,i=a 100 ( σ d s,i σ s)m[((dN⋅i) + Nls )b− ((dN⋅(i−1)) + Nls )b](2) where Δ ε p b,i is ballast permanent strain increment; Δ ε p s,i is subgrade permanent strain increment; i is iterative step; σ d b,i is ballast dynamic deviatoric stress relevant to traffic load (in MPa); σ d s,i is subgrade dynamic deviatoric stress relevant to traffic load (in Pa); σ s is soil compressive strength (in Pa); a, m, and b are material parameters; Nlb is the number of load cycles after the last ballast renewal/tamping; Nls is the number of load cycles after the last subgrade replacement; and dN is the frequency of load application. Note that Eqs. (1) and (2) can be used for single or mixed traffic scenarios. The deviatoric stresses are computed for any arbitrary vehicle and then used to update the settlement after each cycle, considering the previous iteration. Therefore they are compatible with an arbitrary combination of traffic. Train-track dynamic interaction model To simulate railway vehicle dynamics, a variety of vehicle modelling approaches, each with different levels of complexity have been proposed in the literature [44], often with the goal of minimising computational requirements for their application. This goal is particularly relevant when simulating large numbers of axle passages for the purpose of differential settlement analysis. Therefore to investigate the validity of simplifying rolling stock models for the purpose of settlement calculations, a variety of common simplifications are compared. Three approaches for passenger vehicle modelling are analysed, followed by two approaches for freight. Their dynamic responses are compared because this is dominant in differential track settlement. Firstly, to simulate the interaction between the vehicle and the track, a compliance procedure formulated in a moving frame of reference, subjected to a moving train is used [45,46]. Three common simplifications for modelling a passenger train are illustrated in Fig. 2. The first option, as shown in Fig. 2(a), is a complete vehicle model which takes into account the main structural aspects of the train dynamics. In contrast, Fig. 2(b) ignores the car box, instead only considering the dynamic motion of the bogies (semi-sprung masses) and the wheelsets (unsprung masses). Further, Fig. 2(c) only considers the presence of the wheelsets (unsprung masses) for dynamic computation. Fig. 1. Engineering-economic model overview. Δ ε p b,i=0.375( σ d b,i)2× [(1+0.4log10((dN⋅i) + Nlb ) ) − (1+0.4log10((dN⋅(i−1)) + Nlb ) ) ] (1) C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 4 Freight vehicles typically contain only one level of suspension, meaning only two modelling options are considered. The first option shown in Fig. 3(a) is a complete vehicle model with one level of suspension. The simplified option shown in Fig. 3(b) is where only the bogies and the wheelsets are considered. The formulation of the dynamic interaction force in the frequency domain is: {Fdyn(Ω)}= − ([Vc] + [VH]+ [Tc])−1{Δu(Ω) } (3) where Δu(Ω) = the vertical track irregularity; Tc is the flexibility the track compliance; Vc is the flexibility of the vehicle compliance; and VH Fig. 2. Passenger vehicle modelling approaches: (a) complete vehicle model, (b) simplified model containing bogies and wheelsets, (c) simplified model containing only wheelsets. C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 5 is the contact flexibility. The details of the track compliance and the vehicle compliance, including the mass and stiffness matrices of both passenger and freight vehicles are given in Appendix A. A simplified 2D model based on beam on elastic foundation theory [47] is used to calculate the response due to train-track dynamic interaction. An infinite Euler-Bernoulli beam is used to represent the rail which is supported by a single continuous elastic layer. It has the following material properties (single rail): Young’s modulus E=2.1× 1011 N/m2; second moment of area I=30.55 ×10−6m4; cross section area A=0.00763m2; density ρ =7850 kg/m3; and support stiffness s=100 ×106N/m2. An artificial track irregularity profile for wavelengths in the range 3 <λ≤35m is generated using the PSD (Power Spectral Density) function defined by FRA [48] but the constants modified to generate specific SD profiles at higher speeds (>177 km/h). The properties of passenger and freight vehicles are summarised in Table 1, where the properties of passenger vehicle are taken from [49] and freight vehicle adapted from [50]. The speeds of passenger and freight vehicles are 200 and 97 km/h respectively. Fig. 4 compares the peak displacements due to dynamic loading along the track length, for the complete vehicle and simplified vehicle models. As shown in Fig. 4(a), the displacements due to the different vehicle assumptions show significant differences, however are in a similar range in terms of magnitude. In contrast, the freight vehicle shows a large discrepancy in magnitude, comparing the full and simplified vehicles - Fig. 4(b). This large difference is justified by the higher stiffness of the primary suspension and suspensions. Considering the errors associated with using simplified passenger and freight vehicles, for the purpose of differential settlement modelling, this shows it is important to use full vehicle models (i.e. avoid reducing the number of degrees of freedom). Quasi-static versus dynamic excitation Rolling stock moving at constant speed on plain line induces both quasi-static and dynamic excitation. The track’s quasi-static deflections, stresses and strains are not influenced by track irregularities meaning they move uniformly with the load [13]. Considering the well established relationships between stress and settlement, such loading therefore also induces uniform settlement along the track. In contrast, dynamic excitation is influenced by track irregularities and is thus the source of differential settlement evolution. To illustrate this concept more clearly, Fig. 5 shows the quasi-static and dynamic deviatoric stresses measured on the track centreline, 0.1 m below the sleeper, for an entire 200 m track length. Excitation is from the full freight train model interacting with a track irregularity of FRA’s Fig. 3. Freight vehicle modelling approaches: (a) complete vehicle model, (b) simplified model of bogies and wheelsets. Table 1 Parameters of passenger and freight vehicles. Parameter Passenger Freight Number of cars 11 40 Number of axles 44 160 Axle spacing (m) 2.9 1.7 Bogie spacing (m) 19 9.7 Car body mass (kg) 329 ×10 2 864 ×10 2 Car body pitching moment of inertia (kg.m 2 ) 208 ×10 4 102 ×10 4 Bogie mass (kg) 4932 2800 Wheelset mass (kg) 1538 2000 Bogie pitching moment of inertia (kg.m 2 ) 5150 2020 Primary suspension stiffness (kNm −1 ) 3420 – Primary suspension viscous damping (Nsm −1 ) 360 ×10 2 – Secondary suspension stiffness (kNm −1 ) 1320 2660 Secondary suspension viscous damping (Nsm −1 ) 360 ×10 2 25 ×10 2 C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 6 class 4 (suitable for maximum speed at 130 km/h). The corresponding ballast settlement is shown in Fig. 6. It is seen that the quasi-static excitation induces uniform settlement along the track, whereas the dynamic excitation induces non-uniform settlement along the track. This illustrates the concept that dynamic excitation is the primary driver of differential railway track settlement. Differential settlement model validation The ability of the model to predict the evolution of differential settlement with increasing axle passages is validated by comparing against historical track geometry data from a UK track section. The standard deviation of vertical track geometry profile over a 200 m track length is analysed, considering wavelengths in the 3–35 m range. Fig. 7 shows the finite element mesh and dimensions of the track cross-section. The material properties of the rails, rail pads, sleepers, ballast, sub-ballast and subgrade are described in Appendix B. The subgrade properties are based upon site investigation data. The line operates almost exclusively with passenger rolling stock, and thus the vehicle properties are solely based upon the passenger vehicle shown in Table 2. The traffic volume per year is 19.58 million gross tonnes (MGT) with an operational line speed of 125mph. To facilitate validation, track geometry obtained via a Track Recording Vehicle (TRV/TRC) on 5 dates: 10-November-2014, 23-March-2015, 07-March-2016, 20-March-2017 Fig. 4. Maximum rail deflection along track (dynamic excitation only): (a) passenger vehicle models (b) freight vehicle models. Fig. 5. Track-bed deviatoric stress due to quasi-static and dynamic excitation. C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 7 and 19-March-2018, is used. Analysis of maintenance records shows that tamping was not performed between any of these dates. The geometry data recorded on 10-Nov-2014 is used as the track starting condition. Then the model individually simulates every axle load passage until 19-March-2018. The geometry SD evolution is compared against the TRV data, as shown in Fig. 8. The rectangular markers represent the geometry SD data recorded in-situ, and the triangle marker is the SD of the initial 2014 track profile. The dashed line represents the predicted geometry SD, evolving after every axle passage. The strong correlation between the predicted geometry SD curve and the real data confirm the model is capable of predicting the evolution of differential track settlement. Economic model The engineering model provides the inputs for the economic model. The generalised, coupled modelling approach for direct cost calculation is: (a) Simulate the damage from increasing the linespeed or running more trains using the engineering model (b) Determine the timeframe until damage correction is required Fig. 6. Track-bed settlement due to a single axle passage. Quasi-static and dynamic components. Fig. 7. Finite element mesh of the track model for validation. C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 8 (c) Calculate a unit cost of damage correction to give additional cost of increased linespeed or extra traffic on the network The cost calculations only consider changes in tamping intervals and do not include investments in the capability of the line, i.e. fixed inputs in the production of infrastructure services. The calculated costs therefore comprise short-run marginal costs rather than long-run marginal costs. This makes the estimates consistent with the concept of direct costs for the purposes of setting track access charges. Assuming tamping is the primary form of maintenance used to improve the SD of track, the present value cost (PVC) of an infinite time series of tamping activities at time T with constant intervals ¯ T is: lim n→∞PVC =c1 (1−e−r ¯ T)(4) where c is the tamping cost per track kilometre, and r is the annual discount rate. Following the marginal cost calculation for track renewals in [51], a track section that is observed at time t is considered, which is before the first tamping activity at T. The time until the next tamping activity is thus T−t. The PVC of the track section analysed in time t is: PVC t=ce−r(T−t)1 (1−e−r ¯ T)(5) To calculate a marginal cost of additional train movements, a temporary increase in tonnes Δq at time t=0 is considered, which will make the first tamping interval shorter, and all the subsequent intervals will be scheduled earlier. The subsequent intervals will have the same length as before since the traffic increase is temporary, i.e. the calculation considers the effect of one extra traffic unit running on the line. This is illustrated in Fig. 9 where T1 is the first tamping time for the baseline case (no change to traffic), T1+¯ T is the second interval etc., whilst T2 and T2+ ¯ T are the corresponding intervals for the scenario where traffic (temporarily) increases. The tamping activities are typically determined by a track geometry SD limit, as shown in Fig. 9. Note that railway administrations may also monitor the rate of change of track geometry SD and intervene if it increases suddenly, rather than only focusing on a limit value. This type of increase is more ad-hoc and can be associated with a variety of alternative deterioration mechanisms, including shrink-swelling of clay embankments. Therefore this paper focuses upon planned interventions. The marginal cost is expressed as: MCq= ∂ PVC t ∂ q= ∂ PVC t ∂ T ∂ T ∂ q= − cr e−r(T−t) (1−e−r ¯ T) ∂ T ∂ q(6) In this calculation, an increase in traffic during the first tamping interval is considered which can be expressed as the change in the average annual tonnes, ∂ ¯ q1 ∂ q. As noted by [51,52], this change can be Table 2 Ballast, sub-ballast and subgrade properties (case studies). Component Parameter Value Ballast Height (m) 0.3 Length in transversal direction (m) 2.8 Young’s modulus (Pa) 180 ×10 6 Density (kg/m 3 ) 1650 Poisson’s ratio 0.27 Hysteric damping coefficient 0.06 Sub-ballast (Sand-gravel) Height (m) 0.5 Length in transversal direction (m) 3.5 Young’s modulus (Pa) 180 ×10 6 Density (kg/m 3 ) 2300 Poisson’s ratio 0.3 Hysteric damping coefficient 0.05 Settlement parameter a 0.52 Settlement parameter b 0.15 Settlement parameter m 1.49 Compressive strength (kPa) 350 Subgrade (Lean clay) Young’s modulus (Pa) 70 ×10 6 Density (kg/m 3 ) 1900 Poisson’s ratio 0.35 Hysteric damping coefficient 0.03 Shear wave speed (mph) 260 Settlement parameter a 1.10 Settlement parameter b 0.16 Settlement parameter m 2.0 Compressive strength (kPa) 250 Fig. 8. Validation of vertical rail profile SD over time: predicted vs field data. C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 9 approximated as ∂ ¯ q1 ∂ q≈1 T, which has a decreasing error the more stable the traffic volume is. That is, the change in tamping interval due to a (percentage) change in tonnes is: ∂ T ∂ q= ∂ T ∂ ¯q1 ∂ ¯q1 ∂ q= ∂ T ∂ ¯q1 1 T(7) This is expressed in terms of a deterioration elasticity, γq= ∂ T ∂ ¯ q1 ¯ q1 T. The marginal cost in Eq. (4) is expressed as: MCq= ∂ PVC t ∂ q= − cr e−r(T−t) (1−e−r ¯ T) γq ¯q1 (8) Note that c is cost per track-km, which is multiplied by one over tonnage ( 1 ¯ q1), and the marginal cost is thus a cost per tonne per track-km, i.e. cost per tonne-km. The marginal cost calculation for a temporary change in the speed (v) for traffic is like the calculation for an increase in tonnage. The deterioration elasticity γv is the percentage change in tamping interval after a percentage change in speed, and ¯ v1 is the annual speed during the first tamping interval. This marginal cost calculation also uses cost per trackkm (c), which is multiplied by one over speed (1 ¯ v1)and the marginal cost is thus expressed as a cost per v per track-km (i.e. cost per v-km). In other words, it is the extra cost per v (km/h) for a train running on one km of track. MCv= ∂ PVC t ∂ v= ∂ PVC t ∂ T ∂ T ∂ v= − cr e−r(T−t) (1−e−r ¯ T) ∂ T ∂ v= − cr e−r(T−t) (1−e−r ¯ T) γv ¯v1 (9) Case studies The validated model is used to perform two analyses. Firstly it is used to analyse the effect of increasing line speed on differential track settlement, comparing speeds of 201, 209, 217 and 225 km/h (125, 130, 135 and 140mph). Secondly, it is used to investigate the effect of adding additional passenger or freight trains on the settlement of an existing track. Track parameters The finite element track mesh is the same as Fig. 7, however to reduce the number of potentially influential variables, the thin upper soil layer is assigned the material properties of the supporting soil layer below it. The track parameters including the characteristics of the rails, rail pads and sleepers are the same as the validation case, and are shown in Appendix B. The properties of ballast, sub-ballast and subgrade are summarised in Table 2. Considering the two applications under investigation (linespeed and additional trains), the only track difference is the vertical geometry track profile. Operational parameters The operational parameters are kept similar to the validation case, with a traffic volume per year of 19.58 million gross tons (MGT), and an average traffic volume per day of 0.054 MGT. The traffic volume is converted to passenger trains assuming an 11-car train with 17 tonne axle loads. Number of trains per MGT =106 (44*17)≈1337 trains The MGT per passenger train is thus approximately (1 1337 ≈) 0.00075. For freight trains, the calculation is based on a 50-car train with 25 tonne axle weights. Number of trains per MGT =106 (160*25)≈250 trains The MGT per freight train is thus (1 250 =)0.004. Fig. 9. Idealised tamping interval concept for two scenarios. C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 16 [Kv] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2Ks 0 −Ks 0 −Ks 0 0 0 0 0 0 2Ks⋅lb2 −Ks⋅lb 0 Ks⋅lb 0 0 0 0 0 −Ks −Ks⋅lb Ks +2Kp 0 0 0 −Kp −Kp 0 0 0 0 0 2Kp⋅lw2 0 0 −Kp⋅lw Kp⋅lw 0 0 −Ks Ks⋅lb 0 0 Ks +2Kp 0 0 0 −Kp −Kp 0 0 0 0 0 2Kp⋅lw2 0 0 −Kp⋅lw Kp⋅lw 0 0 −Kp −Kp⋅lw 0 0 Kp 0 0 0 0 0 −Kp Kp⋅lw 0 0 0 Kp 0 0 0 0 0 0 −Kp −Kp⋅lw 0 0 Kp 0 0 0 0 0 −Kp Kp⋅lw 0 0 0 Kp ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (18) Mass and stiffness matrices of the freight vehicle which contains only one set of suspension: [Z] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 lw 0 0 001−lw 0 0 0 0 0 0 1 lw 0 0 1 0 1 −lw ⎤ ⎥ ⎥ ⎥ ⎥ ⎦(19) [Mv] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Mc 0 0 0 0 0 0Jc 0 0 0 0 0 0 Mb +2Mw 000 0 0 0 Jb +2Mw⋅lw20 0 0 0 0 0 Mb +2Mw 0 0 0 0 0 0 Jb +2Mw⋅lw2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (20) [Kv] = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2Ks 0−Ks 0−Ks 0 0 2Ks⋅lb2−Ks⋅lb 0Ks⋅lb 0 −Ks −Ks⋅lb Ks 000 0 0 0 0 0 0 −Ks Ks⋅lb 0 0 Ks 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (21) where Mc is mass of the car box; Mb is mass of the bogie; Mw is mass of the wheelset; Jb is the rotational inertia of the car body; Kp is the complex stiffness of the primary suspension; Ks is the complex stiffness of the secondary suspension; lb is half the distance between the bogie’s centre of gravity; and lw is half the wheelbase that shares the same bogie. Kp and Ks are defined as: Kp =kpri +i ω cpri (22) Ks =ksec +i ω csec (23) where kpri is the spring stiffness of the primary suspension; ksec is the spring stiffness of the secondary suspension; cpri is the viscous damping of the primary suspension; and ksec is the viscous damping of the secondary suspension. Appendix B. . Ballasted track properties (validation case) Component Parameter Value UIC 60 Rail (single rail) Height (m) 0.172 Length in transversal direction (m) 0.015 Section area (m 2 ) 7.677 ×10 3 Moment of Inertia y-y (m 4 ) 3.038 ×10 -5 Moment of Inertia z-z (m 4 ) 0.512 ×10 -5 Young’s modulus (Pa) 2.11 ×10 11 Density (kg/m 3 ) 7850 Poisson’s ratio 0.3 Hysteric damping coefficient 0.01 Railpad (spring element) Continuous stiffness (N/m 2 ) 255 ×10 6 Viscous damping (Ns 2 /m 2 ) 22.5 ×10 3 Sleeper (G44) Height (m) 0.2 (continued on next page) C. Charoenwong et al. Transportation Geotechnics 37 (2022) 100845 17 (continued) Component Parameter Value Length in transversal direction (m) 2.5 Sleeper spacing (m) 0.65 Young’s modulus (Pa) 3 ×10 10 Density (kg/m 3 ) 2500 Poisson’s ratio 0.15 Hysteric damping coefficient 0.01 Ballast Height (m) 0.3 Length in transversal direction (m) 2.8 Young’s modulus (Pa) 180 ×10 6 Density (kg/m 3 ) 1600 Poisson’s ratio 0.27 Hysteric damping coefficient 0.061 Sub-ballast Height (m) 0.5 Length in transversal direction (m) 3.5 Young’s modulus (Pa) 212 ×10 6 Density (kg/m 3 ) 1913 Poisson’s ratio 0.3 Hysteric damping coefficient 0.054 Subgrade layer 1 Height (m) 0.5 Young’s modulus (Pa) 30 ×10 6 Density (kg/m 3 ) 2000 Poisson’s ratio 0.3 Hysteric damping coefficient 0.03 Settlement parameter a 0.64 Settlement parameter b 0.10 Settlement parameter m 1.70 Compressive strength (kPa) 100 Subgrade layer 2 Height (m) 5 Young’s modulus (Pa) 80 ×10 6 Density (kg/m 3 ) 2000 Poisson’s ratio 0.3 Hysteric damping coefficient 0.03 Settlement parameter a 0.64 Settlement parameter b 0.10 Settlement parameter m 1.70 Compressive strength (kPa) 220 References [1] Fr¨ ohling RD. 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