Modeling of track stability in crossings with a finite element model
Michiel Maertens
Supervisors: Prof. Jan Mys, Prof. dr. ir. Hans De Backer Counsellor: Dr. ir. Amelie Outtier
Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering
Department of Civil Engineering Chair: Prof. dr. ir. Peter Troch Faculty of Engineering and Architecture Academic year 2017-2018
Modeling of track stability in crossings with a finite element model
Michiel Maertens
Supervisors: Prof. Jan Mys, Prof. dr. ir. Hans De Backer Counsellor: Dr. ir. Amelie Outtier
Master's dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering
Department of Civil Engineering Chair: Prof. dr. ir. Peter Troch Faculty of Engineering and Architecture Academic year 2017-2018
Copyright license
“The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation."
Michiel Maertens
June 1, 2018
Preface
My fascination for trains and railway tracks dates back to my childhood. Travelling by train towards a new destination was always a breath-taking journey. That fascination was reflected in constructing railway scale models when I was a child, a hobby shared with my grandfather and father. From an early age, I indulged myself in the world of railroads and trains.
So, when I started the master program of civil engineering at Ghent University, the choice to follow the elective course of railway engineering was an evidence. Professor Jan Mys taught many aspects of railway engineering with great enthusiasm, which certainly raised my interest in this niche of engineering. I truly believe that railway transport is the most sustainable solution for the contemporary transport problems not only in Belgium, but also other densely populated regions around the world. Therefore, society should invest in safe, reliable and high-quality railway transport. This can only be realised if many actors make constructive efforts on this subject, going from politicians to commuters, from engineers to labourers.
By choosing for this master dissertation subject, I did not only opt for a research subject that completely suits my interest, but I hope this can also contribute to a deeper insight in the safety of railway tracks. Besides that, the research is performed with the help of finite element software, a skill which is highly appreciated in my further professional career. I am grateful to professor Jan Mys and professor Hans De Backer to have been part of this research, as this subject not only has an academic value, but moreover has a real life application within the studies of Infrabel. From the start, I worked on this research with great conviction, with the aim to deliver a high-quality reference work.
i
Acknowledgements
In the first place, my gratitude goes to professor Hans De Backer and professor Jan Mys to provide this highly interesting master dissertation subject. Throughout the year, they provided me with feedback about the obtained results and directed me towards the right solutions and research steps. They helped me to focus on the essential aspects of the research. The high-quality dissertation of Bram Landuyt, who investigated the stability of turnouts, served as a reference for my research. I would like to thank him for the provision of his entire master dissertation and for giving me an introduction to the finite element software Siemens NX. The way he modeled the railway tracks in the finite element software was very clear and was a huge step forward in this master dissertation. As such, I could rapidly analyse the results into much more detail.
Furthermore, I want to thank Ben Ferdinande and Annelies Stevens of Infrabel for providing me the technical documents to be able to perform this study. During the several meetings, they gave me the helpful feedback on the obtained results. Additionally, I would like to mention the appreciated help of Evy Van Puymbroeck with her introduction to Siemens NX and answering my questions related to this program. It required some time to fully understand the software, because it is much more complex than the common programs being used in the academic program of civil engineering. Last, but not least, I would like to express my gratitude towards my parents, family, friends and girlfriend, for their strong support and trust, not only this academic year, but during my entire studies.
ii Abstract
Title: Modeling of track stability in crossings with a finite element model
Author: Michiel Maertens
Supervisors: Prof. Jan Mys, Prof. dr. ir. Hans De Backer
Master’s dissertation submitted in order to obtain the academic degree of Master of Science in Civil Engineering
Summary:
In this dissertation, the buckling behaviour of continuously welded crossings applied in Belgian railway lines is investigated. Compared to plain track, the crossings are more vulnerable to buckling because of the structural discontinuity in rail geometry. In this regard, finite element models are made in the software Siemens NX to analyse the structural behaviour of crossings under thermal loads. The critical buckling temperatures and the development of displacements and internal forces are determined to assess the safety of the continuously welded track models. In the context of this dissertation, the stability of track is investigated under the application of thermal loads only. This means that the influence of train loads and dynamic buckling is not studied. The dissertation by Landuyt, who analysed the stability of turnouts, is used as reference to implement the rail geometries and the structural properties in the finite element software Siemens NX and the Nastran solver. The characteristics of the created finite element models are outlined first.
The models being investigated are a diamond crossing, a diamond crossing with double slips and a switch diamond. The plans of respectively the H4V4H4, TJD EUH4 and (XZX)1/8 crossings, provided by Infrabel, are used to set up the finite element models. Firstly, the critical buckling temperatures are determined and the models are validated by means of the resulting displacements, internal forces, stresses and strains. The safety assessment of the continuously welded crossings is based on the critical temperature loads, the lateral displacements, the permissivity of the rails in the fastening systems and the peak compressive stresses in the rails. Subsequently, different modifications in the geometries are made to get a better understanding in which parameters influence the buckling behaviour of the crossings. In this respect, also parameter studies are performed to determine the relative influence of the structural resistances on the buckling strength. Finally, finite element models of the old types of crossings, which were connected by means of fishplated joints, are created. The structural behaviour of these models are compared with the continuously welded crossings.
Keywords:
Track stability, continuously welded railway track, buckling, crossings, finite element model, ballasted track
iii Modeling of track stability in crossings with a finite element model
Michiel Maertens
Supervisors: Prof. Jan Mys, Prof. dr. ir. Hans De Backer
Abstract – In this dissertation, the buckling behaviour of result in smaller critical buckling temperatures than the ones continuously welded crossings applied in Belgian railway lines is obtained for static buckling under thermal loads only. investigated. Finite element models are made to determine the critical buckling temperatures, together with the development of II. LATERAL TRACK BUCKLING displacements and internal forces. The models being investigated are a diamond crossing (H4V4H4), a diamond crossing with double slips (TJD EUH4) and a switch diamond ((XZX)1/8). A. Thermal loading Firstly, the input characteristics of the structural parameters In continuously welded railway track, buckling is induced by and the implementation of the crossings in a finite element model the developed thermal compressive forces in the rail due to the in Siemens NX are discussed. The study by Landuyt served as a restrained expansion: reference for the modeling in the finite element software. FEATT= − (1) Subsequently, the resulting buckling curves are used to assess the x ( 0 ) safety of the continuously welded crossings. The development of The Young’s modulus E is set to 206 GPa and the coefficient the displacements, strains and forces are used to validate the of thermal expansion α equals 1,152⋅10-5 for steel in rail behaviour of the models. Finally, parameter studies and different applications [5]. The cross-sectional area of the rail A is modifications in the geometries of the crossings are made to get a dependent on the used profile and the neutral rail temperature better understanding in which parameters determine the buckling T is typically situated between 20°C and 30°C in Belgium. The strength of the investigated crossings. 0 maximal expected rail temperature is in Belgium equal to 60°C, Keywords – Track stability, continuous welded railway track, for track where the use of magnetic braking systems are buckling, crossings, finite element model, ballasted track prohibited. This is the case for the considered crossings. The maximal expected temperature increase above the neutral I. INTRODUCTION temperature is thus estimated at 40°C. The application of continuously welded rails for ballasted track became one of the greatest technical achievements in B. Lateral track buckling phenomenon th railway engineering in the 20 century. The related problem of The buckling phenomenon of railway track is characterised lateral track buckling due to the restrained thermal expansions by the load-displacement curve, plotting the temperature and the accompanying high compressive forces did not get the increase above the neutral rail temperature versus the lateral full attention of engineers at first. Consequently, several displacements of track. This curve typically exhibits an upper accidents with instabilities of railway track occurred throughout (ΔTb,max) and a lower (ΔTb,min) critical buckling temperature. For history. Arnold D. Kerr [1] was the first to publish a critical a level 1 safety approach [6], the allowable temperature increase survey of results of buckling tests and to perform an improved for safe operation is equal to ΔTb,min. Track can buckle at this analysis of static track buckling. At present, several theoretical temperature if sufficient external energy is applied, such that the and numerical buckling analyses have been made of plain and track evolves to a post-buckling equilibrium state. curved track. Among others Esveld [2] and Van [3] discussed in Buckling generally occurs from a small misalignment in detail the parameters affecting the buckling behaviour of track. track. This is described by a sinusoidal curve with a wavelength The software program CWERRI, developed by the TU Delft, is L and an amplitude δ0. Maximal initial misalignments of 20 mm used to model and quantitatively solve the track buckling can be expected for in-service track. The initial track defect phenomenon for straight and curved tracks and the interaction influences the critical buckling temperature significantly. In with bridge structures. Zacher [4] and Landuyt [5] created finite general, the buckling resistance decreases with decreasing element models to investigate the buckling resistance of defect wavelength and increasing misalignment amplitude. turnouts for static thermal loading. Other parameters affecting the buckling strength of railway However, no available study exists about the buckling track are the lateral and longitudinal ballast resistance, torsional behaviour of crossings. In this dissertation, finite element fastener resistance, longitudinal fastener resistance, rail type and models of a diamond crossing, a diamond crossing with double curvature radius. The structural resistances typically have a non- slips and a switch diamond are made in the software Siemens linear force-displacement curve. In the literature, the structural NX to determine the buckling resistance of the crossings. The characteristics of these parameters have been tested. The geometries of respectively the H4V4H4, TJD EUH4 and obtained results are used to implement the characteristics in the (XZX)1/8 crossings applied in Belgian railway lines are used to finite element models. create the models. The resulting forces and displacements are The lateral ballast resistance is the most important parameter examined to validate the models. These can be compared with contributing to the buckling strength. The shape of the load- the existing studies. The aim of this dissertation is to assess the displacement relationship is dependent on the quality of the safety of continuously welded crossings. Only thermal loads are ballast bed. Consolidated track exhibits a large peak resistance applied to the track, so that static buckling is investigated. Train and improves the buckling strength. Recently tamped track or loads are not implemented in the models and a dynamic track with low ballast quality does not show a clear peak buckling analysis is not performed. These aspects may however resistance and results in a progressive buckling curve.
iv C. Safety assessment of continuously welded railway track which respectively the wavelength and amplitude are To be safe for operation, the critical buckling temperature of designated. the model must be higher than the maximal expected rail temperature. Also the lateral displacements before buckling and B. Materials and structural properties the relative displacements of the rails in the fastening systems To obtain the desired elastic response of the track, the rail must be smaller than the limitations. Finally, also the peak elements are assigned a high grade steel quality, with the yield stresses are important to assess the fatigue life of the rail. strength set at 1200 MPa and the ultimate tensile strength 1280 MPa. As such, no effects of plastic yielding are expected. The III. CHARACTERISTICS OF THE FINITE ELEMENT MODEL E-modulus and coefficient of thermal expansion have been The real build-up of the track needs to be simplified to a defined in the previous section. The rails are assigned 60E1 or numerical model with sufficient accuracy. Landuyt [5] 50E2 profiles, in accordance with the plans. The elements performed a detailed study to determine the adequate properties representing the check rails have a 33C1 profile assigned. of the finite element model in Siemens NX using the Nastran Wooden sleepers are applied in the plans of the crossings. In solver. Most of the assumptions and characteristics are also the finite element models, the mesh of the sleepers are assigned applied in the finite element models of this study. the same steel properties as for the rails. The reason for this can be explained as follows. In reality the ballast bed is spread in A. Geometry and mesh aspects of the finite element model between the sleepers, while in the finite element model the The geometry of the crossings, provided by the plans of longitudinal ballast resistance is represented by only two springs Infrabel, is simplified to a three-dimensional line model. The at both bends. The sleepers could thus behave as a simply basic concept is illustrated in Figure 1. The rails, check rails and supported beam, which is not the case in reality. By assigning sleepers are simplified to their centre lines and meshed with the high stiffness properties of steel, the deflections are thus beam elements. The CBEAM element type is used in Siemens prevented. The actual material properties of the sleepers are not NX and a mesh density of approximately 0,2 m is applied. The of importance for the buckling behaviour of the finite element movable parts of switchblades are not modeled, as these rail model. The sleepers are assigned rectangular cross-sections, sections do not transfer significant forces to the sleepers and are with the standard dimensions 150 mm x 260 mm. The width and free to expand. The frogs and common crossings are modeled height differ in some sleepers of the TJD EUH4 and (XZX)1/8 by connecting the intersecting rails with additional beams in the and are modified accordingly in the finite element models. transverse direction. Finally, about 100 meters of plain track are The input characteristics of the load-displacement relation- added at both sides of the crossing. ships of the non-linear springs representing the lateral and The rails and sleepers are connected by fasteners, which are longitudinal ballast resistances are illustrated in respectively modeled as non-linear springs. The lateral and longitudinal Figure 2 and Figure 3. These are based on the data for standard ballast resistances are also implemented in the model by non- sleeper cross-sectional dimensions cited by Zacher [4] and are linear springs. The CBUSH elements in Siemens NX are applied dependent on the sleeper length, for which the conversion for these elements. Two springs are connected at both ends of formula proposed by Zacher is applied. The resistance the sleeper to simulate the longitudinal ballast resistance, which corresponds to recently tamped track, being conservative. For means that each spring takes half of the stiffness of the actual varying widths and heights of the sleepers, the input longitudinal resistance of the sleeper in the ballast bed. characteristics are modified according to similar conversion formulas, based on the contribution factors of the faces of the sleepers cited by Zacher. The fastener characteristics are in accordance with a Vossloh type fastening system, based on test results cited by Zacher [4] and In Jae [7]. The longitudinal fastener resistance is modeled by a bilinear characteristic, with a maximal asymptotic force resistance of 13 kN at a displacement of 0,2 mm. The lateral resistance is high, with an assumed linear relationship with a stiffness of 500 kN/mm. The non-linear relationship of the torsional fastener resistance is illustrated in Figure 4. The other degrees of freedom of the fastening system are assigned large stiffnesses, such that they are physically restrained.
Figure 1: Schematical representation of concept finite element model 14 570 cm 12 540 cm Defects with a prescribed wavelength and amplitude are 510 cm 10 implemented in the model. A 1-cos(x) curve is applied to the 480 cm initial straight lines of the rails. This defect is applied to all rails 8 450 cm within a particular section, which physically means that the 420 cm 6
sleepers of the track panel exhibit a misalignment. In the finite 390 cm Force [kN] Force element models, the rails and structural resistances are stress- 4 360 cm free at the neutral temperature with the misalignment 330 cm implemented. The most critical wavelength for lateral buckling, 2 300 cm being 8 m, is applied in the studies. Misalignment amplitudes of 0 270 cm 8 mm and 20 mm are used to investigate the buckling strength 0 1 2 3 4 5 of the track model. The designation of the defects are in Displacement [mm] accordance with following codes: 8m8mm and 8m20mm, in Figure 2: Input characteristics of the lateral ballast resistances
v 12 Zone 3 570 cm 10 540 cm 510 cm 8 480 cm 450 cm Figure 5: Extract of the plan view of the meshed geometry H4V4H4 6 420 cm diamond crossing and indication track defect zone
390 cm Force [kN] Force 4 360 cm B. Results 330 cm 2 The resulting buckling curves for different defect 300 cm configurations in the H4V4H4 diamond crossing are illustrated 0 270 cm in Figure 6. It is concluded that zone 3 of the diamond crossing 0 1 2 3 4 5 has a smaller buckling resistance than the common plain track. Displacement [mm] This is explained by the fact that 4 rails are present on a small Figure 3: Input characteristics of the longitudinal ballast resistances section of track panel, which is more critical than the standard track with two parallel rails. A larger initial misalignment 5,0 results in smaller critical buckling temperatures. In case of the 4,0 8m20mm track defect, the curve evolves to progressive buckling with ΔTb,max = 52,6°C and ΔTb,min = 51,0°C. There is no risk of 3,0 buckling in reality, as the maximal expected temperature increase in the rails amounts to 40°C.
2,0 Torque [kNm] Torque 1,0 140
Track defect 8m8mm in plain track
C] °
[ 120 Track defect 8m8mm in zone 3 of crossing
0,0 0 0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035 Track defect 8m20mm in zone 3 of crossing 100 Angle [rad] Figure 4: Input characteristic of the torsional fastener resistance 80 C. Boundary conditions and thermal loads 60 The rails are clamped at their ends, representing the fixed 40 behaviour of an ongoing continuously welded railway track.
The free end nodes of the non-linear springs, visible in Figure 20 Temperature increase above T above increase Temperature 1, are also clamped. The rails and sleepers are restrained in the 0 vertical translational direction, because only lateral buckling is 0 10 20 30 40 50 60 investigated. The sleepers are moreover only allowed to rotate Lateral displacement [mm] around the vertical axis, for the same reason. Figure 6: Buckling curves for different defect configurations in the Thermal loads are applied uniformly to all rail elements. The H4V4H4 diamond crossing neutral rail temperature is set at 20°C and a sufficiently high temperature increase is applied to ensure the upper critical The development of the lateral displacements for different buckling temperature is reached in the non-linear calculation. temperature increases is illustrated in Figure 7, for the most critical defect configuration. At +41°C, the displacements are D. Solver parameters maximal 2,5 mm. This means the total misalignment can reach The non-linear solution type SOL 106 with global constraints a value of 22,5 mm, which may become critical for safe together with the arc-length method is used in the Nastran solver operation. When buckling, the lateral displacements are to calculate the model. This type allows to increase the thermal growing rapidly and the buckling influence zone is broadening, load step by step and to obtain the results at every loadstep. A which is in accordance with the principles presented in [8]. The number of increments of 120 is applied in the solver parameters. buckled shape is a symmetric mode (shape III) with 3 half waves as described by Samavedam et al. [9]. IV. MODEL OF A DIAMOND CROSSING 10 A. Geometry and implementation track defect 5 0 The first model being investigated is the diamond crossing, -5 (1) 1,7°C for which the plan of the H4V4H4 crossing is used to create a -10 finite element model. An extract of the plan view of the meshed -15 (2) 41,0°C geometry is illustrated in Figure 5. The indicated zone 3 is -20 (3) 52,6°C assumed to be the weakest zone of the diamond crossing. Here, -25 the sleepers are rather short, meaning the sleepers have smaller -30 (4) 51,0°C lateral ballast resistances, and the track panel is not stiffened by [mm] displacement Lateral -35 (post-buckling) the larger cross-sections of frog and check rail. Moreover, the -40 sleeper spacing in this zone is larger than the other zones in the 110 115 120 125 130 135 140 crossing, which also means that the lateral ballast resistance is Distance along railway track [m] reduced. A track defect with length 8 m perfectly fits in this zone between the frog and common crossing. A reference model in Figure 7: Lateral displacements along the track. H4V4H4 8m20mm which the defect is implemented in plain track is also created.
vi The slip in the fastening system is defined by the relative Before the upper critical buckling temperature is reached, the longitudinal displacements between the rail and the sleeper drop in axial forces in the defect area is much less pronounced, underneath. The results in Figure 8 for rail A (see designation in but is still there because the initial misalignment is already Figure 9) show that the slip in the fastener is limited to 0,1 mm growing slightly. The courses of the individual rail forces are at a temperature increase of 41°C. This is lower than the similar as for the post-buckling configurations. At +41°C, a limitation of 30 mm for newly installed fasteners, such that this peak compression force of 701 kN is obtained, corresponding to aspect does not form any problem for the application of a uniform stress of 91,4 MPa. A maximal bending moment Mzz continuously welded diamond crossings. The fluctuations at 116 of 2 kNm is also present in the defect area, which corresponds m and 140 m are explained by the expanding check rails, which to an additional stress of 30 MPa at the most compressed fiber tend to move the sleepers longitudinally and yield local slip of of the cross-section of the rail. These values do not result in the rails in the fastening systems connected to these sleepers. fracture of the rail (typical minimal yield strength of rail in order of magnitude 600-800 MPa), but may become important in 2,0 assessing the fatigue life of the rail. (1) 1,7°C The integration of the course of the axial strains over the entire 1,5 length of the model is equal to the total elongation or shortening (2) 41,0°C of the rail. This should be approximately zero at pre-buckling 1,0 (3) 52,6°C temperatures and slightly increasing when the track is buckling. The course of the strains in rail A in Figure 11 and calculations 0,5 (4) 51,0°C revealed that this is indeed the case. (post-buckling) 0,0 0,0010
Relative long. displ. [mm] displ. long. Relative (1) 1,7°C 0,0006 -0,5 100 110 120 130 140 150 0,0002 (2) 41,0°C Distance along railway track [m] -0,0002 Figure 8: Relative long. displacements rail A. H4V4H4 8m20mm (3) 52,6°C -0,0006
Axial [mm/mm] strainAxial -0,0010 (4) 51,0°C (post-buckling) -0,0014 115 120 125 130 Figure 9: Naming convention of the rails. H4V4H4 Distance along railway track [m] The courses of the axial forces after buckling are given in Figure 11: Axial strains in rail A. H4V4H4 8m20mm Figure 10. The average rail force is calculated after summation It must be noticed that the way of modeling the frog and of the axial forces in each rail and dividing these values by 4. common crossing has an important influence on the course of The course of this average rail force allows to validate the the axial forces. If these elements are implemented in the finite behaviour of the track. In the defect zone, the axial forces drop element model with a larger cross-section (for example 170 mm locally, which is explained by the expansion of the rails when x 170 mm), the axial forces increase locally and the adjoining the lateral misalignment is growing. Indeed, expansions of the rails 60E1 face additional compression forces. Moreover, the rails result in a decrease in compressive forces [8]. critical buckling temperatures are reduced with 5°C in this case. The courses of the individual rail forces are fluctuating in accordance with the buckling shape of the defect, which can be related to the axial strains as plotted in Figure 11 for rail A. C. Parameter study Positive strains correspond to expansions and negative ones to A parameter study is performed on the H4V4H4 model with shortenings. This is in accordance with the decrease in axial track defect 8m20mm in zone 3, in which the structural forces (expansion) and increase in axial forces (shortening) in parameters are varied with a percentage change. The summary Figure 10. The drops in axial forces at 114 m and 140 m are of the results is presented in Figure 12, from which it can be related to the local expansions of the check rails. concluded that the lateral ballast resistance is the most influential parameter on the buckling strength of the diamond -940 crossing. For decreasing ballast quality, the buckling behaviour -920 evolves to more pronounced progressive buckling. These conclusions are in accordance with results in different literature -900 [2][5][9][10]. The longitudinal fastener resistance is the second -880 most influencing parameter, which is explained by the fact that -860 lateral displacements of the rails can only occur if the rails are -840 allowed to slip in the fastening system. -820 Average rail force Axial force [kN] force Axial -800 Rail A D. Improving the buckling resistance of the diamond crossing Rail B -780 Rail C Kish [11] concludes that dynamic vibrations and uplift of -760 Rail D track can reduce the ballast resistance with more than 30%. In some configurations, the critical buckling temperature increase 90 100 110 120 130 140 150 160 can reduce to values close to +40°C. Different modifications can Distance along railway track [m] be proposed to improve the buckling resistance of the H4V4H4 Figure 10: Axial forces at +51,6°C (post-buckling). H4V4H4 8m20mm diamond crossing with track defect 8m20mm. One possibility is
vii to lengthen the sleepers in the critical zone of the crossing, like diamond crossing. The resulting buckling curves for different it is illustrated in Figure 13. Lengthening the sleepers to 450 cm defect configurations in the TJD EUH4 diamond crossing with increases the critical buckling temperature increases to ΔTb,max double slips are illustrated in Figure 15. In any case, the critical = 59,4°C and ΔTb,min = 55,7°C. For sleepers with length 390 cm, temperatures are higher than the ones obtained for the H4V4H4 the results are ΔTb,max = 55,6°C and ΔTb,min = 53,2°C. Increasing model. For the most critical track defect 8m20mm, the critical the width of the sleepers from 260 mm to 300 mm additionally temperature increases are ΔTb,max = 58,9°C and ΔTb,min = increases the critical temperatures with 2°C to 3°C. The lateral 56,0°C. The lateral displacement at +40°C amounts to 1,3 mm. displacements at +40°C are reduced to values of about 1,5 mm. The buckling shape and development of lateral displacements are similar as the results obtained for the diamond crossing. If Lateral Ballast Resistance: Tb,max the ballast resistances are adjusted to the larger sleeper widths, Lateral Ballast Resistance: Tb,min Longitudinal Ballast Resistance: Tb,max the resulting critical temperatures are about 2°C higher. Longitudinal Ballast Resistance: Tb,min Torsional Fastener Resistance: Tb,max
0 140 Torsional Fastener Resistance: Tb,min TJD EUH4 - Longitudinal Fastener Resistance: Tb,max Track defect Longitudinal Fastener Resistance: Tb,min 120 8m8mm in plain C] 62 ° 100 61 track TJD EUH4 -
60 80C]
° Track defect 59 [ 60 8m8mm in zone 58 3 of crossing 57 40 TJD EUH4 - 56 Track defect
Temperature increase above T above increase Temperature 20 55 8m20mm in zone 54 0 3 of crossing 53 0 10 20 30 40 50 60 52 Lateral displacement [mm] 51 Figure 15: Buckling curves for different defect configurations in the 50 TJD EUH4 diamond crossing with double slips 49 48 The course of the relative longitudinal displacements between
Temperature increase above neutral temperautre [ temperautre neutral above increase Temperature 47 -100% -80% -60% -40% -20% +0% +20% +40% rails and sleepers is similar to the ones obtained in Figure 8 for Percentage change in parameter value the diamond crossing, such that the limitation of 30 mm is not reached anywhere. The ends of the junction rails are expanding, Figure 12: Summary of the parameter study. H4V4H4 8m20mm as these evolve in the tongues. The courses of the axial forces in the two curved outer rails of the TJD EUH4 at +40°C are plotted in Figure 16. The overall drop in the forces is explained by the transition from 60E1 rails to the smaller 50E2 sections. The latter have smaller nominal Figure 13: Increased length of the sleepers in critical zone of H4V4H4 forces, so that a transition zone is present. The transition occurs by expansion of the 60E1 rails (decreasing compressive forces) A more economical solution exists in decreasing the sleeper and shortening of the 50E2 rails (increasing compressive spacing from 705 mm to 550 mm in the critical defect zone. This forces). This is also noticed in the plot of the axial strains in the yields ΔTb,max = 60,6°C and ΔTb,min = 57,8°C. Applying safety junction rail C given in Figure 17, in which positive strains caps on every third sleeper results in ΔTb,max = 57,0°C and for suddenly change in negative values at the transition point from every second sleeper ΔTb,max = 61,5°C. A comparable increase 60E1 to 50E2 profiles. In the same plot, it is observed that this in buckling strength can be obtained when a small retaining wall rail is expanding at its free end. is installed next to the track. Small drops in rail forces are observed at 114 m and 140 m, which are contributed to the expansion of the check rails. The V. MODEL OF A DIAMOND CROSSING WITH DOUBLE SLIPS central drop in the plot is explained by the presence of additional The TJD EUH4 track model is used to assess the safety of a rails in the heart of the crossing, as visible in Figure 14. These diamond crossing with double slips. The geometry is given in have the same effect as the check rails, such that the outer rails Figure 14, which shows that the crossing itself exists out of locally expand and drop in compressive axial forces. 50E2 rails and mainly sleepers with an increased width of 300 mm. The track defect is again implemented in the same zone 3. -750
-710
-670
-630 Rail A
Figure 14: Meshed geometry of the TJD EUH4 diamond crossing with Axial force [kN] force Axial double slips -590 Rail B A. Results The results presented below are obtained from the model in -550 80 100 120 140 160 180 which the ballast resistances are not adjusted to the increased Distance along railway track [m] sleeper width of 300 mm, to allow a clear comparison with the Figure 16: Axial forces at +40°C. TJD EUH4 8m20mm
viii 2E-04 After the gaps are closed, the axial forces are transferred through the joint and the rails are at risk of buckling. The lateral 1E-04 displacements at +40°C are limited to only 0,25 mm thanks to the allowance for expansion of the rails in the joints. The critical 0E+00 buckling temperature increases are calculated as being ΔTb,max (2) 40,0°C = 73,5°C and ΔTb,min = 69°C, which means that the buckling -1E-04 resistance is much higher compared to the continuously welded Zero-strain Axial [mm/mm] strainAxial TJD EUH4 crossing, as expected. -2E-04 50 70 90 110 130 VI. MODEL OF A SWITCH DIAMOND Distance along railway track [m] The last model being investigated is the switch diamond, for Figure 17: Axial strains in rail C at +40°C. TJD EUH4 8m20mm which the plan of the (XZX)1/8 crossing is used. The geometry of the switch diamond is given in Figure 19, in which the The peak compression stresses at +40°C are calculated at 105 different sleeper dimensions are indicated. The outer rails are MPa in the 50E2 rail, which is slightly higher than observed in kinked and the Z-block in the heart of the crossing is modeled the diamond crossing. The additional compressive stress due to with a rectangular cross-section with dimensions 170 x 170 mm. bending is comparable in magnitude. The movable parts of the switchblades are omitted in the model.
B. Differences between H4V4H4 and TJD EUH4 models The increased buckling strength of the TJD EUH4 compared to the H4V4H4 can be explained by the denser sleeper spacing in the critical zone, the larger widths of the sleepers and also the application of the 50E2 rail sections. The latter have a smaller Figure 19: Meshed geometry of the (XZX)1/8 switch diamond second moment of area, but the smaller nominal thermal forces have a beneficial effect on the critical temperature load. In Again, a defect 8m20mm is implemented in the zone 3 of the addition, the added rail sections in the heart of the crossing crossing. The shape of the resulting buckling curves are similar contribute to an increased stiffness of the track panel. Finally, to the ones obtained for the common diamond crossing. The the difference in the rail geometry between the H4V4H4 and critical buckling temperature increases are however higher with TJD EUH4 crossings has only a minor effect on the critical values ΔTb,max = 63,4°C and ΔTb,min = 57,7°C. This means the buckling temperatures. switch diamond has a significantly higher buckling strength than the diamond crossing, which can be contributed to C. Model with fishplated joints increased sleeper dimensions and denser sleeper spacing in the The old types of diamond crossings with double slips were critical zone and moreover to the presence of only two rails in connected with fishplated joints, as illustrated in the sketch in the considered defect zone. The development of lateral Figure 18. A finite element model is made in which these joints displacements are again similar to those of the H4V4H4 are implemented to compare the behaviour under thermal loads diamond crossing. with continuously welded track. This is done by using the Figure 20 illustrates the axial forces in the rails at +40°C. The CGAP element in Siemens NX, which defines a gap and friction axial forces in the interrupted junction rails, which change in element. It allows to model an initial gap opening of 3 mm and tongues, are going to zero at their ‘free’-expanding ends. The to assign an axial stiffness when the gap is closed in a non-linear compressive forces in the kinked outer rails are first decreasing calculation. The fishplates are modeled by beam elements with and subsequently increasing, which is explained by the a £600 cross-section. The connection between fishplates and transition from 4 rails to 2 rails as visible in Figure 19. This rails must be able to accommodate for longitudinal expansions behaviour is similar to what is observed in the stock rail of a without restraints. This is foreseen by connecting the fishplates classical turnout, as for example described by Landuyt [5]. The to the rails by spring elements, which have no stiffness in the larger cross-section of the Z-block results in a local increase in longitudinal direction and a very high stiffness in the lateral axial forces at the heart of the crossing. This means that the direction. The contour plot in Figure 18 shows the expected 60E1 rails near the Z-block are facing increased compressive behaviour of the rails when they are expanding. forces. A maximal peak uniform axial stress of 117 MPa is observed, which is comparable to previous models. This peak is dependent on the modeled cross-section of the Z-block. -1000
-800
-600 Rail A Rail B -400 Rail C
Axial force [kN] force Axial Rail D -200 Rail E Rail F 0 Figure 18: Contour plot longitudinal displacements at +10°C in TJD 0 50 100 150 200 250 EUH4 model with fishplated joints. Sketch: dashes = fishplated joints; Distance along railway track [m] dots = aluminothermic welds Figure 20: Axial forces at +40°C. (XZX)1/8 8m20mm
ix VII. CONCLUSIONS It is concluded that the investigated continuously welded crossings are safe for operation, as the critical buckling temperature increases are more than 10°C higher than the maximal expected rail temperature increase of 40°C. These are determined for the most critical track defect with wavelength 8 m and misalignment amplitude of 20 mm implemented in the weakest zone of the crossings. However, if dynamic vibrations and uplift of track are taken into account through a reduction in ballast resistance, the buckling temperatures of the diamond crossing may reduce to critical values. Therefore, different solutions to improve the buckling strength of the diamond crossing were proposed. The permissivity of the rails in the fastening systems and the peak axial forces are limited, so that these do not form any limitation for in-service track.
ACKNOWLEDGEMENTS I would like to thank professor Hans De Backer and professor Jan Mys for providing this interesting master dissertation subject and their feedback and guidance throughout the research process. Additionally, I would like to thank Bram Landuyt for sharing his obtained knowledge in creating finite element models of railway track and providing me the results of his qualitative research. Finally, I want to thank Ben Ferdinande and Annelies Stevens for the feedback on the obtained results and Evy Van Puymbroeck for teaching me the basic features of the finite element software Siemens NX.
REFERENCES [1] Kerr AD. Lateral Buckling of Railroad Tracks due to Constrained Thermal Expansions - A Critical Survey. In: Kerr AD. Railroad Track Mechanics and Technology: Proceedings of a Symposium Held at Princeton University, 21-23 April 1975, Princeton, New Jersey. Pergamon Press; 1978. p. 141-169. [2] Esveld C. Improved Knowledge of CWR Track. [3] Van MA. Stability of Continuous Welded Rail Track. Delft: Technische Universiteit Delft; 1997. [4] Zacher M. Calculation of the critical temperature for track buckling in a switch P3550 - XAM 1/46 on the line Liège-Brussels. München: DB Netz AG; 2011. [5] Landuyt B. Modelling of track stability in turnouts with a finite element model. Ghent: Ghent University; 2017. [6] Esveld C. Modern Railway Track. TU Delft; 2001. [7] In Jae , Sin Choo Y, Sang Chul H. Study on Behavior Characteristics of Turnout and Bridge. In: Proceedings, 8th world congress on railway research: towards a global railway. Seoul, 2008. [8] U.S. Department of Transportation. Track Buckling Prevention: Theory, Safety Concepts and Applications. Washington, D.C.: U.S. Department of Transportation - Federal Railroad Administration; 2013. [9] Samavedam G, Kish A, Purple A, Schoengart J. Parametric Analysis and Safety Concepts of CWR Track Buckling. Washington, D.C.: U.S. Department of Transportation, Federal Railroad Administration; 1993. [10] Choi DH, Na HS. Parametric Study of Thermal Stability on Continuous Welded Rail. IJR International Journal of Railway. 2010;3(4): p. 126- 133. [11] Kish A. On the Fundamentals of Track Lateral Resistance. Peabody, MA, USA: AREMA; 2011.
x Table of contents
Preface i
Acknowledgements ii
Abstract iii
Table of contents xi
List of figures xvi
List of tables xxv
List of abbreviations and symbols xxvi
Chapter 1 Introduction 1
Chapter 2 Literature overview 3
2.1 Introduction ...... 3
2.2 Thermal loads on railway track ...... 5
2.3 Buckling mechanism and response ...... 6 2.3.1 Track instability...... 6 2.3.2 Buckling mechanism in relation to track imperfections ...... 8 2.3.3 Safety approach ...... 9 2.3.4 Buckling influence zone ...... 10 2.3.5 Buckling modes ...... 11 2.3.6 Radial breathing ...... 11 2.3.7 Stress state and stress distribution in (buckled) rails ...... 12
2.4 Static versus dynamic buckling ...... 13
2.5 Parameters influencing track buckling ...... 14 2.5.1 Neutral rail temperature ...... 15 2.5.2 Sleepers ...... 15 2.5.3 Ballast ...... 16 2.5.4 Rail fasteners ...... 19
xi 2.5.5 Rail type and material ...... 21 2.5.6 Rail curvature ...... 21 2.5.7 Initial imperfections ...... 22 2.5.8 Creep...... 24
2.6 Structural parameters ...... 24 2.6.1 Lateral ballast resistance ...... 24 2.6.2 Longitudinal ballast resistance ...... 31 2.6.3 Rail fasteners ...... 31 2.6.4 Track defects ...... 34 2.6.5 Rail types ...... 36
2.7 Overview of existing finite element models ...... 36 2.7.1 CWERRI ...... 36 2.7.2 CWR-BUCKLE ...... 36 2.7.3 Zacher and Landuyt ...... 36
2.8 Limitations and solutions to prevent buckling in summer ...... 37 2.8.1 Prevention of buckling ...... 37 2.8.2 Stress limitations ...... 37 2.8.3 Relative displacements in the fastening systems ...... 37
Chapter 3 Basics of the finite element model 38
3.1 Introduction ...... 38
3.2 Geometry of the models ...... 38 3.2.1 Line model of the structural model...... 38 3.2.2 Boundaries of the model ...... 39 3.2.3 Track defect ...... 40
3.3 Aspects related to finite element modeling ...... 41 3.3.1 Beam elements (rails, check rails, fishplates, sleepers) ...... 41 3.3.2 Spring elements (ballast resistances and fasteners) ...... 42
3.4 Materials and structural properties ...... 44 3.4.1 Materials and sections ...... 44 3.4.2 Ballast resistances ...... 45 3.4.3 Fastener resistances ...... 50
3.5 Boundary conditions ...... 53 3.5.1 Rails and sleepers ...... 53 3.5.2 Springs ...... 53
3.6 Thermal loads and safety assessment ...... 53
xii 3.7 Solver setup...... 54 3.7.1 Solution type ...... 54 3.7.2 Non-linear parameters ...... 55 3.7.3 Arc-length method parameters ...... 55
3.8 Buckling definition ...... 55
Chapter 4 Model of a diamond crossing 57
4.1 Introduction ...... 57
4.2 Properties of the finite element model ...... 57 4.2.1 Geometry ...... 57 4.2.2 Track defects ...... 59
4.3 Results and safety assessment ...... 61 4.3.1 Track defect 8m8mm – comparison between defect locations ...... 61 4.3.2 Track defect 8m20mm ...... 63 4.3.3 Influence of different parameters on the buckling strength ...... 87
4.4 Parameter study ...... 89 4.4.1 Influence of track defect ...... 89 4.4.2 Influence of lateral and longitudinal ballast resistance ...... 89 4.4.3 Influence of fastener characteristics ...... 92 4.4.4 Summary and comparison of parameter study ...... 92
4.5 Improving the buckling resistance of a diamond crossing ...... 94 4.5.1 Influence of the ballast quality ...... 94 4.5.2 Increasing the sleeper lengths and widths ...... 94 4.5.3 Denser sleeper spacing ...... 97 4.5.4 Installing anchor caps on the sleepers ...... 98
4.6 Comparison between H4V4H4 and H3V3V3 ...... 98
4.7 Model with fishplated joints ...... 100 4.7.1 Modeling the fishplated joint ...... 102 4.7.2 Results and comparison with continuously welded diamond crossing ...... 104
4.8 Conclusions ...... 107
Chapter 5 Model of a diamond crossing with double slips 108
5.1 Introduction ...... 108
5.2 Geometry ...... 109
5.3 Results and safety assessment ...... 112
xiii 5.3.1 Comparison between defect configurations ...... 112 5.3.2 Lateral displacements ...... 114 5.3.3 Longitudinal displacements and axial strains ...... 116 5.3.4 Forces in the lateral ballast springs ...... 121 5.3.5 Axial forces ...... 122
5.4 Parameter study ...... 129 5.4.1 Influence of the misalignment amplitude ...... 130 5.4.2 Influence of the lateral ballast resistance ...... 130 5.4.3 Influence of the longitudinal fastener resistance ...... 130
5.5 Model with fishplated joints ...... 131 5.5.1 Principles ...... 131 5.5.2 Results ...... 132
5.6 Conclusions ...... 134
Chapter 6 Comparison between diamond crossing and diamond crossing with double slips 135
6.1 Introduction ...... 135
6.2 Differences in track geometry and structural parameters ...... 135 6.2.1 Track geometry ...... 135 6.2.2 Rail types ...... 136 6.2.3 Sleeper geometry ...... 136
6.3 Investigation of the differences in buckling strength ...... 137 6.3.1 Model configurations ...... 137 6.3.2 Results and comparison in buckling curves ...... 138
6.4 Comparison between parameter studies ...... 139 6.4.1 Sensitivity to a change in lateral ballast resistance ...... 139 6.4.2 Sensitivity to a change in longitudinal fastener resistance ...... 140
Chapter 7 Model of a switch diamond 142
7.1 Introduction ...... 142
7.2 Geometry ...... 143
7.3 Results and safety assessment ...... 144 7.3.1 Buckling curve ...... 144 7.3.2 Lateral displacements ...... 146 7.3.3 Longitudinal displacements and axial strains ...... 148 7.3.4 Axial forces ...... 151
xiv 7.4 Conclusions ...... 155
Chapter 8 Conclusions and further research 156
8.1 Aim of the dissertation and validation of the results ...... 156
8.2 Safety assessment of the considered continuously welded crossings ...... 156 8.2.1 Critical buckling temperatures ...... 156 8.2.2 Permissivity of the rail fastening systems and lateral displacements before buckling ...... 157 8.2.3 Axial forces and stresses in the rails ...... 157 8.2.4 Parameters influencing the buckling behaviour and applications to improve the buckling resistance ...... 158
8.3 Critical inference on the obtained results ...... 158 8.3.1 Reasons why the critical buckling temperatures could be lower in reality ...... 158 8.3.2 Reasons why the obtained results could be too conservative ...... 159
8.4 Further research ...... 160 8.4.1 Bundle of different crossings/turnouts ...... 160 8.4.2 3D modeling of the rails ...... 160 8.4.3 Implementation of dynamic train loads ...... 161 8.4.4 Different implementation of the initial misalignments in a finite element model 161 8.4.5 Verification of the obtained results with tests on site ...... 161
References 163
Annexes 168
Annex 1: Rail profiles ...... 168
Annex 2: Plans of crossings/turnouts ...... 171
Annex 3: Results – Diamond crossing ...... 175
xv List of figures
Chapter 1 Introduction
Figure 1.1: Examples of lateral track buckling occurred throughout history (1) ...... 1
Figure 1.2: Examples of recent lateral track buckling accidents (2) ...... 1
Figure 1.3: Example of static lateral buckling of railway track (6) ...... 2
Chapter 2 Literature overview
Figure 2.1: Example of a rail fracture due to high tensile stresses in cold weather (10) ...... 4
Figure 2.2: Example of rail fracture occurring during the night of 25-26/1/2012 at Remersdaal leading to the derailment of a boxcar (11) ...... 4
Figure 2.3: (A) Typical load-displacement curve for track buckling. (B) Buckling energy concept (17) ...... 7
Figure 2.4: Typical buckling shape (17) ...... 7
Figure 2.5: Temperature increase in function of the lateral displacement in case of good ballast characteristics (A) and poor ballast characteristics (B) (2) ...... 8
Figure 2.6: Track geometry before and after buckling (6) ...... 8
Figure 2.7: Typical buckling response curves for continuously welded railway track (6). (A) buckling at the upper critical buckling temperature. (B) Buckling at a temperature lower than the upper critical buckling temperature (snap-through) ...... 9
Figure 2.8: Comparison between the distribution of the longitudinal rail forces before and after buckling has occurred (6) ...... 10
Figure 2.9: Possible buckling modes for continuously welded railway track (4) ...... 11
Figure 2.10: Influence of radial breathing on the critical buckling temperature increase for increasing curvature (4) ...... 12
Figure 2.11: (A) Cross-sectional distribution of longitudinal stresses in a bended continuously welded rail (21). (B) Resulting longitudinal stresses due to bending in a continuously welded rail in function of the misalignment amplitude and wavelength of the misalignment (21) ...... 13
Figure 2.12: Distribution of residual stresses as measured by ORE D 148 (23) ...... 13
Figure 2.13: Difference in buckling response curve for static and dynamic buckling (6) ...... 14
Figure 2.14: Comparison between the buckling response for wooden versus concrete sleepers in average ballast quality conditions (6) ...... 15
Figure 2.15: Buckling curves for varying lateral ballast resistance (A) and longitudinal ballast resistance (B) (3) .... 16
Figure 2.16: Influence of the lateral peak ballast resistance (A) and longitudinal ballast resistance (B) on the critical buckling temperatures (23) ...... 17
xvi Figure 2.17: Influence of weak/average/strong ballast resistance on the buckling curves (6) ...... 17
Figure 2.18: Influence of the longitudinal ballast resistance on the critical buckling temperatures (4) ...... 18
Figure 2.19: (A) Influence of the torsional stiffness of the fastening systems on the critical buckling temperatures (4). (B) Buckling curves for varying torsional stiffness of the rail fasteners (3) ...... 20
Figure 2.20: Influence of the longitudinal fastener resistance on the upper critical buckling temperature for different track defects in case of a turnout (7) ...... 20
Figure 2.21: Buckling curves for varying rail sizes (3) ...... 21
Figure 2.22: Buckling curves for varying degree of track curvature (3)...... 22
Figure 2.23: Influence of the track curvature radius on the critical buckling temperatures (23)...... 22
Figure 2.24: Buckling curves for varying misalignment amplitude (A) and wave length (B) (3) ...... 23
Figure 2.25: Influence of the misalignment amplitude (A) and half wave length of the misalignment (B) on the critical buckling temperatures (23) ...... 23
Figure 2.26: Typical lateral resistance vs displacement response curve of a sleeper in the ballast for consolidated and tamped track (25) ...... 25
Figure 2.27: Designation of the base, shoulder and crib areas of the sleeper (25) ...... 26
Figure 2.28: Input characteristics of the lateral ballast resistances in the finite element model by Zacher (12) ...... 28
Figure 2.29: Safety caps for wooden sleepers by Vossloh: (A) SN type (33). (B) SV type (34) ...... 29
Figure 2.30: Influence of safety caps on the lateral ballast resistance of wooden sleepers (27) ...... 29
Figure 2.31: (A) UNINAPOLI type sleeper anchor system. (B) rail anchor system (35) ...... 30
Figure 2.32: Overview of techniques to increase the lateral ballast resistance. (A) Retaining wall. (B) Gluing of ballast in the shoulder. (C) Gluing of the ballast in between the sleepers ...... 30
Figure 2.33: Input characteristics of the longitudinal ballast resistances in finite element model by Zacher (12) ... 31
Figure 2.34: (A) Illustration of Nabla fastening system (38). (B) Illustration of Pandrol e-Clip fastening system (39) ...... 32
Figure 2.35: Results for the torsional resistance of a 60E1 rail with a Vossloh fastening system with different rail pads, conducted by TUM (12) ...... 33
Figure 2.36: Vossloh fastening system for wooden sleepers (40) ...... 33
Figure 2.37: Illustrations of K-type fastening system (41) ...... 34
Figure 2.38: Results of the analysis of the maximum track defect measurements in Germany (12) ...... 35
Chapter 3 Basics of the finite element model
Figure 3.1: Two possible ways to model the frog in the finite element model. (A) By means of transverse elements connecting the intersecting rails. (B) By means of a larger cross-section assigned to the rails ...... 39
Figure 3.2: Schematical representation of the imported line model of the track with its structural meaning ...... 41
Figure 3.3: Definition of the CBEAM’s axes for the rail elements, resulting in the correct inclination ...... 42
Figure 3.4: PBUSH entry characteristics in Siemens NX: (A) Nominal stiffness values. (B) Non-linear stiffness characteristics ...... 43
xvii Figure 3.5: Definition of the axes of the non-linear spring elements representing the lateral ballast resistance (A) and longitudinal ballast resistance (B) ...... 44
Figure 3.6: Definition of the axes of the non-linear spring elements representing the rail fasteners ...... 44
Figure 3.7: Input data for the lateral ballast resistances of the sleepers (sleeper lengths 270 cm – 570 cm) ...... 47
Figure 3.8: Input data for the lateral ballast resistances of the sleepers according to the contribution factors of NMBS ...... 49
Figure 3.9: Input data for the lateral ballast resistances for modified sleeper cross-sections: width B = 300 mm and depth D = 200 mm, respectively for sleepers with length 270 cm and 450 cm ...... 49
Figure 3.10: Input data for the longitudinal ballast resistances of the sleepers (sleeper lengths 270 cm – 570 cm) ...... 50
Figure 3.11: Input data for the torsional fastener resistance = DOF4 (Vossloh type fastener) ...... 51
Figure 3.12: Photograph of a check rail placed on a chair (in case of a turnout) ...... 52
Figure 3.13: Boundary conditions for the non-linear springs representing the lateral and longitudinal ballast resistance ...... 53
Figure 3.14: Definition of the non-linear parameters (A) and arc-length method parameters (B) in Siemens NX ..... 56
Chapter 4 Model of a diamond crossing
Figure 4.1: Photograph of a scale model of a diamond crossing (46)...... 57
Figure 4.2: Extract of the simplified line model of the H4V4H4 diamond crossing, imported in Siemens NX...... 58
Figure 4.3: Plan view of the line geometry of the diamond crossing H4V4H4 in Siemens NX ...... 58
Figure 4.4: Plan view of the meshed geometry of the diamond crossing H4V4H4 in Siemens NX ...... 59
Figure 4.5: Three-dimensional view of the meshed geometry of the diamond crossing H4V4H4 in Siemens NX .... 59
Figure 4.6: Naming convention of the considered zones in the diamond crossing H4V4H4 ...... 60
Figure 4.7: Indication of the location of the implemented track defects with wavelength 8 m in plain track and in zone 3 of the diamond crossing H4V4H4 ...... 60
Figure 4.8: Contour plot of the lateral displacements for an implemented track defect 8m8mm in zone 3 of the diamond crossing H4V4H4. Final loadstep (post-buckling – +81,84°C) ...... 61
Figure 4.9: Load-displacement curves for different implemented track defects in the diamond crossing H4V4H4 . 62
Figure 4.10: Designation of the points along the buckling curve where the results are taken. H4V4H4 8m20mm . 64
Figure 4.11: Lateral displacements along the railway track for different temperature increases along the buckling curve. H4V4H4 8m20mm ...... 64
Figure 4.12: Plot of the measured distance along the railway track applied to the H4V4H4 model (contour plot of the lateral displacements) ...... 65
Figure 4.13: Naming convention of the rails in the diamond crossing H4V4H4 ...... 65
Figure 4.14: : Longitudinal displacements of rail A (along global x-axis) along the railway track for different temperature increases along the buckling curve. H4V4H4 8m20mm ...... 66
xviii Figure 4.15: Contour plot of the longitudinal displacements along the global x-axis at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm ...... 66
Figure 4.16: Contour plot of the longitudinal displacements along the axis line of rail A at 훥T = 51,6°C (post- buckling). H4V4H4 8m20mm ...... 67
Figure 4.17: Longitudinal displacements of rail A along the railway track for different temperature increases. Displacements taken along the rail’s axis line. H4V4H4 8m20mm...... 68
Figure 4.18: : Longitudinal displacements of rail C along the railway track for different temperature increases. Displacements taken along the rail’s axis line. H4V4H4 8m20mm...... 68
Figure 4.19: Comparison between the longitudinal displacements of rail A and rail C at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm ...... 69
Figure 4.20: Longitudinal displacements of rail A versus longitudinal displacements of the sleeper underneath rail A at the upper critical buckling temperature (+52,6°C) and at the post-buckling configuration (+51,6°C). H4V4H4 8m20mm ...... 70
Figure 4.21: Schematical sketch showing the influence of expanding check rails on the longitudinal displacements of rails and sleepers. (A) Undeformed configuration. (B) Configuration after a sufficiently high temperature increase ...... 70
Figure 4.22: Relative longitudinal displacements of rail A and sleepers along the railway track at different temperature increases. H4V4H4 8m20mm ...... 71
Figure 4.23: Forces in the springs representing the lateral ballast resistance along the railway track for different temperature increases. H4V4H4 8m20mm ...... 72
Figure 4.24: Forces in the springs representing the longitudinal ballast resistance along the railway track for different temperature increases. H4V4H4 8m20mm ...... 73
Figure 4.25: Torques in the fasteners along the railway track for different temperature increases. H4V4H4 8m20mm ...... 74
Figure 4.26: Longitudinal forces in the fasteners along the railway track for different temperature increases. H4V4H4 8m20mm ...... 74
Figure 4.27: Total axial force after summation of the axial forces in the 4 individual rails of the diamond crossing, for different temperature increases along the buckling curve (pre-buckling). H4V4H4 8m20mm...... 76
Figure 4.28: Total axial force after summation of the axial forces in the 4 individual rails of the diamond crossing, for different temperature increases along the buckling curve (post-buckling). H4V4H4 8m20mm ...... 76
Figure 4.29: Comparison between the summation of the axial forces of the rails for a calculation with and without the axial forces in the check rails. H4V4H4 8m20mm ...... 77
Figure 4.30: Contour plot of the longitudinal displacements of the check rails at a temperature increase of 37,9°C. H4V4H4 8m20mm ...... 78
Figure 4.31: Comparison between the axial forces of the individual rails and the average rail force, post-buckling (+51,6°C). H4V4H4 8m20mm ...... 78
Figure 4.32: Comparison between the axial forces of the individual rails and the average rail force, pre-buckling (+37,9°C). H4V4H4 8m20mm ...... 79
xix Figure 4.33: Bending moments Mzz in rail A along the railway track at a temperature increase of 37,9°C (pre- buckling). H4V4H4 8m20mm ...... 80
Figure 4.34: Rendering of the modeled frog and common crossing, for the original model (A) and modification 1 (B) ...... 81
Figure 4.35: (A) Fabrication of common crossing in Bascoup (50). (B) Photograph of the frog in combination with check rail (51)...... 81
Figure 4.36: Buckling curves for the different modifications of the H4V4H4 model with defect 8m20mm ...... 82
Figure 4.37: Comparison between the course of the total axial forces for different model configurations of the H4V4H4 diamond crossing with implemented track defect 8m20mm in zone 3 (pre-buckling temperatures) ...... 83
Figure 4.38: Comparison between the course of the total axial forces for different model configurations of the H4V4H4 diamond crossing with implemented track defect 8m20mm in zone 3 (post-buckling temperatures) ...... 84
Figure 4.39: Total axial forces along the railway track (summation of rail forces) at 훥T = 48,2°C (post-buckling), for modification 1 of model H4V4H4 8m20mm ...... 85
Figure 4.40: Axial strains in rail A along the railway track for different temperature increases along the buckling curve. H4V4H4 8m20mm ...... 86
Figure 4.41: Contour plot of the axial strains at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm ...... 86
Figure 4.42: Comparison between the buckling curves for the input data of the lateral ballast resistance according to Zacher and NMBS. H4V4H4 8m20mm ...... 88
Figure 4.43: Comparison between the buckling curves for a H4V4H4 diamond crossing with initial misalignment 8m20mm, with and without the presence of check rails ...... 88
Figure 4.44: Influence of increasing/decreasing lateral ballast resistance on the buckling curve. H4V4H4 8m20mm ...... 90
Figure 4.45: Summary of the parameter study for the H4V4H4 diamond crossing (track defect 8m20mm in zone 3) ...... 93
Figure 4.46: Critical zone for buckling in the diamond crossing H4V4H4 ...... 95
Figure 4.47: Contour plot of the lateral displacements of the railway track for a configuration with larger sleepers except at the location of the common crossing. H4V4H4 8m20mm ...... 95
Figure 4.48: Geometry of the diamond crossing for a configuration with sleepers with length 570 cm in the critical zone...... 95
Figure 4.49: Geometry of the diamond crossing for a configuration with sleepers with length 450 cm in the critical zone...... 96
Figure 4.50: Influence of the different modifications in the sleeper geometry on the buckling curve. H4V4H4 8m20mm ...... 96
Figure 4.51: Difference in sleeper spacing in zone 3 of H4V4H4. Left: spacing = 550 mm. Right: spacing = 705 mm...... 97
xx Figure 4.52: Simplified line models of the H4V4H4 (A) and H3V3H3 (B) diamond crossings ...... 99
Figure 4.53: Rendered mesh of the H3V3H3 model in Siemens NX ...... 99
Figure 4.54: Comparison between the buckling curves for the diamond crossing H4V4H4 and H3V3H3, with implemented track defect 8m20mm in zone 3 ...... 100
Figure 4.55: Drawing of a fishplated joint (8)...... 100
Figure 4.56: Detail of the 3 mm gap joint on plan [mm] ...... 101
Figure 4.57: Photograph of a fishplated joint between two rails ...... 101
Figure 4.58: Cross-section and geometry of fishplates (£600) (53) ...... 101
Figure 4.59: Locations were the fishplated joints are implemented in the model of the H4V4H4 diamond crossing ...... 101
Figure 4.60: Input data for the PGAP entry in Siemens NX ...... 103
Figure 4.61: Schematical representation of the implementation of the CGAP element and modeling of the fishplate ...... 103
Figure 4.62: Contour plot of the longitudinal displacements at 훥T = 40°C. H4V4H4 8m20mm with fishplated joints ...... 104
Figure 4.63: Contour plot of the lateral displacements at 훥T = 40°C. H4V4H4 8m20mm with fishplated joints ... 104
Figure 4.64: Comparison between the buckling curves for a continuously welded diamond crossing and a diamond crossing connected with fishplated joints. H4V4H4 8m20mm ...... 105
Figure 4.65: Contour plot of the lateral displacements at 훥T = 53,8°C (post-buckling). H4V4H4 8m20mm with fishplated joints ...... 105
Figure 4.66: Course of the axial forces in rail A along the railway track for different temperature increases. H4V4H4 8m20mm with fishplated joints ...... 106
Chapter 5 Model of a diamond crossing with double slips
Figure 5.1: Photograph of a diamond crossing with double slips (56) ...... 108
Figure 5.2: Plan view of the geometry of the TJD EUH4 diamond crossing with double slips in Siemens NX ...... 109
Figure 5.3: Designation of the different zones considered in the TJD EUH4 ...... 109
Figure 5.4: Part of junction rails (orange) that lie in the defect zone, but have not applied an initial misalignment ...... 110
Figure 5.5: Extract of the plan of the TJD EUH4, showing the abraded rail foot of the stock rail at the location of the tongue (57) ...... 111
Figure 5.6: Photograph illustrating the connection of the stock rail and movable part of the switchblade with gliding chairs in a classical turnout (58) ...... 111
Figure 5.7: Meshed geometry of the TJD EUH4 in Siemens NX ...... 112
Figure 5.8: Three-dimensional view of the meshed geometry of the TDJ EUH4 in Siemens NX ...... 112
Figure 5.9: Load-displacement curves for different defect configurations in the TJD EUH4 diamond crossing with double slips. Comparison with the results for the H4V4H4 diamond crossing ...... 113
xxi Figure 5.10: Comparison between the buckling curves for model configurations of the TJD EUH4, with and without adjustment of the lateral ballast characteristics to the width of the sleeper ...... 114
Figure 5.11: Designation of the points along the buckling curve where the results are taken. TJD EUH4 8m20mm ...... 115
Figure 5.12: Plot of the distance measured along the railway track applied to the TJD EUH4 model (contour plot of the lateral displacements) ...... 115
Figure 5.13: Lateral displacements along the railway track for different temperature increases along the buckling curve. TJD EUH4 8m20mm ...... 116
Figure 5.14: Designation of the naming convention of the rails in the diamond crossing with double slips ...... 116
Figure 5.15: Contour plot of the longitudinal displacements at 훥T = 58,6 °C (post-buckling), according to the global x-axis. TJD EUH4 8m20mm ...... 117
Figure 5.16: Longitudinal displacements of rail C along the railway track for different temperature increases. TJD EUH4 8m20mm ...... 117
Figure 5.17: Axial strains in rail C along the railway track at a temperature increase 훥T = 40°C. TJD EUH4 8m20mm ...... 118
Figure 5.18: Relative longitudinal displacements of rail C and sleepers underneath along the railway track at different temperature increases. TJD EUH4 8m20mm ...... 119
Figure 5.19: Axial strains in rail C along the railway track for different temperature increases along the buckling curve. TJD EUH4 8m20mm ...... 119
Figure 5.20: Longitudinal displacements of rail A along the railway track for different temperature increases along the buckling track. TJD EUH4 8m20mm ...... 120
Figure 5.21: Axial strains in rail A along the railway track for different temperature increases along the buckling curve. TJD EUH4 8m20mm ...... 121
Figure 5.22: Forces in the springs representing the lateral ballast resistance along the railway track for different temperature increases. TJD EUH4 8m20mm ...... 122
Figure 5.23: Total axial forces after summation of the axial forces of the individual rails, along the railway track for different temperature increases (pre-buckling). TJD EUH4 8m20mm ...... 123
Figure 5.24: Total axial forces after summation of the axial forces of the individual rails, along the railway track for different temperature increases (post-buckling). TJD EUH4 8m20mm ...... 124
Figure 5.25: Comparison between the summation of the axial forces of the rails along the railway track for a calculation with and without the axial forces in the check rails. TJD EUH4 8m20mm ...... 124
Figure 5.26: Different zones of rail section in the TJD EUH4 diamond crossing with double slips ...... 124
Figure 5.27: Comparison between the observed total axial force and nominal thermal forces for different configurations, at 훥T = 40,0°C. TJD EUH4 8m20mm ...... 125
Figure 5.28: Comparison between the course of the axial forces of the individual rails (A, B, G and H) and the average rail force, pre-buckling (+1,67°C). TJD EUH4 8m 20mm ...... 126
Figure 5.29: Illustration of the direction of the implemented track defect ...... 126
xxii Figure 5.30: Comparison between the course of the axial forces of the individual rails (A and B) and the average rail force, pre-buckling (+40,0°C). TJD EUH4 8m 20mm ...... 127
Figure 5.31: Comparison between the course of the axial forces of the individual rails (C, D, E and F) and the average rail force, pre-buckling (+40,0°C). TJD EUH4 8m 20mm ...... 127
Figure 5.32: Comparison between the course of the axial forces of the individual rails (A and B) and the average rail force, post-buckling (+58,6°C). TJD EUH4 8m 20mm ...... 128
Figure 5.33: Comparison between the course of the axial forces of the individual rails (C, D, E and F) and the average rail force, post-buckling (+58,6°C). TJD EUH4 8m 20mm...... 128
Figure 5.34: Summary of the parameter study for the TJD EUH4 diamond crossing with double slips (track defect 8m20mm in zone 3) ...... 131
Figure 5.35: Geometry of an old TJD EUH4: dashes = connection with fishplates, dots = welds ...... 132
Figure 5.36: Extract of the line model of the TJD EUH4 with fishplated joints in Siemens NX, with modeling of the entire switchblades ...... 132
Figure 5.37: Contour plot of the longitudinal displacements at a temperature increase of 10°C. TJD EUH4 8m20mm with fishplated joints ...... 133
Figure 5.38: Comparison between the buckling curves for the model of a continuously welded diamond crossing with double slips and the model with fishplated joints. TJD EUH4 8m20mm ...... 133
Chapter 6 Comparison between diamond crossing and diamond crossing with double slips
Figure 6.1: Simplified line model of the diamond crossing H4V4H4 and TJD EUH4 (47) ...... 135
Figure 6.2: (A) Rail geometry of the diamond crossing H4V4H4. (B) Rail geometry of the diamond crossing with double slips TJD EUH4 ...... 136
Figure 6.3: Comparison between the lay-out of the sleepers of the H4V4H4 (blue) and the TJD EUH4 (red)...... 136
Figure 6.4: Line model of TJD EUH4 with removal of the slips (configuration 3) ...... 137
Figure 6.5: Line model of the H4V4H4 diamond crossing with the sleeper layout of the TJD EUH4 ...... 138
Figure 6.6: Comparison between the load-displacement curves for the different considered configurations ...... 139
Figure 6.7: Comparison between the change in critical buckling temperatures for the H4V4H4 diamond crossing and the TJD EUH4 diamond crossing with double slips, subjected to a percentage change in the lateral ballast resistances ...... 140
Figure 6.8: Comparison between the change in critical buckling temperatures for the H4V4H4 diamond crossing and the TJD EUH4 diamond crossing with double slips, subjected to a percentage change in the longitudinal fastener resistances ...... 141
Chapter 7 Model of a switch diamond
Figure 7.1: Photograph of a curved switch diamond (60) ...... 142
Figure 7.2: Meshed geometry of the (XZX)1/8 switch diamond in Siemens NX ...... 143
Figure 7.3: Load displacement curves for different configurations in the (XZX)1/8 switch diamond. Comparison with the results for the H4V4H4 diamond crossing...... 144
Figure 7.4: Designation of the points along the buckling curve where the results are taken. (XZX)1/8 8m20mm 145
xxiii Figure 7.5: Plot of the distance measured along the railway track applied to the (XZX)1/8 switch diamond (contour plot of the lateral displacements) ...... 146
Figure 7.6: Lateral displacements along the railway track for different temperature increases. (XZX)1/8 8m20mm ...... 146
Figure 7.7: Lateral displacements along the railway track in post-buckling regime. (XZX)1/8 8m20mm ...... 147
Figure 7.8: Forces in the springs representing the lateral ballast resistance along the railway track at the post- buckling temperature increase +61,2°C. (XZX)1/8 8m20mm ...... 148
Figure 7.9: Designation of the naming convention of the rails in the diamond crossing with switchblades ...... 148
Figure 7.10: Longitudinal displacements of rail A along the railway track for different temperature increases. (XZX)1/8 8m20mm ...... 149
Figure 7.11: Course of the axial strains in rail A along the railway track at a temperature increase of 40°C (pre- buckling). (XZX)1/8 8m20mm ...... 149
Figure 7.12: Axial strains in rail A along the railway track for post-buckling temperature increases. (XZX)1/8 8m20mm ...... 150
Figure 7.13: Transition zones in axial rail forces in the switch diamond ...... 151
Figure 7.14: Total axial forces after summation of the axial forces of the individual rails along the railway track, for different temperature increases. (XZX)1/8 8m20mm ...... 151
Figure 7.15: Comparison between the course of the axial forces for the models with and without the modeling of the Z-block, 훥T = 40°C. (XZX)1/8 8m20mm ...... 152
Figure 7.16: Axial forces in rail A and B at a temperature increase of 40°C for the model with the Z-block replaced by 60E1 rails. (XZX)1/8 8m20mm ...... 153
Figure 7.17: Axial forces in stock and switch rails of P3350 for 훥T=40°C. Results obtained by Landuyt (7) ...... 153
Figure 7.18: Comparison between the course of the axial forces of the individual rails and the average rail force, pre-buckling (+40,0°C). (XZX)1/8 8m20mm...... 154
Figure 7.19: Comparison between the course of the axial forces of the individual rails and the average rail force, pre-buckling (+57,7°C). (XZX)1/8 8m20mm...... 155
Chapter 8 Conclusions and further research
Figure 8.1: Extract of the plan of the setup in Kinkempois (62) ...... 160
xxiv List of tables
Table 2.1: Allowable temperatures according to safety level 2, second approach (17)...... 10
Table 2.2: Summary of lateral ballast resistances per sleeper found in the literature ...... 26
Table 2.3: Contributions of the different parts of the sleepers to the lateral ballast resistance found in the literature ...... 27
Table 2.4: Intervention limits and immediate action limits for the measured defect amplitudes (measurement principles D1 and D2) according to “Reglementaire Technische Voorschriften Baan – Bundel 2” (42) 35
Table 3.1: Nominal stiffness values and limit resistance forces for the lateral and longitudinal ballast resistances in function of the sleeper length ...... 48
Table 4.1: Critical buckling temperature increases for different model configurations of the diamond crossing H4V4H4 ...... 62
Table 4.2: Comparison axial forces obtained from the finite element model and calculated with formula (4.1). H4V4H4 8m20mm ...... 75
Table 4.3: Total elongation of rail A for different temperature increases along the buckling curve. H4V4H4 8m20mm ...... 87
Table 4.4: Overview of the critical buckling temperature increases and lateral displacements at ≅+40°C for different model configurations. Solutions to improve the buckling strength of the H4V4H4 diamond crossing 97
Table 4.5: Expected increase in lateral ballast resistance for different reinforcing applications ...... 98
Table 5.1: Critical buckling temperature increases for different defect configurations in the TJD EUH4 diamond crossing with double slips ...... 113
Table 5.2: Total elongation of rail A for different temperature increases along the buckling curve. TJD EUH4 8m20mm ...... 121
Table 5.3: Comparison axial forces calculated with (4.1) for 60E1 and 50E2 rail profiles ...... 122
Table 7.1: Critical buckling temperature increases for the (XZX)1/8 switch diamond ...... 145
Table 7.2: Total elongation of rail A for different temperature increases along the buckling curve. (XZX)1/8 8m20mm ...... 150
xxv List of abbreviations and symbols
Abbreviations
BEL Buckling energy level
DOF Degree of freedom
Symbols
Roman upper case letters
A Cross-sectional area of the rail
B Width of the sleeper
D Depth of the sleeper
E Young’s modulus of elasticity
L Sleeper length
L0 Standard sleeper length = 2,7 m
Mzz Bending moment around the z-axis of a beam’s cross-section
T0 The neutral temperature of the rail, the temperature at which the rails are stress-free (no compression or tensile forces present)
T Actual rail temperature
Tallowable The allowable temperature increase of the rails above the neutral temperature to be safe for rail buckling
Upper critical temperature above which buckling will certainly occur Tb,max
Lower critical temperature above which buckling can occur with sufficient external energy Tb,min applied to the rail
Upper critical temperature increase above the neutral temperature, above which buckling will Tb,max certainly occur
Lower critical temperature increase above the neutral temperature, above which buckling can Tb,min occur with sufficient external energy applied to the rail
Roman lower case letters
Coefficient of thermal expansion
xx Axial strain in the rail
0 Initial amplitude of the misalignment
xxvi Chapter 1 Introduction
In the early history of railroad engineering, engineers aimed to decrease the number of expansion joints in the railway tracks. The main reason why the technique of continuous welded railway track was not introduced at first was the concern that the high compressive forces, that would build up during hot summer days because of the restrained thermal expansion, would cause instability of the track. The application of continuous welded rails for ballasted track became one of the greatest technical achievements in railway engineering in the twentieth century. Since the Second World War, many kilometres of continuous welded rail track have been constructed, but the accompanying problem of lateral track buckling did not receive the full attention of engineers at first. As a consequence, several accidents with instabilities of railway track occurred throughout history. Photographs of buckled railway tracks in Figure 1.1 and Figure 1.2 show the results of track instability due to the neglect of a detailed study towards the structural behaviour of continuous welded railway tracks.
Figure 1.1: Examples of lateral track buckling occurred throughout history (1)
Figure 1.2: Examples of recent lateral track buckling accidents (2)
Introduction 1 During the twentieth century, the phenomenon of lateral track buckling increasingly became subject of deeper research. The first performed buckling tests were however not suitable for the determination of buckling temperatures. Arnold D. Kerr (1) was the first author in 1975 to publish a critical survey of the results of track buckling for tangent tracks as well as of analyses of track buckling in the lateral plane. An improved analysis of the static buckling temperatures including the effects of non-linearity followed in 1980 by Kerr (3). In 1985, theoretical and experimental studies on track buckling for tangent and curved tracks were performed by Samavedam et al. (4). Finally, the software program CWERRI for the analysis of buckling of continuous welded railway track was created, of which the properties were discussed by Van in 1997 (5). This program allowed to determine the buckling temperatures for several railway geometries, in plain and curved track, and to model the bridge-rail interaction.
Buckling of track which is induced by a combination of dynamic vehicle loads and thermal loads is called dynamic buckling or vehicle-induced buckling. When the buckling is only caused by thermal loads, it is called static buckling (6) of which Figure 1.3 is an example. In this master dissertation, the focus is on the determination of the critical buckling temperatures for static buckling. However, the majority of accidents with buckling did occur under the influence of train loads. The vehicle loads thus have an important influence on the buckling mechanism, which has always to be taken into account when performing a safety assessment.
Figure 1.3: Example of static lateral buckling of railway track (6)
Landuyt (7) investigated the buckling behaviour of different turnouts with a finite element model created in Siemens NX. The results were compared with existing studies. At present, no available literature exists about the safety for lateral track instability of crossings. Particularly a diamond crossing, a diamond crossing with double slips and a switch diamond are three different track parts that have a complex geometry, in which the structural discontinuities make these models vulnerable for buckling. In this master dissertation, the buckling behaviour of these railway track geometries and the influence of different track parameters on the safety will be investigated. The geometries will also be implemented in the finite element software Siemens NX in this respect. The main goal of this research is to acquire a better insight in the critical temperature increases during hot summer days at which a track instability could occur in certain circumstances. Besides that, the internal forces in the rails at pre-buckling temperatures will be determined, because these are of importance to control the fatigue life of the track.
Introduction 2 Chapter 2 Literature overview
2.1 Introduction
Historically, railway track was constructed with rails with a length of typically 27 m, connected with so called fishplated joints (8). The application of these joints was necessary because of two reasons. Firstly, the fabrication length of rails was limited and secondly, to accommodate the track for thermal expansions without increase of the compression forces in the rail. The joints are accompanied with small margins, that open in cold weather and close when the rails are expanding. The fishplate is therefore prone to wear because of the continuously modifying rail length. Subsequently, these joints require a lot of maintenance. In case of railway track connected by joints, high dynamic forces are introduced in the rails and the comfort of the train passengers is moreover not optimal (8).
With the development of new welding techniques, it became possible to connect almost unlimited rails by welds. The introduction of continuous welded railway track had several advantages over jointed track. Continuous welded track reduces the maintenance costs, increases the service life of both track and vehicles by reducing the impact loads from the wheels, increases the passenger comfort, decreases the traction energy consumption and decreases the emission of noise (5). It also provided the technical possibility for the construction of modern high speed railway lines, offering an ecological solution for transportation problems facing contemporary society. It goes without saying that the application of continuous welded railway track was the most important breakthrough in railway engineering in the past century.
In continuously welded railway track, expansion of the rails due to a temperature increase is restricted. High compressive forces are subsequently the result during hot days, which increases the risk of track instability. At very low temperatures in turn, the rails can break or a permanent inward settlement of the track in low radii curves may occur (9). In the latter case, the rails will most likely break at a welding point where the resistance against fracture is weaker. Two examples of rail fractures are given in Figure 2.1 and Figure 2.2. A similar fracture line is observed in both cases. A large gap is present between both parts of the rail after the failure occurred, but this defect is detected rapidly by the railway signalling system (8) and the risk of derailment is low. This makes fracture of rails a less severe accident than rail buckling.
Literature overview 3
Figure 2.1: Example of a rail fracture due to high tensile stresses in cold weather (10)
Figure 2.2: Example of rail fracture occurring during the night of 25-26/1/2012 at Remersdaal leading to the derailment of a boxcar (11)
Lateral buckling of railway track is particularly dangerous because of the deviation from its initial, correct geometric position. This causes a yaw angle of the wheelsets favouring the emergence of creep. The resulting misalignments can moreover cause train derailments at operating speeds (4). It is of importance to prevent rail instabilities and major displacements to ensure the safety of the passengers and the operation of the railway track. A good understanding and prediction of the critical buckling temperatures for different railway geometries is thus desired. Especially in turnouts and crossings, the rails are submitted to variations in compression forces because of the structural discontinuity in track (8). A small peak in compression force may thus be critical to initiate track buckling. The risk is managed by a sufficient large lateral resistance and the necessary regulations for maintenance (12).
Following literature overview presents the gathered information about the continuously welded rail buckling phenomenon, the parameters influencing the structural behaviour of railway track and how this was implemented in a safety assessment. A better insight in the parameters which influence the track behaviour and buckling was obtained, which forms the basis of the research being the subject of this master dissertation.
Literature overview 4 2.2 Thermal loads on railway track
The neutral temperature T0 of the railway track is the temperature at which the longitudinal stresses in the rail are zero. In continuous welded railway tracks, the longitudinal deformations are prevented at sufficient distance from the free ends. This means that a uniform temperature increase or decrease results in a longitudinal static force developing in the rails:
FEATTx = ( 0 − ) (2.1)
With: E Young’s Modulus A Cross-sectional area of the rail 훼 Coefficient of thermal expansion
T0 Neutral rail temperature T Actual rail temperature
Apart from the outside temperature depending on climatic conditions, the temperature of the rails T also depends on the orientation of the sun, shading, wind speed and direction, interaction with other structures, etc. The maximum rail temperature is higher than the maximal outside climate temperature. Esveld (13) states that the difference between the outside air temperature and the actual rail temperature in sunny weather could be up to 20°C. Samavedam (14) mentions the same temperature difference. In Belgium, the outside air temperature can be as high as 40°C according to Landuyt (7). Eurocode 1 (NBN EN 1991-1-5 ANB) (15) prescribes a maximal outside temperature of 38°C, which can indeed be rounded to 40°C. Adding to this value the difference in air temperature and rail temperature in sunny weather of 20°C, the maximal rail temperature can reach values of up to 60°C.
Brake systems also develop thermal loads on the rail. The eddy current brakes create friction between wheel and rail and also generate eddy currents in the rail, resulting in an increase in rail temperature. Van (5) mentions a study by Keass (1996) that states that a rail heating of 20°C could be expected with a brake load of 90 kN per rail for trains that pass every 10 minutes. For the case of a 180 kN brake load, the increase can even be 40°C. Zacher (12) reports about investigations that showed that the temperature of the rails increases by about 2,8°C in case of a single in-service braking. For six in service brakings per hour, the temperature increase could be up to 21°C high. Adding this to the maximal expected rail temperature during hot summer days of 60°C, the temperature in the rail can even reach a value of 81°C. The maximal temperature increase above the neutral temperature that can be expected in a rail with a neutral temperature of 20°C can thus be as high as 61°C.
The additional temperature increase is generated by the eddy currents of magnetic braking systems. But the use of these are not allowed on every railway section in Belgium according to Jan Mys (16). Especially in crossings, which are the subject of investigation in this dissertation, the in-service speed of passing trains is low and magnetic braking systems are not used. The maximal rail temperature that can be expected in crossings is thus estimated to be 60°C. The reference value for the maximal temperature increase that can occur in a railway track with a neutral temperature of 20°C is set to 40°C.
Literature overview 5 2.3 Buckling mechanism and response
2.3.1 Track instability
To investigate the buckling behaviour of track in a finite element model, it is assumed that it occurs in the lateral plane under the influence of thermal loads only (static buckling). The influence of trains is in this dissertation thus disregarded, because the exact theoretical implementation of dynamic loads in a finite element model is too complex. However, in reality wheel loads can influence the buckling load in two ways (4).
Firstly, the wheels can generate lateral loads on the rails and tend to move the track in the lateral direction, which may contribute to the generation of misalignments. Secondly, passing trains yield a nonuniform pressure on the sleepers. The bottom resistance of the sleepers increases at locations under and near the wheels, while at a distance further away from the wheel loads, the lateral bottom resistance can decrease compared to the static value or even disappear due to uplift of the track. This means that the contribution of the crib and the shoulder to the lateral ballast resistance becomes of great importance in resisting buckling. The zones which are uplifted due to passing trains are thus prone to buckling at a lower temperature than the statically determined one. These considerations are subject of a dynamic buckling analysis, which is not the aim in this dissertation. The majority of the buckles, however, occur under a combination of temperature loads with longitudinal braking or accelerating loads from passing trains.
Track buckling is essentially a local phenomenon. The tendency of the ballasted track to buckle in the lateral direction in a horizontal plane is opposed by its lateral resistance in the ballast, the torsional resistance generated by the fastening systems and also by the longitudinal resistance of the rail fasteners (4). Buckling occurs in general from a small misalignment in track. Straight tracks with no noticeable geometric imperfections could buckle suddenly with a loud bang, whereas the tracks with large horizontal imperfections buckle gradually and more quietly (1). The misalignment significantly reduces the buckling strength compared to a perfectly straight track.
The buckling phenomenon is characterised by a load-displacement curve, plotting the temperature increase of the rail above the neutral temperature versus the lateral displacement of the track. The typical plot of this relationship for a track without imperfections is illustrated in Figure 2.3A. At the neutral temperature T0, no normal forces are present in the rails. When the temperature increases, the compression forces increase elastically up until a point where no further increase in temperature is possible and the track becomes unstable in the lateral direction. In general, buckling starts at a temperature Tb,max, after which the temperature decreases in the post buckling branch for a displacement controlled test. For weak tracks in poor ballast, the shift towards the maximal critical temperature is more gradual. Tb,max is strongly dependent on the initial imperfection and is an upper boundary for the temperature at which buckling will occur. A typical bucking shape of ballasted railway track is illustrated in Figure 2.4. Here, also the forces in the ballast bed which resist lateral buckling are indicated.
Literature overview 6 A B
Figure 2.3: (A) Typical load-displacement curve for track buckling. (B) Buckling energy concept (17)
Figure 2.4: Typical buckling shape (17)
In the post-buckling branch of the load-displacement curve, a minimal temperature Tb,min is present. Below this temperature, there is no risk for buckling. Above this temperature and below the upper critical buckling temperature
Tb,max, buckling can still occur if sufficient external energy or disturbance is applied to the track. This energy could for example originate from a passing train. The track will in this case snap-through towards a post-buckling stable configuration. At the upper critical buckling temperature, there is no extra energy needed for the track to buckle out. The temperature range between the upper and lower critical buckling temperature ( = TTbb,max − ,min ) is designated as the ‘buckling regime’. Obviously, the temperature range between the two critical boundaries is no safe region for the railway to operate.
The external supplied energy required for the track to buckle within the buckling regime is illustrated in Figure
2.3B. At the lower critical buckling temperature, this required energy (designated with Emax in the figure) is maximal and it decreases for higher temperatures. The minimal energy required at the lower critical buckling temperature
(= Emax) increases when the buckling regime becomes larger (6). This means that for track configurations with a large difference in upper and lower critical buckling temperature (high ballast quality, high lateral ballast resistance, only a minor misalignment …) a large amount of external energy is needed before the track buckles at a temperature corresponding with the lower critical buckling temperature. This aspect is of importance to establish safety criteria for track stability.
Literature overview 7 In case of a very low ballast resistance (tamped track, low ballast quality, erosion of ballast, etc) or a track with a small track curve radius, progressive track shifting is found without a snap-through being possible. In this case, the difference between upper and lower critical temperatures is not found and a safety criterium should be based on a limitation of the lateral displacement of the track. The influence of the ballast quality on the buckling curve is schematically illustrated in Figure 2.5. The shape of the buckling curve also depends on the track parameters and strength conditions, as will be clear from the next paragraphs.
A B
Figure 2.5: Temperature increase in function of the lateral displacement in case of good ballast characteristics (A) and poor ballast characteristics (B) (2)
The critical buckling temperatures may be reduced by alternating temperatures (18). The accumulation of lateral track deflections during high temperature cycles may contribute to a lower critical temperature than this given by the analysis with an assumption of a monotonic temperature rise. An initial track defect can thus grow during a number of consecutive hot summer days, which reduces the buckling strength of the railway track.
2.3.2 Buckling mechanism in relation to track imperfections
The initial configuration of the continuous welded railway track is in practice one with a small initial lateral misalignment. It can be described by a sinusoidal curve, with amplitude 0 and wavelength 2⋅L0 as illustrated in Figure 2.6. The compressive force P increases with increasing temperature, yielding growth of the misalignment.
The initial amplitude of the misalignment increases to wB, which is an unstable equilibrium state. At this point the track will buckle suddenly into a new stable configuration with a lateral deflection wC. The sinusoidal curve now spans a larger length of 2⋅L. The lateral deflection after buckling is typically large, ranging from 15 cm to 75 cm.
Figure 2.6: Track geometry before and after buckling (6)
Literature overview 8 The typical buckling response curve for track with an initial misalignment is graphically illustrated in Figure 2.7, in which the points B and C refer to the typical displacements set out in Figure 2.6. Multiple in-between positions between the lower and upper buckling temperature can exist. An equilibrium position along the curve can either be stable or unstable. The dashed line between the neutral temperature and point B (Figure 2.7A) is stable. The positions along the dotted line between the upper and lower critical buckling temperature is in turn unstable and the dashed line after the lower critical temperature is again stable. In real conditions, only the stable positions on the curve can physically be present (6).
Point B is called the bifurcation point and is the common point of stable and unstable positions. Here the track snaps over to point C, which is a stable configuration. With a sudden explosive buckling, the rail forces drop in the buckled zone of the track, caused by the large displacements in the lateral direction which yield an extension of the rail. If sufficient external energy is applied to the track, the track configuration can jump from a point in the pre- buckling branch to a stable post-buckling point, as illustrated by the line 1-2-3 on the graph in Figure 2.7B. The range of temperatures between the lower and upper buckling temperature, the buckling regime, represents the region where track can buckle by snap-through.
A B
Figure 2.7: Typical buckling response curves for continuously welded railway track (6). (A) buckling at the upper critical buckling temperature. (B) Buckling at a temperature lower than the upper critical buckling temperature (snap-through)
2.3.3 Safety approach
With the typical buckling behaviour in relation to the load-displacement curve set out, a safety approach for track stability is determined. The allowable temperature for the railway track to operate safely depends on the desired level of safety:
− Level 1 safety: TTallowable= b,min
− Level 2 safety: TTTallowable= b,min +
The level 1 of safety is a conservative approach: buckling will never occur independent of the amount of energy that is added. Level 2 is less conservative, as the allowable temperature is higher than the lower critical buckling temperature. The determination of ΔT is based on safety considerations as the buckling potential of the track increases rapidly above the lower critical temperature Tb,min, as illustrated in Figure 2.3B. Here, two approaches are applied by Esveld (17).
Literature overview 9 In the first approach for the determination of ΔT, the relationship between the external buckling energy and the temperature increase is used for establishing the criterion. The necessary safety is obtained at a temperature at which a finite buckling energy exists that is larger than zero, but less than the maximal value at Tb,min:
TTallowable= 50% BEL . The buckling energy level (BEL) is determined from a software program and it is suggested to use the 50% BEL (see Figure 2.3B).
The second approach does not require the determination of the buckling energy and an alternative definition of ΔT is based on the prediction of Tb,max and Tb,min. For the different ranges between the upper and lower critical buckling temperature, the allowable temperatures in this safety approach are given in Table 2.1. The case where progressive buckling occurs ( TC 0 ) is not allowable. This is the case if the track is recently tamped in combination with an extremely low ballast quality, where elastic and plastic lateral deformation easily fade into each other. For small buckling regimes, the allowable temperature is even lower than the lower critical buckling temperature.
Table 2.1: Allowable temperatures according to safety level 2, second approach (17)
Range of application Allowable temperature
TTCbb,max − ,min 20 TTTTallowable= b,min +0,25 ( b ,max − b ,min ) (2.2)
5CTTC bb,max − ,min 20 TTallowable= b,min (2.3)
05CTTC bb,max − ,min TTCallowable= b,min −5 (2.4)
TTCbb,max − ,min 0 Not allowable in main lines
2.3.4 Buckling influence zone
Simultaneous with the buckling of track, the compressive forces in the rail drop due to energy release (6). The large lateral displacements yield extension of the rails, causing the decrease of the compressive loads. Together with the lateral deflections in the buckling zone, a longitudinal motion will also take place in the adjoining zones of the track, which can be a long section of the continuous welded railway track. This principle is illustrated in Figure 2.8. Together with the released longitudinal forces, the neutral temperature of the influenced zone of railway track is altered. This means that if buckling has occurred, a large section of continuous welded track has to be repaired and restressed to obtain the desired neutral temperature again.
Figure 2.8: Comparison between the distribution of the longitudinal rail forces before and after buckling has occurred (6)
Literature overview 10 2.3.5 Buckling modes
Field tests revealed that buckling often occurs in the lateral plane. However, if the track is sufficiently restrained in the lateral direction, for example in the case of metal sleepers, buckling can occur out of plane (in vertical direction) or a combination of lateral and vertical displacements can be observed (19). But in general, vertical buckling is rare because of the high rail stiffness in this direction (6). The application of metal sleepers is also not common.
The lateral buckling modes are either symmetrical or anti-symmetrical, mainly depending on the initial imperfection, as illustrated in Figure 2.9. In case of symmetric misalignments, buckling mode shape III is mostly observed in tangent, straight track segments, while curved tracks show shape I modes (4). Calculations (20) have revealed that the differences in critical temperatures between the different modes are rather small.
Figure 2.9: Possible buckling modes for continuously welded railway track (4)
2.3.6 Radial breathing
Radial breathing occurs in high degree weak curves in railway track. With this phenomenon the curves move radially outwards with an increase in temperature and inwards with a decrease in temperature. This effect yields a reduction in axial rail forces and thus has a beneficial effect on the buckling temperature. Figure 2.10 illustrates that larger curvatures result in a higher critical buckling load thanks to the phenomenon of radial breathing (4). However, to obtain the benefit of increasing buckling strength due to radial breathing, the lateral ballast resistance should be low enough to allow breathing. But this is in turn undesirable from a structural point of view (4). It is noticed in the plot that for high degree curvature a smaller lateral ballast resistance (Fp) indeed yields a higher critical buckling temperature. The influence of the lateral ballast resistance on the effect of radial breathing is less pronounced for smaller degrees of curvature. The phenomenon of radial breathing can moreover change the neutral temperature of the track.
Literature overview 11
Figure 2.10: Influence of radial breathing on the critical buckling temperature increase for increasing curvature (4)
2.3.7 Stress state and stress distribution in (buckled) rails
Buckling occurs at locations in track where a small misalignment is present. The bending stiffness of the rail contributes to the resistance to lateral track buckling. This means that the rails do not only face longitudinal axial stresses, but also bending stresses when the misalignment in track is growing during temperature increase. Because the lateral forces in these small misalignment curves are introduced by the fastening systems in the rail foot, the bending of the rail is not straight. The neutral axis of the rail section is shifted, resulting in crooked bending (21). The principle is illustrated in Figure 2.11A. As a consequence, stress peaks are present at one side of the rail that will accelerate fatigue of the rail. Figure 2.11B shows the evolution of the calculated longitudinal stress due to bending in a rail for varying defect amplitudes and wavelengths. The highest stresses are obtained for small wavelengths with large amplitudes, because in these cases the bending stresses become important. For a wavelength of 8 m and an amplitude of 20 mm, the additional stress due to bending is in the order of magnitude of 25 MPa.
As outlined by Wegner (21), the bending stress alone does not result in danger for track instability. However, the eccentrical impact of the resulting longitudinal forces does have an influence on the buckling of a track. The eccentric force indeed favours the lateral displacements and bending of the rail. This is also the reason why larger initial misalignments lead to lower critical buckling temperatures.
The peak stresses can also occur in local, sudden changes in cross-section. This can for example be the case at the location of the tongue of a switch, where the rail foot of the stock rail is abraded. According to Jan Mys (22), the stress concentration at this location can be three times as high than the nominal longitudinal stress in the rail. These considerations show that not only the critical buckling point itself is important in assessing the track safety, but also the peak stresses must be controlled. High peak stresses reduce the fatigue life of the rails.
Finally, one must also take into account the residual stresses introduced during the roller straightening process of the rail. Esveld (23) cites measurements performed by ORE D 148 to determine the distribution of the residual
Literature overview 12 stresses in new rails (as-rolled) and in used rails. The results are given in Figure 2.12. It is noticed that these stresses can reach values as high as 200 MPa. In used rails, the peaks residual stresses are smaller. The distribution of the residual stresses will not influence the critical buckling load, but they may become important in an assessment of the fatigue life of the rail.
A B
Figure 2.11: (A) Cross-sectional distribution of longitudinal stresses in a bended continuously welded rail (21). (B) Resulting longitudinal stresses due to bending in a continuously welded rail in function of the misalignment amplitude and wavelength of the misalignment (21)
Figure 2.12: Distribution of residual stresses as measured by ORE D 148 (23)
2.4 Static versus dynamic buckling
In this master dissertation, the buckling behaviour of continuously welded railway track is investigated for static loads induced by temperature differences only. However, it is important to notice that under train loads, the buckling response curve will show a different behaviour. Kish and Samavedam (6) investigated the difference in response between static and dynamic buckling for a track with wooden sleepers with a 5 degree curvature. The results are illustrated in Figure 2.13.
Literature overview 13 The buckling curve taking into account the effect of dynamic loads exhibits a smaller buckling regime compared to static buckling. The upper and lower critical buckling temperatures are moreover smaller in case of dynamic buckling. Subsequently, static buckling analysis does not yield the most conservative results with respect to the safety assessment of railway track. It must thus be kept in mind that a dynamic buckling analysis will probably result in smaller allowable temperature increases. In the results presented by Kish and Samavedam (6), the difference in upper critical buckling temperatures amounts to approximately 6 to 7°C.
Figure 2.13: Difference in buckling response curve for static and dynamic buckling (6)
2.5 Parameters influencing track buckling
The critical temperatures, shape of the buckling curves and buckling modes described in section 2.3 depend on several characteristics of the ballasted track. Apart from the geometry of the railway track, the parameters effecting the thermal buckling behaviour are:
− Neutral rail temperature − Rail type and sleeper type − Initial misalignment − Curvature radius − Torsional and longitudinal stiffness of the rail fastenings − Vertical ballast stiffness − Lateral and longitudinal ballast resistance − Creep
The influence of these characteristics on the stability of the track is examined in next paragraphs. The focus is on the qualitative influence and not on the exact critical temperatures. As only lateral track buckling under the influence of thermal loads is investigated in this dissertation, the effect of the vertical ballast stiffness on the buckling safety of track is not further pursued.
Literature overview 14 2.5.1 Neutral rail temperature
To assess the safety of the railway track, the critical temperature increase above the neutral temperature is one of the criterions, as was also outlined in section 2.3.3. With increasing neutral temperature, the safety against lateral buckling also increases. However, for large neutral temperatures, the risk for rail fracture in cold weather becomes larger. The measurement of the neutral rail temperature generally has an inaccuracy of ±3 °C (23). Continuous welded railway track composed by wooden sleepers has a less stable neutral temperature during service compared to railway tracks with concrete sleepers (23). Hence, the type of material of the sleeper indirectly influences the safety against buckling by the possible change in neutral temperature.
2.5.2 Sleepers
The main function of the sleepers is to support the rails and traffic loads. Besides that, they need to provide sufficient lateral resistance to prevent lateral movement of the rails in the ballast bed. The ballast resistances are altered by the type of sleeper. Not only the dimensions, but also the material (wood or concrete) influences the characteristics of the force resistance of the sleepers in the ballast. In general, track with concrete sleepers are regarded to have better resistance against buckling than track with wooden sleepers. This is attributed to the larger self-weight, size, shape and better conditions of the ballast quality (6). But the wooden sleepers have a higher friction coefficient with the ballast, smaller sleeper spacing and higher torsional resistance of the fasteners, so that wooden sleepers have their counter advantages.
A comparison between both type of sleepers was made by Kish and Samavedam (6) by keeping other parameters as much as possible constant. The results are presented in Figure 2.14. It is concluded that the wooden sleeper track has less resistance against buckling. The upper critical buckling temperature is higher for concrete tie track, at least. The lower critical temperature is lower in case of concrete sleepers on the contrary. The buckling response for wooden tie track is more gradual and tends towards progressive buckling, which is not desired. It can thus be concluded that the concrete sleepers provide higher safety against track instability compared to the same track constructed with wooden sleepers.
Figure 2.14: Comparison between the buckling response for wooden versus concrete sleepers in average ballast quality conditions (6)
Literature overview 15 2.5.3 Ballast
The ballast provides the most important resistance against track buckling. The lateral displacements of the rails are in the first place restrained by the sleepers that are embedded in the ballast bed. The characteristics of the structural behaviour of the ballast are thus of great importance to investigate the track stability of ballasted track. Next paragraphs give an overview of the influence of the different ballast characteristics on the critical buckling temperatures.
2.5.3.1 Lateral ballast resistance
The lateral ballast resistance describes the force performed by the ballast on the moving sleepers in the lateral direction. In general, for increasing lateral ballast resistance, the critical buckling temperatures increase (3) (4). Figure 2.15A and Figure 2.16A show the influence of changing lateral ballast resistance on the critical buckling temperatures. These plots depict the significant increase in buckling strength for raised lateral ballast resistance.
The shape of the buckling curve, and therefore also the critical temperatures, depends on the quality of the ballast. Figure 2.17 illustrates the characteristics of different ballast qualities and their influence on the load-displacement relationship for track buckling. Progressive buckling can occur for very weak track, which means that there is no difference in upper and lower critical buckling temperature. A well consolidated track has the highest peak lateral resistance and thus a greater resistance against lateral instability with a high upper critical buckling temperature.
A B
Figure 2.15: Buckling curves for varying lateral ballast resistance (A) and longitudinal ballast resistance (B) (3)
Literature overview 16 A B
Figure 2.16: Influence of the lateral peak ballast resistance (A) and longitudinal ballast resistance (B) on the critical buckling temperatures (23)
Figure 2.17: Influence of weak/average/strong ballast resistance on the buckling curves (6)
Literature overview 17 2.5.3.2 Longitudinal ballast resistance
The longitudinal ballast resistance describes the force performed by the ballast on the sleepers to resist their longitudinal displacements. The influence of the longitudinal ballast resistance on the critical buckling temperatures was investigated by Samavedam et al. (4), as illustrated in Figure 2.18. It is concluded that an increase in longitudinal ballast resistance mainly affects the lower critical buckling temperature Tb,min, which increases. The upper critical buckling temperature is almost independent of an increase in longitudinal ballast resistance.
The same conclusions are made by Choi & Na (3) and Esveld (23) in Figure 2.15B and Figure 2.16B. In fact, the buckling strength itself is not affected by the longitudinal ballast resistance as the pre-buckling branch in Figure 2.15 does not change with altering longitudinal ballast conditions. The small influence on the post-buckling branch can be explained by the fact that the longitudinal displacements of the sleepers in the ballast bed start to become important when the lateral deflections of the rails are growing extensively. The large lateral displacements of the sleepers can only occur if some longitudinal displacements of the sleepers take place. A larger longitudinal ballast resistance will thus yield the higher temperature loads for the same lateral displacement of the track in the post- buckling range, as observed in Figure 2.15B.
Figure 2.18: Influence of the longitudinal ballast resistance on the critical buckling temperatures (4)
The longitudinal ballast resistance is more crucial in controlling the neutral temperature variations. A larger resistance will thus yield a more stable neutral temperature and an increased safety against track buckling. It can be concluded that the longitudinal ballast resistance has only a minimal influence on the buckling strength of the track and only influences it indirectly by controlling the neutral rail temperature.
Literature overview 18 2.5.3.3 Effect of sleeper/ballast friction
The bottom surface roughness determines the component of the base resistance and is an important factor contributing to the ballast resistance. In case of wooden sleepers, the roughness increases with the age of the sleeper and the ballast tends to lock itself in the irregular bottom surface of the sleeper. Concrete sleepers tend to become smooth during service life due to the grinding effect between the stones and the sleeper and the base resistance will consequently decrease.
The influence of the roughness of the bottom surface on the buckling strength is studied by Samavedam et al. (4). The roughness factor was expressed by a friction coefficient, defined as the ratio of the measured base resistance to the self-weight of the sleeper. Increasing surface roughness increases the critical buckling temperatures, but the difference in temperature increase is rather small.
2.5.4 Rail fasteners
The rail fastening systems ensure the connection between the rails and the sleeper. The most important parameters characterizing these elements are the torsional resistance, the longitudinal resistance and the vertical uplift resistance. As only pure static lateral buckling is investigated, the vertical resistance of the fastening element is not addressed in this research.
2.5.4.1 Torsional fastener stiffness
The critical buckling temperatures increase with increasing torsional stiffness of the fastener, with the lower critical temperature Tb,min being more sensitive. Figure 2.19A illustrates the influence of this parameter on the thermal buckling strength investigated by Samavedam et al. (4). For high values of the torsional stiffness, the two critical buckling temperatures Tb,min and Tb,max tend to become equal, corresponding to progressive buckling. This occurs however at a higher temperature compared to the upper critical buckling temperature for a track with small torsional stiffness of the fasteners. This means that a higher torsional stiffness does not lead to a track with a smaller buckling strength (4).
The effect of changing torsional fastener resistance on the shape of the buckling curve is illustrated in Figure 2.19B. The same conclusions can be made. In these plots, it is seen that the lateral displacements of the track are almost unaltered in pre-buckling regime. From both figures it can be concluded that the torsional fastener resistance has the largest influence on the lower critical buckling temperature and thus on the post-buckling behaviour of the railway track. The upper critical buckling temperature is affected less, but the influence is nevertheless visible.
Literature overview 19 A B
Figure 2.19: (A) Influence of the torsional stiffness of the fastening systems on the critical buckling temperatures (4). (B) Buckling curves for varying torsional stiffness of the rail fasteners (3)
2.5.4.2 Longitudinal fastener stiffness
The fastening elements are necessary to restrain longitudinal movement of the rails in the sleepers. With buckling and lateral displacement of the track, the rails tend to move relatively to the sleepers. This parameter also plays a role in the transfer of compression forces in the rails during expansion. Landuyt (7) investigated the influence of the longitudinal stiffness of the fastener on the upper critical buckling temperature for a turnout. The results are given in Figure 2.20. The buckling strength decreases with increasing stiffness, which is explained by the increased force transfer to the sleepers triggering the buckling of rails.
Figure 2.20: Influence of the longitudinal fastener resistance on the upper critical buckling temperature for different track defects in case of a turnout (7)
Literature overview 20 2.5.5 Rail type and material
Samavedam et al. (4) concluded that both the upper and lower critical temperature Tb,max and Tb,min decrease with increasing rail size. This is explained by the fact that the longitudinal forces in the rail increase with larger cross- sections according to formula (2.1). The increase in bending stiffness does not outweigh these large compressive forces. It can be concluded that smaller rail sections improve the buckling strength. However small rail sizes are not preferred, because the fatigue life would reduce due to the increased bending stresses from wheel loads. From a theoretical calculation based on thermal buckling equations, Choi and Na (3) came to the same conclusions, as illustrated in Figure 2.21. It is reported that the upper critical buckling temperature is more sensitive to rail size changes.
Figure 2.21: Buckling curves for varying rail sizes (3)
2.5.6 Rail curvature
The results from Samavedam et al. (4) show that the critical buckling temperatures reduce for decreasing curve radii. For recently tamped (weak) ballast, progressive buckling can occur in curves with a radius of about 7 degrees and higher. A combination of small curvatures and recently maintained track is thus very vulnerable for buckling in summer. Choi and Na (3) investigated the buckling curves for varying track curvature, for which the results are presented in Figure 2.22. The same conclusions can be made. Esveld (23) as well performed simulations with CWERRI to determine the influence of the track curvature on the buckling temperatures, see Figure 2.23. From the results it is concluded that a decrease in track curvature radius lowers both the upper and lower critical buckling temperatures. The effect of radial breathing described in section 2.3.6 is less pronounced than the overall decrease in buckling strength for small curvature radii.
Literature overview 21
Figure 2.22: Buckling curves for varying degree of track curvature (3)
Figure 2.23: Influence of the track curvature radius on the critical buckling temperatures (23)
2.5.7 Initial imperfections
Instability of track is initialised by a local imperfection. A region of relatively weak ballast, with a corresponding reduction of the lateral and longitudinal ballast resistance may reduce the critical temperature increase. Also local lift-off of the rail-sleeper structure in front of or behind a wheel will tend to lower the critical buckling temperature (19). Apart from the local reduction in ballast resistance, the initial defect in the rail geometry influences the buckling behaviour.
The critical buckling temperatures decrease with decreasing wavelength of the track defect and with increasing amplitude of the misalignment (4). Figure 2.24 and Figure 2.25 show the results of investigations made by different authors. For increasing amplitude of the track misalignment, the buckling temperatures decrease. In Figure 2.25,
Literature overview 22 the critical buckling temperatures reach a minimum for a certain value of the wavelength, at approximately 12 m. However, defect wavelengths smaller than 8 to 10 m with a large amplitude (>10 mm) are not realistic for in-service track. The influence of wavelengths smaller than 8 m is thus disregarded. It can be concluded that for increasing wavelength and decreasing defect amplitude, the safety against buckling becomes larger.
A B
Figure 2.24: Buckling curves for varying misalignment amplitude (A) and wave length (B) (3)
A B
Figure 2.25: Influence of the misalignment amplitude (A) and half wave length of the misalignment (B) on the critical buckling temperatures (23)
Literature overview 23 2.5.8 Creep
Due to variations in temperature and traffic induced loads, the rails move longitudinally in track. This can either happen by longitudinal displacements of the rails relative to the sleepers or the rails and sleepers relative to the ballasted bed. The longitudinal movements are referred to as creep, which results in the build-up of compressive forces in some sections and tension forces in other sections. This creep yields a deviation of the neutral temperature of the rail and consequently results in areas of the track that are possibly more susceptible to buckling (24).
2.6 Structural parameters
In the previous section, the influence of the different track parameters on the critical buckling temperatures was discussed. In this section, a detailed look is given to the characteristics and values of these parameters, as presented in the literature. The aim is to obtain a critical survey of the existing test results. These findings will subsequently serve as input data for the implementation in the finite element models.
2.6.1 Lateral ballast resistance
2.6.1.1 Tests and characteristics
Results of tests to determine the lateral ballast resistance are rather scarce in the literature and data are rarely presented to isolate resistance due to base, crib and shoulder contact areas. It is often not clear how the resistance is investigated. Several test setups are possible, which have a different effect on the obtained resistance values. The nature of the test can be based on the resistance provided by the whole track system including the ability of the rails to spread the loads or the resistance provided by one single sleeper in the ballast. The resistance can also be determined by a test performed on a series of for example 5 sleepers. To evaluate the buckling behaviour of railway track, large displacements are expected and the tests need to be performed for static failure.
The two most used types of lateral sleeper tests are (25):
− Single sleeper push test: the load/deflection response is recorded of a detached sleeper that is pushed sideways by a machine attached to the rails. − Panel pull method: the load/deflection response is recorded of a section of in-service track that is pulled sideways from the rail head. The lateral resistance of the individual sleeper is then estimated from the results. This test can be performed with the section of track isolated or attached to the rest of the railway line (cut vs. uncut). The cut panel method gives the advantage that it allows an averaging of the individual sleeper contributions. For the uncut panel test however, it is difficult to determine the individual sleeper response as it is difficult to estimate and quantify the effect of the rails in spreading the lateral loads.
The lateral ballast resistance obtained from a single sleeper push test is often smaller than the resistance from a panel pull test. This has to do with to the interaction between the sleepers in the second test method (25).
Literature overview 24 The response of the sleeper in the ballast bed to lateral loads is dependent on many factors: material of the sleeper, spacing between the individual sleepers, the type and quality of the ballast, presence and height of the shoulder ballast, quality of the contact between the sleeper and the ballast, etc. These factors are rarely reported in literature and results of tests. It is therefore difficult to apply the correct values of lateral ballast resistance for the given geometry and characteristics of the railway track in a numerical model.
The characteristic shape of the lateral ballast resistance is illustrated in Figure 2.26. The parameters describing the shape include (26):
− A peak lateral ballast resistance FP with a corresponding lateral displacement wP
− A limit lateral ballast resistance FL with a corresponding lateral displacement wL For weak or tamped tracks, the peak and limit ballast resistances coincide. Under traffic loads, the ballast becomes consolidated and the response curve exhibits a clear peak in resistance, as was illustrated in Figure 2.17. From a buckling point of view, a freshly tamped track is more critical as the resistance to lateral displacements is smaller.
Figure 2.26: Typical lateral resistance vs displacement response curve of a sleeper in the ballast for consolidated and tamped track (25)
Comparable with consolidated or tamped track, the ballast quality has an important influence on the lateral resistance. Good quality of ballast yields a larger resistance. The quality of the ballast is mainly effected by the size of the stones. Dogneton (27) cites a study by Deutsche Bahn that has shown that a ballast of size 25x65 offers a lateral resistance of approximately 40% higher than ballast of size 15x35. Broken or crushed stone has a better resistance than gravel, which is caused by the higher friction between broken stone and the sides of the sleepers (27). Dogneton also concludes that a larger ballast size is generally advantageous, particularly for wooden sleepers. Lastly, polluted ballast tends to show larger lateral resistance compared to clean ballast (27).
2.6.1.2 Typical lateral ballast resistance values
Different idealisations of the response of the lateral ballast resistance have been proposed, such as a constant lateral resistance, a softening/dropping lateral resistance or a full non-linear lateral resistance. In most of the published literature about implementing the ballast characteristics in a theoretical model or a finite element model,
Literature overview 25 the lateral ballast resistance is modeled as a bilinear spring. This corresponds to a maximal resistance for a certain limit displacement and a constant tangential force for larger displacements. This resistance is obtained from tests performed on different configurations. Table 2.2 gives an overview of found values in the literature according to their specific track configurations. It can be concluded that the lateral ballast resistance generally has a maximum value between 6 and 10 kN per sleeper. This value is conservative, taken for the worst possible ballast conditions. Consolidated ballast can exhibit a peak lateral resistance of 15 kN and more.
Table 2.2: Summary of lateral ballast resistances per sleeper found in the literature
Source Type Lateral resistance [kN]
ERRI (28) (25) Peak lateral resistance within 20 mm deflection, 4,2 (min) - 6,9 (max) tamped loose track Peak lateral resistance within 20 mm deflection, 5,4 (min) - 15,7 (max) trafficked track Esveld (23) Peak/limit lateral resistance at 2 mm, low quality 7/7 ballast, wooden sleepers Peak/limit lateral resistance at 2 mm, high quality 15/12 ballast, wooden sleepers Peak/limit lateral resistance at 2 mm, low quality 10/10 ballast, concrete sleepers Peak/limit lateral resistance at 2 mm, high quality 20/16 ballast, concrete sleepers Zacher (12) Average lateral ballast resistance of a standard 6,3 timber sleeper with length 2,6 m in plain track Dogneton (27) Maximal lateral resistance per sleeper, non- ≅ 10 compacted ballast, concrete B70 sleeper Maximal lateral resistance per sleeper, non- ≅ 6 compacted ballast, wooden sleeper Pio Pucillo (29) Average lateral resistance, uncompacted ballast 6 - 8 and unloaded, wooden sleeper RFI 240
2.6.1.3 Contribution factors
An important aspect in describing the resistance of the sleepers in the ballast is the contribution of the faces of the sleeper: base, shoulder and crib (see Figure 2.27). These contributions are useful in predicting the ballast resistance of a sleeper with a different length, width or height than the one obtained from the test site.
Figure 2.27: Designation of the base, shoulder and crib areas of the sleeper (25)
Literature overview 26 Le Pen and Powrie (30) investigated the effects of the bottom (base) resistance, the side (crib) resistance and the end (shoulder) resistance on the total lateral resistance of the sleepers in the ballast bed. Also Lichtberger (31) suggested values. Zacher (12) reports about measurements on standard sleepers by the Technical University of Munich (TUM) that showed different values of the contributions. Lastly, also NMBS provided experimental data on the relative contributions of the sleeper parts, for a consolidated track with an increased shoulder level of 10 cm above the sleepers (7). Van (5) suggested that each face contributes to one third of the total resistance. The results are summarized in Table 2.3.
Table 2.3: Contributions of the different parts of the sleepers to the lateral ballast resistance found in the literature
Source Base resistance Crib resistance Shoulder resistance
Le Pen and Powrie (30) 26-35% 37-50% 15-37%
Lichtberger (31) 45-50% 10-15% 35-40%
Zacher (TUM) (12) 42% 43% 15%
NMBS (7) 22% 20% 58%
Van (5) 33% 33% 33%
From these results, it is clear that the contribution of shoulder, crib and base of the sleepers is uncertain and is depending on many factors, like type and size of sleeper, ballast quality, raised shoulder height, etc. No exact conclusions can be made from the values in the table.
2.6.1.4 Variations in lateral ballast resistance
Kish (32) reports about results of tests performed by UP/Foster-Miller, which reveal that the lateral ballast resistance of wooden sleepers could reduce with 20% in wet ballast during heavy rain. Although dynamic buckling is not the subject of this dissertation, the effect of train induced vibration can be quantified through a variation in the lateral ballast resistance. Kish (32) mentions tests performed by British Rail which reveal that a reduction of 25-50% in lateral ballast resistance can occur due to vibrations, depending on the acceleration and frequency levels. It is noticed that it is not mentioned for which train loading configurations this occurs and it is expected that these large reductions in resistance only happen for large speeds that cause significant vibrations. For the crossings investigated in this master dissertation, it is assumed that the vibrations have a less significant influence because of the limited velocities of the passing trains. Nevertheless, it can be taken into account for a safety assessment of the track.
Lastly, the static train loads themselves are also not taken into account in this dissertation. The effect of uplift forces on the lateral ballast resistance can be quantified although. With uplift of track, it is assumed that the base component of the resistance disappears. Referring to the contribution factors in Table 2.3, this means a reduction of 20-40% in the lateral ballast resistance can be expected under this uplift effect due to passing trains.
Literature overview 27 2.6.1.5 Lateral ballast resistance for different sleeper lengths
Zacher cites in situ measurements of the lateral ballast resistance on timber sleepers with a length of 2,6 m in plain track, conducted by the TUM and the track measurement department of Deutsche Bahn. The tests were performed on different locations, in consolidated and non-consolidated ballast conditions. A mean value of the lateral ballast resistance of 6,3 kN was obtained. The tests were only performed on standard sleepers and no measurements were conducted for sleepers with different lengths. Zacher proposed a formula to estimate the lateral ballast resistance for a different sleeper length L taking into account the results of the contribution factors from Table 2.3:
L LatBR( L) = LatBR( L0 ) 0,85 + 0,15 (2.5) L0 The increased weight of the sleeper, which contributes as well to the lateral ballast resistance, is not taken into account in the formulation. This means that this estimation of the resistance for longer sleepers is conservative. In Figure 2.28, the input relationships for the lateral ballast resistances, which Zacher used in his finite element model of a turnout, are given.
Figure 2.28: Input characteristics of the lateral ballast resistances in the finite element model by Zacher (12)
2.6.1.6 Techniques to increase the ballast resistance
The lateral ballast resistance can be increased by adopting the bearing of the sleepers, for example by adding safety caps (or also called anchor caps). Figure 2.29 shows two examples of safety caps applied on wooden sleepers. Dogneton (27) cites observations by Deutsche Bahn, in which the lateral ballast resistance is increased with 22% if one sleeper out of three is reinforced with a safety cap. Reinforcing one sleeper out of two resulted in an increase of 40% and installing a safety cap on every sleeper increased the resistance with 90%.
Literature overview 28 A B
Figure 2.29: Safety caps for wooden sleepers by Vossloh: (A) SN type (33). (B) SV type (34)
Figure 2.30 illustrates the influence on the lateral ballast resistance of safety caps installed on wooden sleepers, taken from the CFF documents by Dogneton (27). Here, it can be concluded that reinforcing every two sleepers with a safety cap increases the resistance with approximately 50%. This is of the same order of magnitude compared with the results by Deutsche Bahn. This graph shows a less significant increase in resistance for the application of safety caps on every sleeper.
Figure 2.30: Influence of safety caps on the lateral ballast resistance of wooden sleepers (27)
Also Arcoleo and Catena (35) investigated the influence of anchor systems on the lateral ballast resistance. They performed tests on concrete RFI230 sleepers with anchor systems of types SN (Figure 2.29A) and UNINAPOLI (Figure 2.31A). For the SN type anchor system applied on every second sleeper, the average lateral ballast resistance was increased from approximately 5800 N to 7400 kN, which corresponds to an increase of 28%. Tests performed on the same sleepers provided with UNINAPOLI anchor systems revealed an increase in resistance from approximately 760 kg to 1200 kg, which corresponds to an increase of 58%. It is not clear whether this system was applied on every sleeper or on one out of two. The results are comparable with the values obtained by Deutsche Bahn.
Literature overview 29 A B
Figure 2.31: (A) UNINAPOLI type sleeper anchor system. (B) rail anchor system (35)
Arcoleo and Catena (35) also performed tests on a rail type anchor system, like illustrated in Figure 2.31B. This system is not applied directly to the sleepers, but is positioned in the space between two consecutive sleepers and at the location of the rail foot. With this system, the average ballast resistance increased from 760 kg to 1060 kg, corresponding to 40%. Again, this is comparable with other anchor techniques.
Other techniques to improve the lateral ballast resistance are the use of retaining walls along the railway track and gluing of the ballast, both tested by Arcoleo and Catena (35). An overview of these techniques is given in Figure 2.32. For the retaining wall, the results of the tests revealed that the increase in lateral ballast resistance is comparable with the results for a SN anchor system applied to every second sleeper. Approximately 34% in resistance force is gained. For glued ballast in the shoulder, the resistance increased from 760 kg to almost 2500 kg. This corresponds to a gain in strength of approximately 230%. Also with gluing of the ballast in between the sleepers, the resistance increased with 104%. Tests performed on a completely glued ballast profile revealed that the lateral ballast resistance increased to 9000 kg compared to 760 kg for ballast without glue. The effect of gluing on the ballast resistance is thus very high.
A B C
Figure 2.32: Overview of techniques to increase the lateral ballast resistance. (A) Retaining wall. (B) Gluing of ballast in the shoulder. (C) Gluing of the ballast in between the sleepers
Literature overview 30 2.6.2 Longitudinal ballast resistance
The longitudinal ballast resistance response curve is typically bilinear. In buckling phenomena, only displacements in the initial linear part of the longitudinal ballast response will occur, as the longitudinal displacements are expected to be small. Esveld (23) reports about the results of an experimental program conducted to determine the ballast characteristics that are implemented in the models CWERRI, LONGIN and TURN. For concrete sleepers, the typical maximum value of longitudinal ballast resistance was determined to be approximately 11 kN at a displacement of 5 mm.
Zacher (12) cites values for the longitudinal ballast resistances according to Siekmeier. For a single sleeper in plain track, the longitudinal ballast resistance is 13 kN. This in accordance with the value cited by Esveld. Zacher simulates the longitudinal ballast resistance in the finite element model by two springs on both sides of the sleeper, as illustrated in the sketch in Figure 2.33. This means that the actual resistance needs to be halved for the input characteristics of the longitudinal ballast resistance in the finite element model, given in Figure 2.33. This is in accordance with the equivalent stiffness of two springs placed in parallel:
keq =+ k12 k (2.6)
To make an estimation of the longitudinal ballast resistance of longer sleepers, Zacher applied the same methodology used for the derivation of formula (2.5). However, as this is based on the contribution factors of the sleeper faces in a lateral resistance test, it is not believed that the applied formula for longitudinal ballast resistance is completely reliable.
Figure 2.33: Input characteristics of the longitudinal ballast resistances in finite element model by Zacher (12)
2.6.3 Rail fasteners
Fasteners include all the components which together form the structural connection between the sleepers and the rails (17). These elements maintain the position of the rail relative to the sleeper and absorb the rail forces elastically to transfer them to the sleeper, together with providing sufficient damping of vibrations and impacts caused by traffic. The vertical clamping force must be large enough to provide the necessary longitudinal resistance against
Literature overview 31 creep, to limit the breathing length of continuous welded rail track and to limit the gaps that arise when fracture of the rail occurs. The fasteners also have the important function to preserve the track gauge and inclination of the rails within the imposed tolerances. Besides that, they also provide electrical insulation between sleeper and rail.
Rail fastening systems are categorized in two types: direct fastenings (rail and baseplate are fixed to the sleeper with the same fasteners) and indirect fastening (the rail is fixed to an intermediate component first, such as a baseplate, and the intermediate component is fixed by a second fastening system to the sleeper) (17). For the set- up of the numerical model, the determination of the longitudinal and torsional fastener resistance is of importance.
2.6.3.1 Longitudinal fastener resistance
In general, the longitudinal fastener resistance is obtained at an arbitrary limit displacement of 0,2 mm (7). In Jae et al. (36) have measured the longitudinal resistance of a fastener pair for different fastener types: Pandrol, Vossloh and Nabla. The longitudinal fastener resistance for Pandrol amounts to 16 kN, for Nabla 22 kN and for Vossloh 13 kN. Minimum values of 9 kN and 15 kN for the longitudinal fastener resistance were prescribed by the European Railway Agency (37) and Esveld (17) respectively. The results of the tests by In Jae et al. are in accordance with these limit values. The different fastener types are illustrated in Figure 2.34 and Figure 2.36.
A B
Figure 2.34: (A) Illustration of Nabla fastening system (38). (B) Illustration of Pandrol e-Clip fastening system (39)
Van (5) states that it is common that the longitudinal fastener resistance is larger than the longitudinal resistance of the sleeper in the ballast, so that the sleeper will move longitudinally instead of the rail slipping on the sleeper. Van (5) cites an article by Dieterman et al. which shows that vertical stresses by axle loads increase the longitudinal fastener resistance. Also the longitudinal ballast resistance increases due to vertical axle loads or in case of frozen ballast. The increase in longitudinal ballast resistance (expressed in percentages) is in any case higher than the increase in longitudinal fastener resistance. This means that in particular cases, the rails will slip on the sleeper before the sleepers move longitudinally in the ballast bed.
2.6.3.2 Lateral fastener resistance
Zacher (12) estimated the lateral stiffness of a Vossloh fastening system 500 kN/mm. This physically means that the rail is restricted from lateral movement in the connection with the sleeper. As a consequence, the track will always move laterally in the ballast as one piece (sleepers plus rails).
Literature overview 32 2.6.3.3 Torsional fastener resistance
As was concluded from section 2.5, the torsional fastener resistance has an important influence on the critical buckling temperatures. When buckling, the rails are rotating relatively to the sleepers, which the fastening elements have to resist. Landuyt (7) cites Gallati, who states that the torsional fastener resistance depends on the type of fastener, the type of rail pad, the tension in the fastener, the position of the rail in the fastener and the width of the rail itself. The torsional fastener resistance is characterised by a relationship between the torque and the torsion angle. In most references, this behaviour is modeled in a numerical model as linear-elastic (5). The torsional fastener resistance is however a typical non-linear phenomen. Landuyt (7) describes the behaviour as follows:
“A distinction between three different zones is made. In the first part, the resistance increases as more friction is generated. This stops eventually, when the adhesion between rail and railpad is broken. Then the resistance remains almost constant, until it increases sharply in the third part, which is when the rail touches the fastener’s edge.”
Zacher (12) refers to measurements performed by TUM to determine the torsional characteristics of a Vossloh rail fastening system. A typical Vossloh type fastener is illustrated in Figure 2.36. Different rail pads with a thickness of 6 mm were used during the tests. Figure 2.35 illustrates the results of these measurements, in which the typical non-linear behaviour of the fastening elements described above is found.
Figure 2.35: Results for the torsional resistance of a 60E1 rail with a Vossloh fastening system with different rail pads, conducted by TUM (12)
Figure 2.36: Vossloh fastening system for wooden sleepers (40)
Literature overview 33 In case of a linearized relationship, Van (5) mentions torsional resistances of 43 kNm/rad for K-fastenings (illustrated in Figure 2.37) and 72 kNm/rad for W-fastenings (comparable with Vossloh). He also mentions that the torsional fastener resistance for wooden sleepers could be up to 15 times higher compared to the same fastening systems applied to concrete sleepers. In this respect, Samavedam et al. (4) obtained results for tests on fasteners attached to hardwood and found a torsional stiffness between 418 kNm/rad and 836 kNm/rad for a Pandrol system.
Figure 2.37: Illustrations of K-type fastening system (41)
2.6.4 Track defects
One of the most important parameters influencing the buckling response is the track defect or the initial lateral misalignment of the rails. The typical form of the defect used in a numerical buckling calculation and modulation is a wave form sinusoidal curve. This form can be modeled as a 1 – cos(x) wave, as was done earlier in other software calculations as mentioned by Landuyt (7). This yields a tangent transition from ideal track to the defect shape. A small initial lateral misalignment may grow further due to passage of trains in service. The growth of the defect is called track shifting (5).
The applied tolerances for track defect amplitudes in Belgium, found in the Reglementaire Technische Voorschriften Baan – Bundel 2 (42), are summarised in Table 2.4. Herein, two quality levels are described. The intervention limit (IL) correspond to the values above which a corrective maintenance within the foreseeable future to re-establish the original situation is required. The immediate action limit prescribes the allowed maximum value before the situation becomes extremely critical and needs immediate action to prevent derailments or serious accidents. This means the track will be put out of service. A distinction is made between conversion of the measurements for small wavelengths (3m – 25m) and larger wave lengths (25m - 70m), respectively indicated by D1 and D2. Within the framework of this dissertation, the limit values for small in service velocities is important (V ≤ 40 km/h). It is concluded that rail misalignments of up to 22 mm may be expected. The alert limit for small velocities amounts to 10 mm, so that initial misalignments of 8 to 10 mm prescribe a situation in which no actions are undertaken.
Literature overview 34 Table 2.4: Intervention limits and immediate action limits for the measured defect amplitudes (measurement principles D1 and D2) according to “Reglementaire Technische Voorschriften Baan – Bundel 2” (42)
Mean to peak values of the defect amplitude [mm] In service velocity Intervention Limit (IL) Immediate Action Limit (IAL) [km/h] D1 D2 D1 D2 V 40 17 N/A 22 N/A
40V 80 15 N/A 22 N/A
80V 120 11 N/A 17 N/A
120V 160 8 N/A 14 N/A
160V 230 7 24 12 24
230V 300 6 14 10 14
Zacher (12) cites an analysis of the maximum track defect measurements from high speed lines (> 200 km/h) in Germany. Figure 2.38 illustrates the results. It is concluded that for small defect wavelengths (< 30 m), the defect amplitude in the lateral direction is in reality not larger than 6 mm. For a wavelength of 10 m, the maximal amplitude is only 3 mm. This is of course the case for high speed lines, which require a high safety level. For local lines with lower speeds, it is expected that the present defects can be larger.
Figure 2.38: Results of the analysis of the maximum track defect measurements in Germany (12)
To illustrate the typical misalignments used in reference works, the applied track defects in the software application CWERRI are reported (5). For different structures, misalignments with wavelengths 8 m, 10 m and 12 m are applied with magnitudes of 8 mm, 10 mm, 12 mm, 14 mm, 16 mm, 18 mm, 22 mm and 30 mm.
Literature overview 35 2.6.5 Rail types
For the modeled geometries in the following chapters, the rail types presented in Annex 1: Rail profiles are used. The standard rail type applied in Belgian railway lines is 60E1. The standard check rail profile is the 33C1 type (7) and a £600 profile is used for fishplates in fishplated joints. The rail type influences the buckling strength of the track in two ways: the compression forces in the rails are dependent on the rail’s cross-section and the second moment of area contributes to the total bending stiffness of the track in the lateral direction. When buckling, the rails rotate around the vertical axis.
2.7 Overview of existing finite element models
2.7.1 CWERRI
The CWERRI software program (23) was developed in the Netherlands by the Technische Universiteit Delft. The main features of this model were the implementation of longitudinal, lateral and vertical forces, including complete train loads, and the modeling of track/bridge interaction and buckling behaviour of curved tracks. The program contains functionality with respect to both lateral and vertical stability. A longitudinal spring was added at the boundaries in order to model the longitudinal behaviour of linear elastic tangent track up to infinity. For the initial misalignment, a half sine wave was used. An arc-length controlled method is used to deal with the thermal loads in post-buckling configurations. A brief description of examples of calculations with this software program is given in the paper published by Esveld (23).
2.7.2 CWR-BUCKLE
The theory of lateral track buckling and the parameters involved were computerized in the software program CWR- BUCKLE, developed by Kish and Samavedam. It was validated by direct test evaluations in field conditions in the US. Some of the results obtained with this software program are presented in Track Buckling Prevention: Theory, Safety Concepts and Applications (6).
2.7.3 Zacher and Landuyt
Zacher made a finite element model of the switch P3550 – XAM 1/16 Mod.90 (12). Some of the characteristics applied in this model were already outlined in this chapter. Landuyt (7) made a finite element model of the same switch and other types of turnouts in the finite element software Siemens NX, for which the results by Zacher served as validation. Most of the principles applied by Landuyt in the software Siemens NX will also be applied in this dissertation to model the continuously welded crossings.
Literature overview 36 2.8 Limitations and solutions to prevent buckling in summer
2.8.1 Prevention of buckling
The compression forces in the rails, which form the risk of track instability, are dependent on the neutral temperature of the track which needs to be managed (43). Large compressive stresses in summer can be controlled by cutting, destressing and rewelding the rails after adjusting it to the desired neutral temperature (6). As the ballast characteristics influence the buckling behaviour significantly, it is important to ensure a good ballast condition and maintain the alignment of railway track (43). As also train loads and dynamics influence the critical temperatures, speed restrictions could be necessary where the safety margin is small (43).
2.8.2 Stress limitations
Not only the critical buckling temperatures are of importance in assessing the safety of in-service track, also the actual rail stresses need to be within certain limits to prevent a reduction in fatigue life of the rail. Due to the reasons mentioned in section 2.3.7, the stress concentrations in the rails can be much higher than the average longitudinal stress.
According to Mys (8), the Eurocode EN 1991-2 or UIC 774-3 foresees the necessary guidelines to estimate the maximal stress peak. Section 6.5.4.5.1 in EN 1991-2 gives upper boundaries for the permissible additional rail stresses due to the combined response of the bridge structure and track to variable actions. The maximal additional stress in the rails have to be limited to 72 MPa in compression and 92 MPa in tension for applications with ballasted track (8). Note that these values are applicable for the specific design situation of track on bridges and for track configurations with 60E1 rails with a tensile strength of at least 900 N/mm², in plain track, in a ballast bed with concrete sleepers and a sleeper spacing of 65 cm. But these limit design values give a good indication of which order of magnitude of stresses become critical for the fatigue life of the rails. For a 60E1 railway track with a neutral temperature of 20°C, subjected to a temperature increase of 40°C, the nominal stresses amount to approximately 100 MPa. The maximal allowable stress peak is thus 172 MPa, in accordance with section 6.5.4.5.1 in EN 1991-2.
2.8.3 Relative displacements in the fastening systems
Besides the stress limitations, also the relative displacements of the rails in the fastening systems (permissivity) need to be limited in pre-buckling configurations. According to Mys (22), these should be limited to 30 mm in the newly installed fastening systems. In the old types, the limitation is set to 15 mm. Concluding, the criterions to be able to continuously weld the crossings are set by the critical buckling temperatures, the stress limitations and the relative displacements of the rails in the fastening systems.
Literature overview 37 Chapter 3 Basics of the finite element model
3.1 Introduction
To investigate the mechanical behaviour of the railway tracks under the application of thermal loads with the finite element software Siemens NX, the real build-up of the track needs to be simplified to a numerical model with sufficient accuracy. Landuyt (7) performed a detailed study to determine the adequate properties of the finite element model in Siemens NX using the Nastran Solver. Most of the assumptions and model characteristics are also applied to the finite element models made in the context of this master dissertation. The framework that is used to build the finite element models is delineated in this chapter.
In the following sections, the general properties of the finite element model (geometry, mesh characteristics, materials and structural properties) are outlined. Specific aspects related to a particular track model of a crossing are discussed in the corresponding chapter. The goal of this dissertation is to investigate the buckling behaviour of continuously welded crossings, which are more and more installed in the Belgian railway complexes to replace the old types with jointed track.
The old types of turnouts and crossing were constructed using fishplated joints. In chapter 4 and chapter 5, finite element models are discussed in which these fishplated joints are implemented in the geometry. They are used to obtain a better understanding of the difference in behaviour of continuously welded railway crossings and crossings with fishplated joints. The aspects related to implementing these joints in the finite element model are not discussed in the framework of this chapter, but are expounded extensively in the aforementioned chapters.
3.2 Geometry of the models
3.2.1 Line model of the structural model
The composition of structural elements is simplified to a line model, which can be used as input geometry for the Siemens NX finite element software. In this regard, the sleepers and rails are represented by lines going through their centre of gravity of the cross-section in a CAD model, of which the geometry is obtained from the CAD-file of the plans of the crossings. The plans of the crossings are presented in Annex 2: Plans of crossings/turnouts. The model is three-dimensional, which means that the sleepers and rails do not lie in the same plane.
Where switchblades are part of the track model (for instance in the diamond crossing with double slips), the movable parts of these rails do not transfer any significant forces to the sleepers as no fastening system is present. These
Basics of the finite element model 38 parts are thus free to expand under a temperature increase. In this sense, the movable switchblades can be omitted from the model as they do not contribute to the structural behaviour. The switchblades are put in the finite element model from the point that they are connected with fasteners to the sleepers. This point is in accordance with the plans of the crossings. Creep devices (which are for example present in turnouts to prevent differential longitudinal movement of the switchblades relative to the stock rails caused through rail creep (44)), are not installed in the crossings being investigated here, because the tongues are relatively short.
The modeling of the frog can happen in two ways, as illustrated in Figure 3.1. Either the intersecting rails can be connected in the transverse direction over the length of the frog by additional beam elements (configuration A) or the intersecting rails can be assigned a larger cross-section to increase the stiffness significantly (configuration B). At the intersection point of two rails, the node is connected in both assumptions. It is assumed that the first option will be a more realistic representation regarding the stiffness, since the frog is one massive block and behaves as one piece. In configuration B, both rails are connected only at their central intersection point. However, the frog also contributes to larger nominal thermal forces, which is not expressed in configuration A. In chapter 4 the influence of the way of modeling of the frog on the buckling behaviour will be investigated. The base models are created using configuration A.
A B
Figure 3.1: Two possible ways to model the frog in the finite element model. (A) By means of transverse elements connecting the intersecting rails. (B) By means of a larger cross-section assigned to the rails
The rails and sleepers are connected by fasteners, which are modeled by non-linear spring elements. In the CAD- model, these springs are represented by a vertical line connecting the centre lines of rail and sleeper. The lateral and longitudinal ballast resistances are simulated by assigning the properties to non-linear springs as well. Therefore, line elements are added at the end of the sleepers. More details about the exact locations and orientation of these elements are given in the following sections.
It is important that the line model consists out of individual lines in between each intersection of two lines. This facilitates the creation of nodes in the finite element software Siemens NX. As such, the end nodes of different structural elements coincide with each other (for instance common node of sleeper/fastener and fastener/rail) and the meshed geometry is as intended from the CAD line model. If the end points do not coincide in the imported CAD model, the mesh nodes are possibly unconnected, yielding an undesired structural behaviour. More details about the meshing of the individual elements are given in section 3.3.
3.2.2 Boundaries of the model
About 100 meters of plain track are added at both sides of the actual model of the crossing. This is necessary to model the actual behaviour of the track which is continuously welded and to minimise the influence of the clamped boundary conditions (see section 3.5), such that the nominal axial forces equalise at the ends of the model (44). In the region of changing rail geometry, the normal forces in the rails will deviate from the nominal one.
Basics of the finite element model 39 3.2.3 Track defect
Buckling of railway track always occurs at locations where an initial track defect is present that actually ‘triggers’ the buckling phenomenon. A defect with a prescribed wavelength and amplitude thus needs to be implemented in the finite element model. As was discussed in section 2.6.4, railway track defects are mostly modeled in the reference literature with a sinusoidal shape. Here a 1-cos(x) misalignment curve will be applied to the initial straight line of the rails. The defect curve is implemented in the model by making use of a polyline made up of small individual segments. This polyline is created by converting a spline representing the sinusoidal curve. Typically 15 line elements in between two fasteners are present, which makes sure that the curved geometry of the rails is smooth enough without unnecessarily increasing the refinement too much.
In the context of this dissertation, the focus will be put on defects with wavelengths of 8 m and amplitudes of 8 mm and 20 mm. As was concluded from section 2.5.7, the most critical situation regarding track buckling is obtained for decreasing wavelengths up to 8 m and for increasing misalignment amplitudes. Larger defect wavelengths will not decrease the critical buckling temperature. Besides that, the implementation of the defect depends on the geometry of the track model and must fit in the actual structure. In the following chapters, this aspect is given a closer look. An initial misalignment of 8 mm is a realistic value for which no corrective action is undertaken, regarding the alert limit in Belgian railway tracks discussed in section 2.6.4 and results of the survey given in Figure 2.38. The track is close to be put out of service and needs a corrective action in the foreseeable future when the misalignment amounts to 20 mm. Both amplitudes will thus be applied in the finite element models. In what follows, the defects will be given the designation code of for example ‘8m20mm’, which corresponds to an imperfection with wavelength 8 m and amplitude 20 mm.
The way the defect is physically set in the model influences the resulting buckling behaviour. The defect can be seen as an initial deformation of one rail (for instance due to train loading or impact), which physically means that an initial stress is present in the rail. When the sleepers are misaligned together with the positions of the fasteners, an initial deformation of all the rails connected to the sleepers is present. This kind of misalignment is more likely to be present. For this case, the nature of the rails can be either stressed or be stress-free. If the rails are welded together with a misalignment in the sleepers already present, the rails can be assumed to be initially stress-free at the neutral temperature. If the misalignment has occurred due to cyclic train loading, large cyclic temperature variations or other impact forces, the rails will be deformed and have a particular stress state. The influence of bending on the stress distribution in the rail was discussed in section 2.3.7.
For the implementation of the misalignment in the finite element models, the track defect is applied on all rails together in the particular chosen section so that the sleepers have a misalignment. The stress states of the rails, fasteners and ballast resistances are not accommodated for this misalignment, so that all structural elements are stress-free. This situation might be less conservative regarding the development of buckling, since the lateral ballast resistance behaves stiffer for the initial deformations for instance. This is concluded from Figure 2.28 for example. If the sleeper has already displaced in the ballast at the neutral temperature, the resistance will be less stiff. The same is true for the longitudinal ballast resistance and the torsional fastener resistance. This assumption must be taken into account when making the safety assessment for the obtained results of the track model.
Basics of the finite element model 40 3.3 Aspects related to finite element modeling
The line model discussed in the previous section is imported in the finite element software Siemens NX. The structural entities need to be converted to meshes. Figure 3.2 gives an illustration of the line model with its structural meaning. Each line is assigned a mesh property: beam elements for the sleepers and rails and non-linear spring elements for the ballast resistances and fastener elements.
Figure 3.2: Schematical representation of the imported line model of the track with its structural meaning
3.3.1 Beam elements (rails, check rails, fishplates, sleepers)
As discussed by Landuyt (7), the rails, check rails, fishplates and sleepers are modeled with beam elements, because solid elements would increase the complexity too much without increasing the accuracy significantly. The CBEAM element type is used in Siemens NX. The structural properties are assigned to the PBEAM entry corresponding to the CBEAM mesh element.
The CBEAM elements are assigned to the line elements corresponding to the sleepers and rails from the imported CAD geometry (see 3.2.1). Any single line is divided in three mesh elements. This corresponds in general with three rail elements in between two fasteners, which results in dimensions of approximately 20 cm per mesh element. At the location of the defects and intersecting rails, smaller elements are applied. This refinement of the mesh accommodates for the expected stress concentrations at these locations. The beam elements assigned to the sleepers are in general larger, as these do not require a fine mesh. This is in accordance with the applied settings of Landuyt (7). Nodes of the same mesh are automatically merged with a tolerance of 0,1 mm. In the PBEAM entry, the cross-section of the sleepers/rails and the corresponding material properties are assigned to the mesh. These properties are further elaborated in section 3.4.1.
Attention needs to be paid to the orientation of the CBEAM mesh elements when assigning them to the lines of the CAD-model. Because the rail profiles are inclined with an angle 1/20 relative to the vertical – as will be discussed in section 3.4.1.1 – the orientation of the beam’s axis decides the inward or outward inclination of the rail profile. The rails are always inclined inwards towards each other, so that the orientation illustrated in Figure 3.3 is correct.
Basics of the finite element model 41 The sleeper’s cross-section is symmetrical with the vertical, so that only to the correct width/height position of the sleeper CBEAM elements has to be ensured.
Figure 3.3: Definition of the CBEAM’s axes for the rail elements, resulting in the correct inclination
3.3.2 Spring elements (ballast resistances and fasteners)
The geometrically linear element type CBUSH is used in Siemens NX to model the ballast resistances and fastener elements. For this element, the orientation of the spring is not updated with the displacements of the structural model in Nastran (7). According to research by Landuyt (7), one element per spring is sufficient, because the stiffness assigned to the element is entered directly into the stiffness matrix.
The physical properties are assigned to the PBUSH entry. In this entry, the nominal stiffnesses and the non-linear spring characteristics are entered for the six degrees of freedom. The nominal stiffness values are used for the initial tangent stiffness of the spring in a non-linear calculation in Nastran. At the first increment during the calculation, the stiffness of the spring is calculated in accordance with the entered non-linear force-displacement curves. Figure 3.4 illustrates the characteristics of the PBUSH entry in the software Siemens NX.
The positioning of the spring elements in the simplified line model is illustrated in Figure 3.2. The fasteners form the connection between the rails and the sleepers and are represented by a vertical line. The lateral ballast resistance of the sleeper is created by a single horizontal spring attached to one end of the sleeper. As stiffness characteristics can be assigned to both directions for a single translation axis of the spring, this single spring is sufficient. For this orientation of the line, the lateral ballast resistance will thus work along the axis of the spring element.
The longitudinal ballast resistance is implemented in the model by two springs attached to both ends of the sleepers. They are placed vertically and not according to the direction in which the longitudinal ballast resistance works, which is actually horizontally in the plane of the track. The reason for the vertical positioning is found by the ease of construction of the finite element model. As the end of the spring elements need to be clamped (see section 3.5.2), these ends are easier to select in this case compared to a horizontal position. The stiffness properties according to the longitudinal ballast resistance need then to be assigned along the correct defined axis. More details are given in the following sections.
Basics of the finite element model 42 A B
Figure 3.4: PBUSH entry characteristics in Siemens NX: (A) Nominal stiffness values. (B) Non-linear stiffness characteristics
The first axis of a spring element (DOF 1) is the line on which the mesh is created (GA-GB). The second axis (DOF 2) needs to be defined by the user with a reference vector. The third axis (DOF3) and the corresponding rotational degrees of freedom (DOF4, DOF5 and DOF6) are then fixed with the definition of a right-handed coordinate system. The definitions of the element’s axes for the lateral ballast resistances, the longitudinal ballast resistances and the fasteners are illustrated in Figure 3.5 and Figure 3.6. The direction of the reference vector of DOF2 is not the same for every spring element of the same type, as it is dependent on the orientation of the spring element axis GA-GB itself and this changes along the railway track. The created meshes in Siemens NX must thus be grouped according to the same orientation properties. For example, the two added sections of plain track on each side of the crossing have different orientations and consequently also the reference vectors of the lateral and longitudinal ballast characteristics and the fastener elements.
Basics of the finite element model 43 A B
Figure 3.5: Definition of the axes of the non-linear spring elements representing the lateral ballast resistance (A) and longitudinal ballast resistance (B)
Figure 3.6: Definition of the axes of the non-linear spring elements representing the rail fasteners
3.4 Materials and structural properties
3.4.1 Materials and sections
3.4.1.1 Rail sections
The applied rail profiles in the finite element models are 60E1 and/or 50E2, according to the plan of the crossing. Depending on the type of track model also check rails (33C1) and fish plates (£600) are present. In Annex 1, drawings of the cross-sections of these profiles are given. The rail profiles are installed with a slope of 1/20 with respect to the vertical, according to common practice in Belgium (8). The material assigned to these mesh elements is steel, of which the properties are further elaborated in section 3.4.1.3.
3.4.1.2 Sleepers
Wooden sleepers are applied in the models. Wood is an orthotropic material, which makes it difficult to model it in a finite element software. However, Landuyt (7) noticed with good reason that it is not needed to model the sleepers with their correct stiffness properties dependent on the kind of material assigned to it:
Basics of the finite element model 44 − The stiffness of the sleepers themselves does not have a significant influence on the buckling behaviour of the track. The lateral and longitudinal ballast resistances do depend on the type of sleeper, but this is already taken directly into account by the properties of the non-linear springs representing these ballast resistances. − The fastener properties include the failure characteristics of wood, because the reference tests of the different fastener types were conducted with wooden sleepers. Local failure of the wood species is thus already taken into account in the obtained results.
Landuyt (7) applied the properties of steel to the sleeper elements in his finite element model. Because the longitudinal ballast resistance is modeled by two springs attached to the sleeper ends, the sleepers will tend to deflect like a simply supported beam between its supports. In reality, the ballast resistance is spread over the full length of the sleeper and bending of the sleeper is restricted. In this respect, Landuyt assigned a larger stiffness to the sleeper to model the effect of the ballast between the sleepers. The larger stiffness can be foreseen by applying the larger modulus of elasticity of steel compared to wood. In this regard, the sleepers in the finite element models are also assigned the steel material properties in the context of this dissertation.
The standard wooden sleepers have a rectangular cross-section with a height of 150 mm and a width of 260 mm. Some of the sleepers in the models have a modified cross-section in which the width and/or height are changed. This is in accordance with the plans of the modeled crossing and will be outlined in the next chapters.
3.4.1.3 Steel
High strength steel is applied to the rail sections in the finite element model. As discussed by Landuyt (7), it is not the intention that the yield strength of the steel is reached in the calculation, because the plastic behaviour of steel is not what is intended to be investigated and elastic deformations are assumed during buckling. Landuyt therefore set the yield strength to 1200 MPa and the ultimate strength to 1280 MPa. These characteristics are also applied in the finite element models being part of this dissertation. The other essential material characteristics assigned to the steel elements are the modulus of elasticity E, set to 206 GPa, and the thermal expansion coefficient 훼, equal to 1,152⋅10-5 (7).
3.4.2 Ballast resistances
To model the ballast characteristics in the finite element model, the resistance is simulated by non-linear springs. The input characteristics of these springs are based on the obtained values by the studies performed by Zacher (12), which are seen as the most reliable according to the configurations from which they were obtained. In section 2.6.1 it was concluded that the results from Zacher are in accordance with other studies and moreover on the conservative side. These correspond to ballast conditions in recently tamped track, being the most critical track configuration for lateral track buckling. The load-displacement curves in Zacher’s study are purely non-linear, while in other studies with finite element models bilinear characteristics are used. The latter are less detailed, but the difference is not that significant. In the finite element models of this dissertation, the full non-linear characteristics are assigned to the spring properties.
Basics of the finite element model 45 In the PBUSH entry of the CBUSH element, the nominal stiffness and the force-displacement curve representing the non-linear stiffness properties of the spring are entered. The springs representing the lateral and longitudinal ballast resistance only need stiffness properties in one degree of freedom: the translational stiffness in one direction. According to the axis orientations indicated in Figure 3.5, the stiffness properties for the lateral ballast resistance is entered along the x-direction (DOF1) of the element and for the longitudinal ballast resistance along the z-direction (DOF3).
In the crossings modeled in the next chapters, sleepers with different lengths are present. The standard sleeper length is 270 cm. For longer sleepers, the lateral and longitudinal ballast resistance increases. Zacher (12) proposed a formula to determine the resistances for sleepers with a different length than the standard test length for which the literature provides values. This is based on the contribution of each of the sides of the sleeper to the ballast resistance. Different values for these contribution factors were discussed in section 2.6.1.3.
The proposed formula by Zacher for the lateral ballast resistance in function of the sleeper length reads:
L LatBR( L) = LatBR( L0 ) 0,85 + 0,15 (3.1) L0
Herein, L0 is the standard sleeper length for which the lateral ballast resistance is tested. The factors 0,85 and 0,15 are based on the contributions according to Zacher given in Table 2.3. If the length of the sleeper becomes larger, the base and crib resistance increases when the sleeper is forced to move laterally. This yields the summation of the corresponding two contribution factors (0,42 + 0,43 = 0,85). The shoulder area is not changed, which makes this contribution factor (= 0,15) independent from the sleeper length.
An equivalent formula could be established based on the contribution factors determined by NMBS in Table 2.3:
L LatBR( L) = LatBR( L0 ) 0,42 + 0,58 (3.2) L0 It is clear that this formula is more conservative, as the lateral ballast resistance increases less for the same increase in sleeper length compared to formula (3.1).
A similar relationship between sleeper length and longitudinal ballast resistance is mentioned by Zacher (12):
L LongBR( L) = LongBR( L0 ) 0,56 + 0,44 (3.3) L0 Equivalent formulas to determine the lateral ballast resistance for changes in sleeper width (B) or depth (D) can be established, based on the contribution factors:
B LatBR( B) = LatBR( B0 ) 0,57 + 0,43 (TUM) (3.4) B0
B LatBR( B) = LatBR( B0 ) 0,80 + 0,20 (NMBS) (3.5) B0
D LatBR( D) = LatBR( D0 ) 0,58 + 0,42 (TUM) (3.6) D0
D LatBR( D) = LatBR( D0 ) 0,78 + 0,22 (NMBS) (3.7) D0
Basics of the finite element model 46 The contribution factors listed in Table 2.3 are determined for lateral ballast resistances. No tests are available to determine the contribution factors of each of the faces of the sleeper for longitudinal displacements of the sleepers in the ballast. It is therefore not possible to establish reliable formulas to determine the longitudinal ballast resistance for changing dimensions of the sleepers, except for formula (3.3).
However, from section 2.5.3.2 it was concluded that a variation in longitudinal ballast resistance has only a minor influence on the lower critical buckling temperature and almost no effect on the upper critical buckling temperature. It is therefore not needed to obtain a high accuracy on the estimation of the longitudinal ballast resistances according to a change in depth or width of the sleeper, because no significant difference in buckling behaviour will be obtained. Similar formulas as (3.4) -(3.7) can thus be established for the longitudinal ballast resistance according to the contribution factors of Table 2.3:
B LongBR( B) = LongBR( B0 ) 0,85 + 0,15 (TUM) (3.8) B0
D LongBR( D) = LongBR( D0 ) 0,58 + 0,42 (TUM) (3.9) D0
3.4.2.1 Lateral ballast resistances
The input values for the force-displacement curve of the lateral ballast resistance are based on the used characteristic values by Zacher (12) (see Figure 2.28). The resistances for sleepers that are longer than the standard length of 270 cm are estimated using formula (3.1). The input data for the non-linear springs representing the lateral ballast resistance are given in Figure 3.7, which are entered in the field of the x-translation in Figure 3.4B. Note that for larger displacements than plotted in the figure, the force resistance is extrapolated and thus keeps constant.
14 570 cm
12 540 cm
510 cm 10 480 cm
8 450 cm 420 cm 6 Force[kN] 390 cm
360 cm 4 330 cm
2 300 cm
270 cm 0 0 1 2 3 4 5 Displacement [mm]
Figure 3.7: Input data for the lateral ballast resistances of the sleepers (sleeper lengths 270 cm – 570 cm)
Basics of the finite element model 47 The nominal stiffness values for the lateral ballast resistance, dependent on the sleeper length, are given in Table 3.1. These are entered in the X-translation entry from the PBUSH properties given in Figure 3.4A. In the same table, the limit resistance forces are given. These values are important to assess the lateral resistance, as for lateral forces equal or larger than this maximal value, the sleepers can move ‘freely’ in the lateral direction.
Table 3.1: Nominal stiffness values and limit resistance forces for the lateral and longitudinal ballast resistances in function of the sleeper length
Nominal stiffness [N/mm] Limit resistance force [N] Sleeper Lateral ballast Longitudinal ballast Lateral ballast Longitudinal ballast length [cm] resistance resistance (per resistance resistance (per spring) spring) 270 17 375 10 000 6 300 6 500 300 19 016 10 622 6 895 6 904 330 20 657 11 244 7 490 7 309 360 22 298 11 867 8 085 7 713 390 23 939 12 489 8 680 8 118 420 25 580 13 111 9 275 8 522 450 27 221 13 733 9 870 8 927 480 28 862 14 356 10 465 9 331 510 30 503 14 978 11 060 9 736 540 32 144 15 600 11 655 10 140 570 33 785 16 222 12 250 10 544
The applied estimation formula for the resistance of longer sleepers is based on the contribution factors cited as evidence by Zacher. The more conservative formula (3.2) that is based on the contribution factors by NMBS is applied to obtain the input data given in Figure 3.8. The maximal resistances are clearly lower (for instance 9,2 kN instead of 12,2 kN for a sleeper with length 570 cm). However, since the contribution of the larger weight of a longer sleeper is not taken into account in the estimation formulas, it is assumed that the conversion formula by Zacher is already conservative enough and the input data given in Figure 3.7 are taken as the standard approach values for the finite element models. The characteristics in Figure 3.8 will be used to check the influence on the buckling behaviour when comparing both approaches.
The input characteristics listed before are for a standard sleeper with a cross-section B x D = 260 mm x 150 mm. Some sleepers in the modeled railway structures in the next chapters have a different cross-section: respectively sleepers with a larger width B = 300 mm and sleepers with a larger width and depth B x D = 300 mm x 200 mm. The corresponding lateral ballast resistances are determined with formulas (3.4) and (3.6), both by applying the Zacher approach. The input data for these sleepers are illustrated in Figure 3.9, only for the sleepers with length 270 cm and 450 cm in order not to overload the figure. The force-displacement curve of the standard cross-section is plotted for comparison.
Basics of the finite element model 48 10 570 cm 9 540 cm 8 510 cm
7 480 cm
6 450 cm
5 420 cm
Force[kN] 4 390 cm
3 360 cm 330 cm 2 300 cm 1 270 cm 0 0 1 2 3 4 5 Displacement [mm]
Figure 3.8: Input data for the lateral ballast resistances of the sleepers according to the contribution factors of NMBS
450 cm (B=300mm & D=200mm) 270 cm (B=300mm & D=200mm)
450 cm (B=300mm) 270 cm (B=300mm)
450 cm 270 cm 14
12
10
8
6 Force[kN]
4
2
0 0,0 1,0 2,0 3,0 4,0 5,0 Displacement [mm]
Figure 3.9: Input data for the lateral ballast resistances for modified sleeper cross-sections: width B = 300 mm and depth D = 200 mm, respectively for sleepers with length 270 cm and 450 cm
Basics of the finite element model 49 3.4.2.2 Longitudinal ballast resistances
The input values for the longitudinal ballast resistances are again based on the characteristic values used by Zacher (12) (see Figure 2.33). The resistances for sleepers that are longer than the standard length of 270 cm are estimated using formula (3.3). The input data for the non-linear springs representing the longitudinal ballast resistance are plotted in Figure 3.10. The nominal stiffness values, dependent on the sleeper length, are given in Table 3.1. These are entered in the Z-translation entry from the PBUSH properties given in Figure 3.4A. Again, the limit resistance forces are given in the same table. For changing widths and/or depths of the sleepers, the input data are changed in accordance with formulas (3.8) and (3.9), similar to the lateral ballast resistances.
12 570 cm
540 cm 10 510 cm
8 480 cm
450 cm
6 420 cm
Force[kN] 390 cm 4 360 cm
330 cm 2 300 cm
270 cm 0 0 1 2 3 4 5 Displacement [mm]
Figure 3.10: Input data for the longitudinal ballast resistances of the sleepers (sleeper lengths 270 cm – 570 cm)
3.4.3 Fastener resistances
The fasteners applied in the modeled railway structures are of the Vossloh type, of which an illustration is given in Figure 2.36. The manufacturer of these fastening systems does not provide any values for the torsional, longitudinal or transversal resistances. To implement the resistances into the finite element model, an appeal is made to results from tests in the literature, which were discussed in section 2.6.3. The fasteners are modeled by non-linear springs, CBUSH elements, in the finite element software Siemens NX.
Only buckling in lateral direction is investigated, which means that all rail and sleeper elements are kept restraint in the vertical direction. Because the fasteners are modeled as vertical spring elements that make the connection between the rail elements and sleeper elements, the translational DOF in the vertical direction (x-axis of the spring element) does not need a constraint. A representation of the definition of the axes of the spring element of the fastener is given in Figure 3.6.
Basics of the finite element model 50 3.4.3.1 Rail fasteners
The non-linear spring element has six degrees of freedom, for which the stiffness properties have to be entered in the PBUSH entry given in Figure 3.4. DOF1 (translation along x-axis) does not need input data, because of the vertical constraints already present on the rails and sleepers. The second DOF is the translation in the transverse direction (y-axis). The fastener is here provided with a large stiffness and the value of 500 kN/mm, proposed by Zacher (12) for a Vossloh type fastening system, is applied. This is entered in the finite element software as a linear force-displacement relationship. The nominal stiffness value for DOF2 is thus 500 kN/mm.
The third DOF, the translation in the longitudinal direction of the rail (z-axis), is restrained by the longitudinal fastener resistance. In section 2.6.3, a value of 13 kN for a Vossloh-type fastener was found from tests conducted by In Jae et al. (36). In Belgium, this value is according to the limit displacement of 0,2 mm (7). A bilinear load- displacement characteristic is entered as input data, with 13 kN as maximal value at the displacement of 0,2 mm. This corresponds with a nominal stiffness of 65 kN/mm.
The torsional fastener resistance corresponds with DOF4, which is the rotation around the x-axis. For this input data, reference is made to Figure 2.35, in which measurements of the resistance were conducted by TUM. Zacher (12) and Landuyt (7) performed calculations with the most conservative characteristic curve (W14 Skl 14 Zw 900 = black dashed line). With respect to the finite element models of this dissertation however, it is chosen to make use of the results of the red curve in the figure (Lupolen 3510 K Test2), which is still conservative, but more representative. For small rotations, the stiffness is smaller than black dashed curve, which is in this respect even more conservative in pre-buckling configurations of the track. For large rotations, the stiffness becomes larger, being more in coincidence with the other results in the figure. The corresponding non-linear load-rotation characteristic used as input data in the software is given in Figure 3.11. The nominal rotational stiffness value for DOF4 is 80214516 Nmm/rad or 80,214516 kNm/rad.
5,0
4,5
4,0
3,5
3,0
2,5
2,0 Torque Torque [kNm] 1,5
1,0
0,5
0,0 0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035 Angle [rad]
Figure 3.11: Input data for the torsional fastener resistance = DOF4 (Vossloh type fastener)
Basics of the finite element model 51 The other rotational degrees of freedom, being DOF5 (around the y-axis) and DOF6 (around the z-axis) are assigned a large stiffness, respectively 1⋅1014 Nmm/rad and 1⋅1018 Nmm/rad, both as linear load-rotation characteristics. Landuyt (7) mentions that the stiffness may not be too large, as the stiffness is entered directly on the stiffness matrix and can possibly lead to numerical issues.
3.4.3.2 Check rail fasteners
The check rails are placed on chairs, as illustrated in Figure 3.12. They mainly influence the lateral stiffness of the track panel, but their influence on the normal forces in the rails or the buckling behaviour for defects outside the zone of the check rails is minimal. These check rails are assumed to be connected to the sleepers by fastener-type elements, also implemented in the finite element model by means of non-linear springs.
Figure 3.12: Photograph of a check rail placed on a chair (in case of a turnout)
The non-linear springs again need to be assigned stiffness properties in the 6 degrees of freedom. Similar to the common rail fasteners in the previous section, the first DOF can be left unstrained because of the applied boundary conditions. The lateral stiffness (DOF2) and longitudinal stiffness (DOF3) are assigned the same properties as for the rail fasteners above. The lateral stiffness by the chairs is large enough. Due to the small length of the checkrails, the longitudinal stiffness is not of big importance for the behaviour of the model. No actual stiffness properties could also be found for the longitudinal resistance of the chairs in literature.
With respect to the structure of the chairs, the torsional resistance of the check rail fasteners is much larger than the common Vossloh fastener. Instead of the input characteristics for DOF4 given in Figure 3.11, a linear load- rotation characteristic with stiffness 10⋅109 Nmm/rad is assigned for the check fastener, representing the torsional stiff chairs. The other rotational degrees of freedom, DOF5 and DOF6 are assigned comparable stiffnesses as the Vossloh-type fastener, respectively 10⋅109 Nmm/rad and 1⋅1018 Nmm/rad. These make the rotations of the check rails in the chairs practically impossible, which is in accordance with the real behaviour.
Basics of the finite element model 52 3.5 Boundary conditions
3.5.1 Rails and sleepers
The model of the crossing is extended by 100 m of plain track. The ends of these tracks represent an ongoing continuous welded railway track. To model this, the ends of the rails are clamped, simulating the fixed behaviour of a continuous welded railway track outside the discontinuity zone where changes in geometry occur or defects are present in the track. Far away from the buckling region, no longitudinal displacements will indeed occur.
As only lateral track buckling is being investigated and the effects of vertical uplift are disregarded, the rails will be restrained in the vertical translational direction. In this way, buckling is forced to occur purely in the lateral direction. The vertical displacements of all kind of rail type beam elements (rails, check rails, fishplates) are restricted. The sleepers are also restrained in the vertical direction, together with the rotation around the element’s x-axis and y- axis. If rotations around the latter axes would be free to occur, these would imply movements in the vertical direction, which is not the intention of this investigation.
3.5.2 Springs
The fasteners of the rails and check rails have both their ends connected to rail and sleeper. As such, no boundary conditions need to be applied to these elements. The springs representing the characteristics of the lateral and longitudinal ballast resistance have one end connected to the sleepers. These springs transfer the forces in the sleepers towards the ballast, which is assumed to be ‘the outside world’. Therefore, the free end of each spring element is clamped, as illustrated in Figure 3.13.
Figure 3.13: Boundary conditions for the non-linear springs representing the lateral and longitudinal ballast resistance
3.6 Thermal loads and safety assessment
A uniform temperature load is applied to all steel track elements, being rails, check rails, frogs, common crossing and fishplates. This means that there is no assumption of temperature differences along the model. Temperature differences could arise, due to shadow zones along the track or different in-service number of trains passing by. The
Basics of the finite element model 53 latter possibly influences the amount of heating of the rails. It is assumed that a uniform temperature increase on all elements yields the most critical loading situation.
In the results, the focus will be on the temperature increase above the neutral temperature. This means that the exact choice of neutral temperature is not of importance, as only the relative temperature change is investigated. A neutral temperature of 20°C is assumed and the temperature will be increased towards 220°C. This value will be large enough to observe the buckling behaviour with the applied track defects.
In practice, the neutral rail temperature is realised within the range of 20°C and 30°C in Belgium (12). In cold weather, the realised neutral temperature will be closer to 20°C for practical reasons. The chosen temperature is therefore a realistic and conservative value, but is only of importance for the safety assessment. For the results obtained with the calculation of the finite element model, the neutral temperature does not matter as the temperature increase is of importance. To assess the safety of the track, the temperature increase will be added to the conservative value of the neutral temperature of 20°C. With the maximal possible rail temperature of 60°C for a crossing where the use of magnetic brake systems is forbidden (see section 2.2), a maximal temperature increase of 40°C can be expected. This is the reference value for the safety assessment of the obtained results after calculations of the model.
3.7 Solver setup
3.7.1 Solution type
As discussed by Landuyt (7), the solution type SOL 106 Nonlinear Statics – Global Constraints in the Nastran Solver allows for the use of geometric non-linearity. This is needed to model the buckling behaviour of railway track, which is non-linear. This solution type allows to increase the temperature step by step and the solver to recalculate the stiffness matrix for each load step. With the Global Constraints option, the boundary conditions apply for every load step and do not change during the calculation. The analysis type Structural is used in the NX Nastran Solver (7). In the solver parameters, the Large Displacements setting is turned on, as with buckling behaviour of railway track large displacements are expected for the solution to converge.
The arc-length method is the most efficient method for solving non-linear systems of equations when the considered problem exhibits critical points, which are points at which the loaded structure cannot support an increase in the applied loads and instability occurs (45). The method is proven to cope well with limit points or problems with snapping behaviour. This method can thus provide solutions past the upper critical buckling temperature, which is desired to make a prediction of the safety against buckling. As was also discussed in section 2.3.3, the post-buckling lower critical buckling temperature is determinative in the safety approach of continuously welded railway track.
NX Nastran also provides the solution type SOL 106 Nonlinear Statics – Subcase Constraints, in which the boundary conditions could be adapted for different subcases of the solution. This type of solver could be used in combination with a first subcase that implies a forced displacement, representing the track defect at a certain location. With the
Basics of the finite element model 54 application of a forced displacement, the rails have an initial stress state contrary to an implemented track defect in the rail geometry. The latter is the standard misalignment implementation used in the finite element models of the next chapters, as explained in section 3.2.3.
An alternative method can thus be proposed with the application of subcases in the solution. In the first subcase the temperature is increased together with the forced displacement. In the second subcase, the stiffness matrix of the last step of the first subcase is used as starting point. Here, the rails are in a stressed state according to the lateral displacement. The forced displacement can be removed and the temperature further increased. This method to implement the track defect in the finite element model is however not possible with the NX Nastran solver, because the arc-length method cannot be used in combination with forced displacements as defined constraints. The only possible method to implement the misalignment in the finite element model thus remains the stress-free state in combination with the solution type SOL 106 Nonlinear Statics – Global Constraints.
3.7.2 Non-linear parameters
The non-linear parameters are defined in the window illustrated in Figure 3.14A. The number of increments is set to 120, which should be sufficient to obtain a smooth transition in the buckling curve and a good accuracy in the solution. The expected temperature increases are in the range of 40°C – 200°C. The maximum iterations per load increment are kept on the default value of 25. The intermediate output flag is set to ‘all’, which means that for every increment, the stresses, forces and displacements are given in the results. Efficient and fast calculations are obtained with the ‘AUTO’ stiffness update method, which forces the solver to perform a stiffness update based on the convergence rate (7). The number of iterations before update are set to 1, which improves the calculation time. The tolerances for load and work are set to 0,001 and 1⋅10-7 respectively, according to a guideline mentioned by Landuyt (7). A high accuracy should be obtained. The maximum number of bisections is set to 10 to increase the capacity of the Nastran solver to find convergence (7). The other parameters in Figure 3.14A are kept at their default values.
3.7.3 Arc-length method parameters
The arc-length method parameters are defined in the window shown in Figure 3.14B. The arc-length adjustment ratios defines the allowable change in arc length in the non-linear solution. If the solution does not show the desired accuracy, these parameters could be changed to increase it. The default values are kept initially. The maximum increments are set to 120 in accordance with the number of increments of the non-linear parameters.
3.8 Buckling definition
The goal of the finite element models is to get more insight in the buckling behaviour of the railway models and to determine the safety levels of the particular track geometries. A useful instrument to determine the level of safety is the resulting buckling curve after calculation, which plots the temperature increase above the neutral temperature in function of the lateral displacement of the defect. Figure 2.5 shows a typical buckling curve for railway track.
Basics of the finite element model 55 From the literature study in chapter 2, it is clear that the upper and lower critical buckling temperature are important parameters to assess the buckling risk. Table 2.1 gives an overview of the existing safety level regulations based on the critical temperatures. For cases where progressive buckling takes place, not only the critical temperature increases, but also the absolute displacements become important to assess the instability of the railway track. Too large displacements includes a risk for derailments of trains. The value of the lower critical buckling temperature gives the most conservative allowable temperature increase (level I safety) at which lateral track buckling could not occur. This parameter will thus serve as reference for the determination of the buckling risk of the investigated railway track models. Also the relative longitudinal displacements between rail and sleepers and the course of the axial forces in the rails will be determined.
A
B
Figure 3.14: Definition of the non-linear parameters (A) and arc-length method parameters (B) in Siemens NX
Basics of the finite element model 56 Chapter 4 Model of a diamond crossing
4.1 Introduction
The basic principles to build up a finite element model in Siemens NX have been explained in chapter 3. The first model being investigated in this chapter is the common diamond crossing. The geometry of a H4V4H4 diamond crossing installed in Belgian railway tracks is used to make the finite element model. A photograph of a scale model of a typical diamond crossing is given in Figure 4.1.
Figure 4.1: Photograph of a scale model of a diamond crossing (46)
4.2 Properties of the finite element model
4.2.1 Geometry
The model is based on the drawings of the diamond crossing H4V4H4 provided by Infrabel (47), illustrated in Figure A.5 in Annex 2. As explained in section 3.2.1 of chapter 3, the rails and sleepers are simplified to lines passing through the centre of gravity of their cross-section. Before and after the diamond crossing, approximately 100 meters of plain track is added, for the reasons mentioned in chapter 3. A sleeper spacing of 0,6 m is applied for these track segments.
Model of a diamond crossing 57 The sleepers of the diamond crossing are positioned according to their real positions on plan (see Figure A.5), which is different from Landuyt (7) who modeled the sleepers with a uniform distance of 0,6 m between their centrelines for the finite element models of a P3550 – XAM 1/46 turnout and a P215 – XA 1/8 turnout. Landuyt expected that this assumption does not have a great influence on the buckling behaviour. However, since the lateral ballast resistance is the most influential parameter for track buckling behaviour, it is expected that a denser spacing of sleepers has a non-neglectable influence on the critical buckling temperatures. Also vice versa, when the spacing of the sleepers is larger than the common 60 cm, the buckling strength of the track is decreased. The sleeper spacing in the H4V4H4 crossing varies between 435 mm and 705 mm.
As outlined in chapter 3, the non-linear springs are represented in the CAD model by individual lines. The rails and sleepers lie in a different horizontal plane. An extract of the simplified line model that is imported in the finite element software is illustrated in Figure 4.2.
Figure 4.2: Extract of the simplified line model of the H4V4H4 diamond crossing, imported in Siemens NX
Figure 4.3: Plan view of the line geometry of the diamond crossing H4V4H4 in Siemens NX
All the sleepers in the diamond crossing have a standard cross-section of 150 mm x 260 mm. The characteristic load-displacement curves of the lateral and longitudinal ballast resistance in Figure 3.7 and Figure 3.10 are thus applied. Check rails are present at the location of the outer frogs. These are also implemented in the finite element model with the principles described in chapter 3. The rails have a 60E1 cross-section and the check rails a 33C1
Model of a diamond crossing 58 section. Figure 4.3 gives an overview of the entire geometry of the model in the finite element software. The plan view and three-dimensional view of the rendered meshed geometry in Siemens NX are illustrated in respectively Figure 4.4 and Figure 4.5.
Figure 4.4: Plan view of the meshed geometry of the diamond crossing H4V4H4 in Siemens NX
Figure 4.5: Three-dimensional view of the meshed geometry of the diamond crossing H4V4H4 in Siemens NX
4.2.2 Track defects
First, a model is made without track imperfections and the temperature is raised to a sufficiently high value (>400°C). The results showed that the first lateral disturbance appears in the region between the check rails/frog and the common crossing, indicated as zone 3 in Figure 4.6. In zone 2, the check rails and frog contribute to a section of track panel with large lateral stiffness, which gives this zone a larger buckling strength. The same is true for zone 4, where the presence of the common crossing also makes the track panel stiff in the lateral direction. Zone 1 does not have this increased stiffness from added rail sections, but here the sleepers have lengths ranging between 450 cm and 570 cm contributing to large ballast resistances. The safety against buckling is thus also large in this zone.
Regarding the aforementioned reasons, zone 3 is indeed the weakest part of the diamond crossing. Here, no extra stiffness from frogs or check rails is present and the sleepers are rather short with lengths ranging between 300 cm and 390 cm. The lateral ballast resistance is thus limited compared to other sections of the track and the sleeper spacing is moreover larger than the other zones of the diamond crossing. In zone 3, a distance of 70,5 cm between the sleepers is foreseen in the plan (Figure A.5), while the spacings in the other zones are ranging between 43,5 cm and 60 cm. It is clear that the larger spacing provides less lateral resistance. Finally, the compression forces are
Model of a diamond crossing 59 present in four rails within a limited width of track panel, which makes the zone more prone to buckling compared to plain track with only 2 parallel rails.
Zone 1 Zone 2 Zone 3 Zone 4
Figure 4.6: Naming convention of the considered zones in the diamond crossing H4V4H4
A track defect with a wavelength of 8 m perfectly fits in the indicated zone 3 in Figure 4.6, which is in between the frog and the common crossing. The sinusoidal imperfection curve, of which the characteristics were discussed in chapter 3, is applied to the four rails in zone 3 with the imperfection oriented towards the same direction. This is the most critical situation and represents a stress-free initial state with an initial misalignment of the whole track panel.
Besides the implementation of the misalignment in the diamond crossing itself, another model with a track defect in the added plain track is made to make a comparison between the buckling behaviour for both defect locations. The conclusion that zone 3 of the diamond crossing is more prone to lateral track buckling can be confirmed in this way by calculation results. Firstly, track defects with an amplitude of 8 mm and a wavelength of 8 m are implemented in the finite element model, representing a realistic track condition for which no immediate corrective action is undertaken. Afterwards, the defect amplitude is increased to the most critical value of 20 mm, according to the guidelines mentioned in section 2.6.4. For larger defects, the railway track will be put out of service out of precaution. Figure 4.7 illustrates the locations of the implemented track defect for which the buckling behaviour is investigated. Note that the wavelength is drawn exactly in the figure, but the amplitude is not on scale.
Figure 4.7: Indication of the location of the implemented track defects with wavelength 8 m in plain track and in zone 3 of the diamond crossing H4V4H4
The diamond crossing is symmetric in two directions, which means that the buckling behaviour does not depend on the location of the track defects being before or after the central part of the crossing. The direction in which the defect is orientated is also not of importance due to the symmetry. The direction of the defect in plain track as shown in Figure 4.7 must be upwards, since the ballast resistance is much higher in the downward direction between the two railway tracks. Here the separating rail tracks are close to each other, so the real ballast resistance is large and buckling in the downward direction of this track segment is prevented naturally.
Model of a diamond crossing 60 4.3 Results and safety assessment
The calculations are performed for the finite element models of the H4V4H4 diamond crossing with the different implemented track defects. The results of the calculations encompass properties like displacements, forces and stresses for each load step of the non-linear calculation. These are used to post-process the results in contour plots and graphs. The obtained results also serve as a verification of the reliability of the finite element model.
4.3.1 Track defect 8m8mm – comparison between defect locations
Figure 4.8 illustrates the lateral displacements in the finite element model of the H4V4H4 diamond crossing with a track defect of 8m8mm in the crossing. The results are according to the last loadstep of the calculation, which is in a post-buckling configuration at +81,84 °C. It can be noticed that the track is unstable at the location of the implemented defect. The influenced zone is larger than the triggered part of the track panel according to the defect wavelength of 8 m. Parts of the frog and common crossings are also facing displacements. This is in accordance with Figure 2.8, which states that the buckled zone is larger than the original length of the misalignment.
Figure 4.8: Contour plot of the lateral displacements for an implemented track defect 8m8mm in zone 3 of the diamond crossing H4V4H4. Final loadstep (post-buckling – +81,84°C)
The buckling curves (load-displacement curves) are created by plotting the lateral displacements of a single node across the loadsteps of the calculation. The end node of the sleeper located at the crest of the implemented sinusoidal track defect is taken as reference, which is indicated by the circle in Figure 4.8. The graphs in Figure 4.9 are obtained for the different model configurations. The shape of the curves is in accordance with results found in the literature, which was discussed in section 2.3.2.
It is noticed that the curves corresponding to misalignments with an amplitude of 8 mm did not yet reach the lower critical buckling temperature. By increasing the number of load increments, the final solution step was not changed. It must be the case that the solver cannot find a solution past the last calculated deformation state. Adapting the arc-length method parameters, like the Minimum and Maximum Arc-Length Adjustment Ratio in Figure 3.14B, does also not change the final solution state. As mentioned by Landuyt (7), in complex finite element models it becomes
Model of a diamond crossing 61 difficult for the solver to find any stable solution that has an equilibrium between loads and displacements past the last solved step. This is especially the case when the load-displacement curve becomes almost horizontal, because for only a small load increase, very large displacements are needed to find an equilibrium.
When the curves for the two locations of the track defect 8m8mm are compared, it is seen that the crossing has lower safety against buckling than the plain track. The critical buckling temperatures are lower and the curves tend to evolve to progressive buckling, which means that the displacements in the pre-buckling branch already start to become large.
In Table 4.1, the resulting critical buckling temperature increases are summarized for the different model configurations. For a track defect 8m8mm in the crossing, the upper and lower critical buckling temperature increases above the neutral temperature are respectively 87°C and approximately 67°C. The lower value is estimated based on the trend of the buckling curve. In practice, no risk of buckling exists in this case as the maximal expected temperature in the rails is estimated to be 40°C for a neutral temperature of 20°C.
140 Track defect 8m8mm in plain track 120 Track defect 8m8mm in zone 3 of crossing Track defect 8m20mm in zone 3 of crossing
100
C] ° 80
60 temperature[ 40
Temperature increase Temperatureincrease above neutral 20
0 0 10 20 30 40 50 60 Lateral displacement [mm]
Figure 4.9: Load-displacement curves for different implemented track defects in the diamond crossing H4V4H4
Table 4.1: Critical buckling temperature increases for different model configurations of the diamond crossing H4V4H4
Critical buckling temperature increase Model configuration Upper critical buckling temp Lower critical buckling temp
횫Tb,max 횫Tb,min H4V4H4 – track defect 8m8mm in 123 °C ≅ 70 °C plain track H4V4H4 – track defect 8m8mm in 87 °C ≅ 67 °C zone 3 of crossing H4V4H4 – track defect 8m20mm in 52,6 °C 51,0 °C zone 3 of crossing
Model of a diamond crossing 62 4.3.2 Track defect 8m20mm
4.3.2.1 Buckling curve and critical temperatures
The track defect with a wavelength of 8 m implemented in zone 3 of the crossing yields the most critical buckling configuration. Standard plain track has a higher buckling reserve, which is concluded from the obtained results. Subsequently, the defect amplitude was increased to 20 mm, close to the ‘immediate action limit’ value for in- service track. The resulting buckling curve is given in Figure 4.9 and the critical temperature increases are presented in Table 4.1. As could be expected, the safety against lateral track buckling is lower compared to the smaller misalignment amplitude. The upper and lower critical buckling temperature increases above the neutral temperature are respectively 52,6°C and 51,0°C. This still provides enough safety regarding a maximal rail temperature of 60°C in combination with a neutral rail temperature of 20°C. The buckling regime is however small and the safety level 2 approach in Table 2.1 of section 2.3.3 prescribes in this case that the allowable temperature increase is equal to ΔTb,min – 5°C, or 46°C. This allowable temperature increase is close to the maximal expected value of 40°C.
The obtained buckling behaviour tends to evolve to progressive buckling. This means that the lateral displacements already become large for temperatures below the upper and lower critical buckling temperature. For the configuration with a track defect 8m20mm, the lateral displacement is approximately 2,5 mm at a temperature increase of 40°C. During a hot day in summer with an initial imperfection of 20 mm, the total misalignment of the track could thus reach 22,5 mm. This exceeds the immediate action limit value for small velocities prescribed in Table 2.4.
4.3.2.2 Lateral displacements
In Annex 3, contour plots of the lateral displacements for different temperature increases are given in Figure A.9 up to and including Figure A.12. The lateral displacements during buckling are the same for the rails and the sleepers at a certain location along the railway track, because the lateral fastener resistance is high. As a matter of fact, the lateral displacements of the track in the ballast can be examined along one rail, which represents the overall movement of the track. In Figure 4.11 the lateral displacements in function of the distance along the railway track are plotted for different temperature increases along the buckling curve. Figure 4.10 illustrates the convention of the points along the buckling curve where the results are taken. The model of the H4V4H4 diamond crossing in relation to the distance along the track is shown in Figure 4.12, which can serve as a reference for the obtained results. The initial track defect with wavelength 8 m is situated between the distances 118,5 m and 126,5 m.
For a small temperature increase, the lateral displacements of the track are negligible. At an increase of 41°C above the neutral temperature, the track defect is already growing and a maximal lateral displacement of about 2,5 mm is found. When the temperature is further increased, the track starts to buckle and no extra temperature increase is possible to obtain a stable solution. The displacements start to grow rapidly for an almost constant temperature increase, as noticed in the plots. It is observed that the influence zone of the defect increases when going further along the buckling curve, as could be expected from the principles visible in Figure 2.8.
Model of a diamond crossing 63 60
50 (3) +52,6°C (4) +51,0°C (5) +51,6°C
C] 40 ° (2) +41,0°C 30
20 temperature[
10
(1) +1,66°C Temperature increase Temperatureincrease above neutral 0 0 10 20 30 40 50 60 Lateral displacement [mm]
Figure 4.10: Designation of the points along the buckling curve where the results are taken. H4V4H4 8m20mm
The buckling shape is symmetrical, corresponding to a shape III mode in Figure 2.9. This shape is mainly noticed in straight tracks as was discussed in the literature review. As such, the observed shape in Figure 4.11 is a logical result, because the H4V4H4 geometry consists of intersecting straight rails. Once the track has started to buckle and reached the lower critical buckling temperature increase at 51,0°C, it is seen that the first and third individual waves of the instability do not increase further, but the defect as a whole is shifting in the lateral direction. The largest misalignment wave is still increasing in magnitude.
10
0
-10
-20 (1) 1,7°C
(2) 41,0°C -30 (3) 52,6°C
-40 (4) 51,0°C (post-buckling) Lateral displacement [mm] Lateral displacement
(5) 51,6°C (post-buckling) -50
-60 110 115 120 125 130 135 140
Distance along railway track [m]
Figure 4.11: Lateral displacements along the railway track for different temperature increases along the buckling curve. H4V4H4 8m20mm
Model of a diamond crossing 64 Distance along railway track [m]
Distance along railway track [m]
Figure 4.12: Plot of the measured distance along the railway track applied to the H4V4H4 model (contour plot of the lateral displacements)
4.3.2.3 Longitudinal displacements
The longitudinal fastener resistance is an important factor in controlling the longitudinal displacements of the rails and the sleepers. Contrary to the lateral fastener resistance, the longitudinal resistance of the fastening system is not that large and relative displacements between sleepers and rails can be expected. In the results from this and the next sections, the rails are designated according to the convention in Figure 4.13.
global x-axis
Figure 4.13: Naming convention of the rails in the diamond crossing H4V4H4
In Annex 3, the contour plots of the longitudinal displacements for different temperature increases along the buckling curve are given in Figure A.13 up to and including Figure A.16. The longitudinal displacements are here defined as the displacement along the global x-axis, which is along the symmetry axis of the diamond crossing as indicated in Figure 4.13. These plots seem to have the same shape as the lateral displacements. This has to do with the definition of the global x-axis, since the rails are positioned at a small angle compared to this axis. This means that part of the lateral displacements infiltrate in these plots.
Figure 4.14 gives the longitudinal displacements of rail A along the railway track, taken along the global x-axis. Indeed, the shape of the curve follows more or less the shape of the lateral displacements in Figure 4.11. This means that the lateral displacements of the rails are infiltrated in the results and that these plots are not completely valuable for interpretation.
Model of a diamond crossing 65 1,0
0,0
-1,0
(1) 1,7°C -2,0 (2) 41,0°C (3) 52,6°C
Longitudinal Longitudinal displacements [mm] -3,0 (4) 51,0°C (post-buckling) (5) 51,6°C (post-buckling)
-4,0 90 100 110 120 130 140 150 160
Distance along railway track [m]
Figure 4.14: : Longitudinal displacements of rail A (along global x-axis) along the railway track for different temperature increases along the buckling curve. H4V4H4 8m20mm
Figure 4.15 below shows a close-up of the results in Annex 3 for the post-buckling configuration at ΔT = 51,6°C. The track is shown at the location of the implemented defect. When comparing the displacements of the sleepers and the rails, it is already noticed that these are no longer equal. This means that some significant relative displacement is present and the rails are sliding in the fastening systems. The displacements in rail A and B and rail C and D respectively have opposite signs, which means that they are displaced in the opposite direction. This is explained by the infiltration of the lateral displacements.
Figure 4.15: Contour plot of the longitudinal displacements along the global x-axis at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm
Model of a diamond crossing 66 The longitudinal displacements have to be taken along the local coordinate system of the rails, with the x-axis defined along the axis line as illustrated in Figure 4.13. Figure 4.16 illustrates the displacements taken along the axis line of rail A. The difference with Figure 4.15 is immediately visible for the values of rail A and B. The longitudinal displacements of rail C and D are obviously even more influenced by the lateral displacements, as the reference axis is again inclined compared to their axis lines.
Figure 4.16: Contour plot of the longitudinal displacements along the axis line of rail A at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm
Figure 4.17 and Figure 4.18 give the longitudinal displacements along the axis line of respectively rail A and rail C. The results for both rails take the same shape, with positive displacements before the defect location and negative displacements after. As the positive sense is from left to right in Figure 4.12, this means that the rails are expanding towards the location of the defect. This is what could be expected, as the rails are moving in the lateral direction at the defect, making space for the rails to expand longitudinally.
At the centre (amplitude) of the defect, the longitudinal displacement becomes zero. It is noticed that this location of zero displacement is different for both rails. This is declared by the different angle of their centre lines, which causes a difference in force transfer. A twist in the curves is appearing for both rails, but it appears for rail A and C on a different location. In both plots, the influence zone of expanding rails is growing for increasing loadsteps, according to previous conclusions.
Model of a diamond crossing 67 1,0 (1) 1,7°C (2) 41,0°C
0,5 (3) 52,6°C (4) 51,0°C (post-buckling) (5) 51,6°C (post-buckling)
0,0
-0,5 Longitudinal Longitudinal displacements [mm]
-1,0 80 90 100 110 120 130 140 150 160 170 Distance along railway track [m]
Figure 4.17: Longitudinal displacements of rail A along the railway track for different temperature increases. Displacements taken along the rail’s axis line. H4V4H4 8m20mm
1,0
0,0
(1) 1,7°C (2) 41,0°C -1,0 (3) 52,6°C
(4) 51,0°C (post-buckling) Longitudinal Longitudinal displacements [mm] (5) 51,6°C (post-buckling)
-2,0 80 90 100 110 120 130 140 150 160 170
Distance along railway track [m]
Figure 4.18: : Longitudinal displacements of rail C along the railway track for different temperature increases. Displacements taken along the rail’s axis line. H4V4H4 8m20mm
The absolute values of the longitudinal displacements are larger for rail C compared to rail A. A comparison is made in Figure 4.19 at the post-buckling temperature increase of 51,6°C. A conclusive reason for this difference could not be found, but the difference in approach angle can be one of the underlying causes. In any case, the integration of the curves over the total length of the model around the ‘zero-line’ represents the total longitudinal displacement of the rail. As the rails are clamped at the ends, this must equal approximately zero, which is for both rails the case.
Model of a diamond crossing 68 1,0
0,0
-1,0 Rail A (51,6°C)
Rail C (51,6°C) Longitudinal Longitudinal displacements [mm]
-2,0 90 100 110 120 130 140 150 160 Distance along railway track [m]
Figure 4.19: Comparison between the longitudinal displacements of rail A and rail C at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm
As discussed earlier, the fasteners control the relative displacements between the rails and the sleepers. As the resistance by the fastening system is limited, relative displacements different from zero can be expected. In Figure 4.20, the longitudinal displacements of rail A are plotted against these of the sleepers underneath the rail for two different points along the path of the buckling curve. It is immediately noticed that the rail and sleepers are moving relatively to each other at the defect location. The absolute value of the relative displacements is increasing for increasing displacements along the buckling curve.
Also at the location of the check rails (zone 114 m – 118 m and 138 m – 142 m), small relative displacements are present as shown in the close-up. This can be explained as follows. The small check rails have no constraints at their ends and are thus able to expand. The rails nearby are however continuously welded and longitudinal expansions are hindered. The expanding check rails tend to move the sleepers longitudinally due to their connection by means of the fastening elements in the finite element model, being outwards from the central point of the check rails on. This is schematically illustrated in Figure 4.21, in which the deformed state after expansion of the check rails is represented by the dashed elements. The rails themselves do not undergo these longitudinal displacements due to their restraints and small relative displacements between rail and the sleeper underneath are the result.
The sleepers left and right with respect to the central point of the check rails are moving respectively in the negative and positive sense of the global x-axis. This behaviour is also observed in the plots in Figure 4.20, as the longitudinal displacements of the sleeper are first lower and then higher compared to these of the rail. The sleepers that are not connected to the check rails are not influenced by the expansions and here the relative displacements become zero again. Note that this behaviour is strongly influenced by the longitudinal resistance of the check rail fasteners in the finite element model. If they would have zero resistance, the described effect would disappear. If the resistance of the check rail fastener would in turn be much higher, the influence would be more distinct.
Model of a diamond crossing 69 (3) 52,6°C - Rail A (3) 52,6°C - Sleeper Side A
(5) 51,6°C (post-buckling) - Rail A (5) 51,6°C (post-buckling) - Sleeper Side A 4,0
3,0
2,0
1,0
0,0 Longitudinal Longitudinal displacements [mm]
-1,0 90 100 110 120 130 140 150 160 Distance along railway track [m]
Figure 4.20: Longitudinal displacements of rail A versus longitudinal displacements of the sleeper underneath rail A at the upper critical buckling temperature (+52,6°C) and at the post-buckling configuration (+51,6°C). H4V4H4 8m20mm
A B
Figure 4.21: Schematical sketch showing the influence of expanding check rails on the longitudinal displacements of rails and sleepers. (A) Undeformed configuration. (B) Configuration after a sufficiently high temperature increase
The values of the calculated relative longitudinal displacements are plotted in Figure 4.22 at different load configurations along the buckling curve. At the location of the local instability, the relative displacements are increasing. The increasing lateral displacements of the rails and sleepers could only occur if the rail is slipping longitudinally in the fastening system. At the realistic temperature increase of 41°C, the slipping of the rail in the fastener is limited to a value of approximately 0,1 mm. When the track would theoretically buckle, the relative displacements would increase fast: being for example already 3 mm at a temperature increase of 51,6°C.
Model of a diamond crossing 70 Referring to section 2.8.3, the limitation for the relative longitudinal displacements of the rails in the fastening system is 30 mm. It is concluded that this limit displacement will never be reached in the diamond crossing before buckling takes place.
Again, the same variation in relative displacements at the location of the check rails is observed, as was discussed before. For the check rails at a distance of about 140 m, this variation is perfectly symmetric around the zero-line. For the check rails at approximately 117 m, close to the misalignment, this is only the case for pre-buckling temperatures. In post-buckling configurations, the variation shifts towards negative displacements. This is explained by the fact that the misalignment is growing into the zone of the check rails and the rails are here already facing relative displacements. The variation is thus in equilibrium around the global trend-line of the relative displacements. These observations are illustrated in the close-up of Figure 4.22, in which the dashed line represents the zero-line.
3,5
3,0 (1) 1,7°C
2,5 (2) 41,0°C
(3) 52,6°C 2,0 (4) 51,0°C (post-buckling) 1,5 (5) 51,6°C (post-buckling)
1,0
0,5
Relative longitudinal Relative longitudinal displacement [mm] 0,0
-0,5 90 100 110 120 130 140 150 160
Distance along railway track [m]
Figure 4.22: Relative longitudinal displacements of rail A and sleepers along the railway track at different temperature increases. H4V4H4 8m20mm
4.3.2.4 Forces in lateral ballast springs
Figure 4.23 illustrates the forces present in the non-linear springs representing the lateral ballast resistance. These springs take up the forces generated by the lateral movement of the track. Outside the influence zone of the growing track defect, the lateral forces are zero as the track is not moving laterally there (see Figure 4.11). For a small temperature increase of 1,7°C, the forces in the ballast are limited. Note that the sign of the force is corresponding with the direction of lateral movement, being respectively upwards or downwards in plan view. In absolute values, no difference in interpretation of the forces has to be made.
Model of a diamond crossing 71 10
8
6
4
2
0
Force[kN] -2 (1) 1,7°C
-4 (2) 41,0°C
-6 (3) 52,6°C
-8 (5) 51,6°C (post-buckling)
-10 100 105 110 115 120 125 130 135 140 145 150 Distance along railway track [m]
Figure 4.23: Forces in the springs representing the lateral ballast resistance along the railway track for different temperature increases. H4V4H4 8m20mm
The limit displacement of the lateral ballast resistance was defined at 2 mm in the characteristics load-displacement curve given in Figure 3.7. For larger displacements, the ballast can no longer take extra lateral forces and the sleeper is ‘free’ to move in the lateral direction. When looking at the buckling curve for the track defect 8m20mm in Figure 4.9, it is seen that at the crest of the misalignment the limit displacement is reached at a temperature increase of approximately 38°C. For sleepers with length 300 cm, 330 cm, 360 cm and 390 cm, the maximal forces at the limit displacement are respectively 6,90 kN, 7,49 kN, 8,09 kN and 8,68 kN, according to Table 3.1. These are the type of sleepers present in zone 3 of the diamond crossing where the initial misalignment is present. In Figure 4.23, it is noticed that at the centre of the track defect (at 122,5 m) the limit force for the sleeper with length 330 cm is indeed reached for a temperature increase of 41°C.
When the temperature is further increased and the displacements grow along the buckling curve, the number of sleepers that reach the limit displacement and corresponding maximal resistance force is augmenting. This is also observed in Figure 4.23, where the zone with horizontal plateaus is broadening. This also means that a larger region of the railway track is ‘free’ to move in the ballast and the defect amplitude can grow quickly. Note that the length of the sleepers change in the defect region and because of that the horizontal plateaus are shifting in their limit forces.
The ever growing region of free movement can also be related to the behaviour observed in Figure 4.11. As discussed in section 4.3.2.2, the individual waves in the buckling mode are at a certain moment no longer growing individually in an opposite direction, but the whole buckled zone is shifting laterally in one single direction. This can be explained by the fact that a large number of sleepers have reached the maximal resistance force in the ballast and the whole defect area is able to move laterally without any restraints.
Model of a diamond crossing 72 4.3.2.5 Forces in longitudinal ballast springs
Similar like the previous section, Figure 4.24 illustrates the forces in the springs representing the longitudinal ballast resistances. Again, the sign is corresponding to the direction of longitudinal movement. In this respect, the same shape as Figure 4.17 and Figure 4.18 is found. The forces are taken for one of the two springs on each end of the sleepers. The somewhat erratic curves can be explained in this respect, as the longitudinal forces from the sleepers on the ballast are distributed to the two springs on each end. Depending on the small variations in the rotation of the sleepers around the vertical axis, the distribution in forces can fluctuate slightly.
According to the increase in absolute longitudinal displacements and broadening of the influence zone for increasing loadsteps concluded in Figure 4.17 and Figure 4.18, the same evolution is observed in the course of the longitudinal ballast forces. Also the small fluctuations at the position of the check rails are in accordance with the behaviour observed for the relative displacements in Figure 4.20 and Figure 4.22.
8
6
4
2
0
-2 Force[kN]
-4 (2) 41,0°C
-6 (3) 52,6°C
-8 (5) 51,6°C (post-buckling)
-10 90 100 110 120 130 140 150 160 Distance along railway track [m]
Figure 4.24: Forces in the springs representing the longitudinal ballast resistance along the railway track for different temperature increases. H4V4H4 8m20mm
4.3.2.6 Torques in the fasteners
The torques present in the fasteners along the railway track are given in Figure 4.25. The shape is in accordance with previous results such as the lateral displacements. Before buckling, at a realistic temperature increase of 41°C, the torques are limited to values of approximately 0,2 kNm. Referring to Figure 3.11, this value is in the first branch of the torsional fastener characteristic. This means that the resistance is still elastic, coming from generated friction. The adhesion between rail and railpad is consequently not broken. For the post-buckling situation given in Figure 4.25, the maximal absolute torques are larger than 2,0 kNm, being already in the third part of the characteristic resistance curve in Figure 3.11. This means that the rail is touching the fastener’s edge as described in section 2.6.3.3.
Model of a diamond crossing 73 3,0
2,0 (2) 41,0°C (5) 51,6°C (post-buckling) 1,0
0,0
Torque Torque [kNm] -1,0
-2,0
-3,0 100 105 110 115 120 125 130 135 140 145 150
Distance along railway track [m]
Figure 4.25: Torques in the fasteners along the railway track for different temperature increases. H4V4H4 8m20mm
4.3.2.7 Longitudinal forces in the fastener
Finally, also the longitudinal forces in the fasteners were examined. These are plotted in Figure 4.26. For increasing temperatures, the absolute values of the forces are increasing. Referring to the definition of the longitudinal fastener resistance in section 3.4.3, a maximal resistance force of 13 kN is obtained at a relative displacement of 0,2 mm. This maximum force is nowhere reached for a pre-buckling temperature increase of 41°C. For the post- buckling temperature of 51,6°C instead, the longitudinal forces in the fastener have the maximal value of 13 kN in the zone between 120 m and 125 m which is at the location of the local track instability. Here, the rails are thus slipping in the fastening system as was already concluded in paragraph 4.3.2.3. The effect of expanding rails towards the buckled zone, as discussed earlier, is also visible in the plot for the post-buckling configuration.
15,0
(2) 41,0°C 10,0 (5) 51,6°C (post-buckling)
5,0
0,0 Force[kN] -5,0
-10,0
-15,0 100 105 110 115 120 125 130 135 140 145 150 Distance along railway track [m]
Figure 4.26: Longitudinal forces in the fasteners along the railway track for different temperature increases. H4V4H4 8m20mm
Model of a diamond crossing 74 4.3.2.8 Axial forces
When the temperature is increased, the compression forces in the rails are growing because of the restrained expansion at the end nodes of the model. For a perfectly straight track without imperfections, the axial forces would increase with no lateral displacements until the buckling load is reached and the rails become unstable. When buckling, the rail forces will drop in the buckling influence zone as was already explained from Figure 2.8. This has to do with the elongation of the rails when they can move laterally in a buckled configuration. For this finite element model with the implemented track defect 8m20mm, the track already faces small lateral displacements before the upper critical buckling temperature is reached. This was concluded from Figure 4.11. It is therefore expected that a drop in rail forces is already taking place before buckling.
To assess the global behaviour of the axial forces during buckling, an equivalent total force is obtained by making the summation of the axial forces in the four rails designated in Figure 4.13. As such, the total longitudinal force present in the railway track is obtained. Subsequently, the axial forces in the individual rails are examined and compared. The resulting axial forces along the railway track of the diamond crossing, after summation of the forces in the individual rails, are given in Figure 4.27 and Figure 4.28. The forces have a negative sign as these represent compression forces. For increasing temperatures, the axial forces also increase, which is in accordance with formula (4.1) that expresses the thermal force in a rail that is restrained from any expansions:
FEATTEATx = ( 0 −) = (4.1)
At both ends of the model, the axial forces in the results represent the thermal force for a clamped rail at both ends for which expansions are completely restrained. The obtained results in Siemens NX and the calculated values with formula (4.1) are compared in Table 4.2. The values for the E-modulus, the coefficient of expansion 훼 and the cross- sectional area A of a 60E1 rail profile are respectively 206 GPa, 1,152⋅10-5 and 76,70 cm². It is concluded that the differences are neglectable and that the obtained results from the finite element model are reliable.
Table 4.2: Comparison axial forces obtained from the finite element model and calculated with formula (4.1). H4V4H4 8m20mm
Axial force per rail [kN] Temperature increase above neutral temperature Obtained from model Calculated with (4.1)
1,7°C -30,32 -30,25 37,9°C -689,24 -689,20 52,6°C -958,15 -958,18 51,0°C -930,44 -930,48 51,6°C -939,63 -939,59
Model of a diamond crossing 75 -4000
-3000
-2000 (1) 1,7°C Axial Axial [kN] force (2) 37,9°C -1000 (3) 52,6°C
0 0 50 100 150 200 250 Distance along railway track [m]
Figure 4.27: Total axial force after summation of the axial forces in the 4 individual rails of the diamond crossing, for different temperature increases along the buckling curve (pre-buckling). H4V4H4 8m20mm
-3900
-3800
-3700
-3600
Axial Axial [kN] force -3500 (3) 52,6°C (4) 51,0°C (post-buckling) -3400 (5) 51,6°C (post-buckling)
-3300 0 50 100 150 200 250 Distance along railway track [m]
Figure 4.28: Total axial force after summation of the axial forces in the 4 individual rails of the diamond crossing, for different temperature increases along the buckling curve (post-buckling). H4V4H4 8m20mm
In each graph, two small drops in axial forces are observed at distances of approximately 117 m and 140 m. This is exactly at the location of the check rails. The reason for this is found by looking at the free expansions of the check rails and sleepers, which also induce small expansions in the rails. That is to say, a local decrease in compression. This is also related to the small fluctuation observed in Figure 4.22 for the relative longitudinal displacements, which show that the rails are slipping in the fasteners and some expansion of the rails could take place here. The explained theory in the corresponding section can thus also be applied here.
Model of a diamond crossing 76 The influence of the check rails is examined in Figure 4.29, in which the axial forces of the individual check rails are added to the summation of the four other rails and compared to the results shown before. Now the total axial force increases locally at the location of the check rails, which is due to the added cross-section of the check rail. The rail force in the check rail could not be as high as would be the case for a clamped rail, because the ends of the check rail are free to expand. The expansion of the check rail is only restrained by the connection with the sleepers formed by the check rail fastening systems in the finite element model.
-2800
-2750
-2700 Axial Axial [kN] force
-2650 (2) 37,9°C (Check rails added) (2) 37,9°C (Without check rails)
-2600 100 105 110 115 120 125 130 135 140 145 150 Distance along railway track [m]
Figure 4.29: Comparison between the summation of the axial forces of the rails for a calculation with and without the axial forces in the check rails. H4V4H4 8m20mm
The contour plot of the longitudinal displacements at a realistic temperature increase of 37,9°C are illustrated in Figure 4.30, that immediately visualises the expansion of the check rails. Note that the longitudinal displacements of the sleeper relative to the rails in Figure 4.21 are exaggerated, but the actual ‘expansions’ of the sleepers are nevertheless slightly visible in this contour plot. At this temperature increase, the maximal rail force that is present in each of the check rails is -43,56 kN. The axial force that would be expected for a clamped check rail (A = 42,02 cm²) calculated with formula (4.1) amounts -376,33 kN. It is clear that the present force in the check rails does not reach this value and that consequently the increase in the total axial force in Figure 4.29 is limited.
Finally, in Figure 4.27 and Figure 4.28 the overall drop in rail forces when the track becomes unstable is visibly present, as expected. Also before the upper critical buckling temperature is reached, a minor drop in axial compression forces is noticed as the track defect is already growing laterally. The drop stays however limited up to and including the upper critical buckling temperature, after which the axial forces are decreasing more significantly. The largest decrease in compression force is found right at the centre of the defect, located at approximately 122,5 m. In the post-buckling branch of the solution, the buckling influence zone is enlarging.
Model of a diamond crossing 77
Figure 4.30: Contour plot of the longitudinal displacements of the check rails at a temperature increase of 37,9°C. H4V4H4 8m20mm
With the behaviour of the total axial forces in the railway track being investigated, the courses of the individual rail forces are plotted in Figure 4.31 and Figure 4.32, respectively for the post-buckling temperature increase of 51,6°C and the pre-buckling temperature increase of 37,9°C. Together with the actual rail forces also the average rail force is given, which is calculated as the total axial force from the previous results divided by four. In both graphs, the small drops in forces are again found at the location of the check rails.
-940
-920
-900
-880
-860
-840 Axial Axial [kN] force Average rail force -820 Rail A Rail B -800 Rail C Rail D -780
90 100 110 120 130 140 150 160 Distance along railway track [m]
Figure 4.31: Comparison between the axial forces of the individual rails and the average rail force, post-buckling (+51,6°C). H4V4H4 8m20mm
Model of a diamond crossing 78 -705 Average rail force Rail A Rail B Rail C -695 Rail D
-685 Axial Axial [kN] force
-675
-665 100 105 110 115 120 125 130 135 140 145 150
Distance along railway track [m]
Figure 4.32: Comparison between the axial forces of the individual rails and the average rail force, pre-buckling (+37,9°C). H4V4H4 8m20mm
The forces in the rails are fluctuating around the average rail force, which is a similar behaviour as found in the course of the longitudinal displacements in Figure 4.17, Figure 4.18 and Figure 4.19. The relation between both results is thus discernible. The alternating decreases and increases in compression forces correspond to local elongations and shortenings of the rails respectively. This behaviour is observed in the growing misalignment curve. This can also be related to the courses of the axial strains, which will be outlined in the next section. From a safety point of view, the absolute values of the rail forces at a temperature increase of 37,9°C are important. This is approximately the maximal temperature increase that could be expected in reality for railway track with a neutral temperature between 20°C and 25°C. It is seen in Figure 4.32 that the maximal compression force is about 10 kN higher than the nominal rail force of 689 kN.
In section 2.8, limit values in the Eurocode EN 1991-2 for bridge applications were adduced. In compression, a limit peak value of approximately 172 MPa is prescribed at a temperature increase of 40°C. The value is not set as a criterion for applications here, but gives a representation of the order of magnitude of the allowable compression stresses. In the diamond crossing being investigated in this dissertation, an axial peak force of 701 kN present in a 60E1 rail section corresponds to a stress of 91,4 MPa. Also for the nominal force of 689 kN, the rail stress already amounts 89,8 MPa. Taking into account the stress distribution in a bended rail as discussed in section 2.3.7 (see Figure 2.11A), the stress peak in the most compressed fibre in the defect region of the track will be higher.
This additional compression stress due to bending can be assessed by plotting the bending moments Mzz in rail A of the crossing at the location of the instability for a temperature increase of 41°C. The results are given in Figure 4.33. At the centre of the misalignment, the bending moment reaches a maximal absolute value of 2 kNm. It is
Model of a diamond crossing 79 noticed that the springs representing the fastening systems in the finite element model are connected to the rails in the centroid of the rail beam. This is contrast with reality, in which the forces are introduced at the rail foot. Therefore, the bending moments will not completely coincide with what could be expected in reality and outlined in Figure 2.11A, but the difference is further neglected. As a matter of fact, the bending moments in the finite element model are not crooked, but symmetrical with respect to the centroid. The maximal stress due to bending in the most compressed fiber is calculated with formula (4.2), in which the characteristics of the 60E1 rail section are in accordance with Figure A.1 in Annex 1.
My 2 106 Nmm 75,0 mm =zz max = = 29,28 MPa bending 44 (4.2) Izz 512,3 10 mm As such, the expected maximal compressive stress in the rail section is the summation of the normal stress and the maximal bending stress. The value of 29,3 MPa needs thus to be added to the maximal normal stress peak of 91,4 MPa, yielding a value of 120,7 MPa. This stress peak accelerates the fatigue of the rail and needs to be taken into account by the designers. It does not exceed the reference limit value prescribed in the Eurocodes (section 2.8).
1,5
1,0
0,5
0,0
-0,5
-1,0 Bending Bending moment [kNm]
-1,5
-2,0 110 115 120 125 130 135 Distance along railway track [m]
Figure 4.33: Bending moments Mzz in rail A along the railway track at a temperature increase of 37,9°C (pre-buckling). H4V4H4 8m20mm
The frog and common crossing were modeled by two intersecting rails which are connected by lateral rail elements, as shown in Figure 4.34A and outlined in section 3.3.1. It is expected that this has an influence on the distribution of rail forces. Therefore, two modifications were made on the way the frog and crossing were modeled in the finite element model, in order to investigate the influence on the course of the axial forces and buckling behaviour:
− Modification 1: The lateral rail elements that form the connections between the intersecting rails are removed and the rails at the location of the frog and common crossing are meshed with a rectangular cross-section with dimensions 170 mm x 170 mm. The rendered mesh in the finite element model for this modification is illustrated in Figure 4.34B. This way of modeling represents the larger section and high
Model of a diamond crossing 80 stiffness that is accompanied with the real configuration of the frog and common crossing. The fabrication and appearance in track of a frog are illustrated in Figure 4.35. The fabrication of these elements often happens by a single casting of manganese steel (8). The coefficient of thermal expansion of this material is approximately 2,2⋅10-5 (48) (49), which is about two times higher than the coefficient for steel. The corresponding compression forces for restrained expansion would for this material thus also be significantly larger. The material in the finite element model is kept steel. Considering the larger cross- section, an increase in thermal forces is expected and as a consequence also the buckling curve will be influenced. − Modification 2: The check rails and the lateral connecting elements for the frog and common crossing are removed from the model. Only four intersecting 60E1 rails are remaining. Because there is in this case no variation in cross-section, the course of the axial forces should be fluent.
A B
Figure 4.34: Rendering of the modeled frog and common crossing, for the original model (A) and modification 1 (B)
A B
Figure 4.35: (A) Fabrication of common crossing in Bascoup (50). (B) Photograph of the frog in combination with check rail (51)
Firstly, the influence on the critical buckling temperatures is investigated. The resulting buckling curves for both modifications are given in Figure 4.36 and compared with the load-displacement curve of the original finite element model of the H4V4H4 diamond crossing with implemented track defect 8m20mm. As expected, modification 1 with a larger cross-section of the frogs and common crossing has a significant influence on the critical buckling temperature increases, that are decreased with about 5°C. The upper and lower critical buckling temperature increases for this modification are respectively 47,3°C and 45,5°C, compared with 52,6°C and 51,0°C for the original
Model of a diamond crossing 81 finite element model. The influence of the way of modeling of the frogs and common crossing can therefore not be neglected. But still, there is enough safety margin with respect to the maximal expected rail temperature increase of 40°C in reality. However, taking into account the level 2 safety approach in Table 2.1 for small buckling regimes, the allowable temperature increase above the neutral rail temperature would be 40,5°C. This becomes critical in theory. It is also noticed that the lateral displacements before buckling have increased slightly. At a temperature increase of 40°C above the neutral rail temperature, the displacement can be up to 3,5 mm. This means the lateral displacements become critical for safe operation of track.
For modification 2 of the finite element model, the influence on the buckling curve is much smaller. Indeed, as the cross-section of the rails is not modified, the axial forces will not change much and the critical buckling load is not affected. There is some influence on the lower critical buckling temperature, which is about 1°C lower than the original model. This has to do with the smaller contribution to the lateral stiffness of the frogs and common crossings due to the removal of the connecting rail elements and check rails for this modification. As was concluded from the previous sections, the influence zone of lateral displacements and axial forces is broadening and affecting the region outside the defect in the post-buckling behaviour. It is therefore logical that the decreased stiffness in these regions also yields a reduction in the lower critical buckling temperature.
60
50
C] 40 °
30 H4V4H4 8m20mm 20 temperature[ H4V4H4 8m20mm - Modification 1
10 H4V4H4 8m20mm - Modification 2 Temperature increase Temperatureincrease above neutral 0 0 20 40 60 80 100 Lateral displacement [mm]
Figure 4.36: Buckling curves for the different modifications of the H4V4H4 model with defect 8m20mm
A closer look can now be given to the influence of the modifications on the courses of the axial forces along the railway track. Figure 4.37 and Figure 4.38 give a comparison between the results of the total axial forces present along the railway track for pre-buckling and post-buckling temperature increases. As the three models have a different solution and buckling curve, the exact temperature increases are not the same as these are dependent on the loadsteps followed by the solver. A larger temperature gives a larger nominal axial force at the ends of the model. However, the focus is placed on the course of the forces and not on the comparison between the absolute values.
Model of a diamond crossing 82 For modification 1, a local increase in total axial forces is observed in the central region of the track model for the pre-buckling temperature increase of 39,9°C. This is due to the larger cross-section of the rails that are present at the frogs and common crossing, which yield a larger thermal force according to formula (4.1). The transition in axial force does not happen suddenly, but is spread over a certain length of rail. This means that the 60E1 rails also have to face larger axial forces, which is expressed in the lower buckling strength at the location of the implemented track defect. To allow the transition in axial forces, the 60E1 rails are shortened to increase in compressive stress, meaning creep is developing.
In rail C of the finite element model with modification 1, the maximal compression force in the 60E1 cross-section even amounts -903 kN at the temperature increase of 39,9°C. This corresponds to a stress of 117,7 MPa in compression, which is significantly larger than the value of 91,4 MPa in the original model. It can be concluded that a larger cross-section assigned to the frog and common crossing not only has a significant influence on the critical buckling temperatures, but also on the maximal stresses in compression. The effect on the fatigue life of the rail is thus even larger. Two small drops in axial forces are again present at the location of the check rails for the course of axial forces of modification 1, for which the explanation is given in the previous paragraphs. After buckling, the rail forces drop in the buckling influence zone, in accordance with the previous results.
For modification 2, which is a model with only four intersecting rails, the influence of the check rails and frog has disappeared in the course of the axial forces. A smooth transition is observed without local drops or fluctuations. At a pre-buckling temperature of 35,2°C, a small drop in rail forces at the location of the track defect is observed as the misalignment is already growing.
-3400
-3200
-3000
-2800
-2600 Axial Axial [kN] force
-2400 H4V4H4 8m20mm (37,9°C) H4V4H4 8m20mm - Modification 1 (39,9°C) -2200 H4V4H4 8m20mm - Modification 2 (35,2°C)
-2000 0 50 100 150 200 250 Distance along railway track [m]
Figure 4.37: Comparison between the course of the total axial forces for different model configurations of the H4V4H4 diamond crossing with implemented track defect 8m20mm in zone 3 (pre-buckling temperatures)
Model of a diamond crossing 83 -4000
-3800
-3600
-3400 Axial Axial [kN] force
H4V4H4 8m20mm (51,6°C) -3200 H4V4H4 8m20mm - Modification 1 (53,1°C)
H4V4H4 8m20mm - Modification 2 (50,5°C) -3000 0 50 100 150 200 250
Distance along railway track [m]
Figure 4.38: Comparison between the course of the total axial forces for different model configurations of the H4V4H4 diamond crossing with implemented track defect 8m20mm in zone 3 (post-buckling temperatures)
A final remark is given about the transition of the axial forces from pre-buckling to post-buckling temperatures in modification 1 of the finite element model. At +39,9°C a local increase in compression forces is observed, while at the post-buckling temperature increase of +53,1°C the forces have dropped in the buckling influence zone. Figure 4.39 illustrates the course of the axial forces at an intermediate stage between both situations. It is observed that locally at the location of the track instability, the forces have already dropped. Further away from the defect location, the compression forces are still increased. This points at the fact that in this configuration, the buckling influence zone is still limited and at +53,1°C the latter has widened significantly.
It is also noticed that the difference in upper and lower critical buckling temperature increases for modification 1 is slightly larger than the original model. This has to do with the increased lateral stiffness of the frog and common crossing, which withstands the buckling longer (in terms of displacements along the buckling curve). This is in accordance with the behaviour in Figure 4.39 that shows that the common crossing still has increased axial forces, which means that the rails are not yet elongating there. This points to increased strength against lateral and longitudinal displacements of the rail.
Model of a diamond crossing 84 -3700
-3600
Axial Axial [kN] force -3500
-3400 0 50 100 150 200 250 Distance along railway track [m]
Figure 4.39: Total axial forces along the railway track (summation of rail forces) at 훥T = 48,2°C (post-buckling), for modification 1 of model H4V4H4 8m20mm
4.3.2.9 Axial strains
The axial strains give an indication of the elongations or shortenings of the rails and can serve as validation of the structural behaviour of the finite element model. Figure 4.40 gives the results in rail A for the considered temperature increases in pre- and post-buckling configurations along the buckling curve. The absolute values of the strains are increasing for increasing lateral displacements, which is as expected regarding the previous results.
A positive strain means an elongation in rail, a negative value corresponds to shortening of the rail. Figure 4.41 illustrates the contour plot of the axial strains in the whole defect area. It is remarked that rail A/B and rail C/D have opposite signs of strains. This is explained by the different angle in which both rails are orientated compared to each other, in combination with the defect which is orientated in the same sense for all rails. As the lateral distance between the rails is fixed because of the large lateral fastener resistance, the alternating elongation and shortening between the rail seems to be logical. For rail A, the rail is compressed in the central region of the defect and elongated sideways of this compressed central section. The course of the strains is also in accordance with the course of the axial forces in rail A in Figure 4.31. Locations with positive strains correspond with locations of decreased compressive forces relative to the average rail force and vice versa for negative strains. The different signs for rail A/B and C/D are also in accordance with different fluctuations of axial forces for these rails in Figure 4.31. The structural behaviour of the model is thus logical and valid.
Model of a diamond crossing 85 0,0016
0,0012
0,0008
0,0004
0,0000
-0,0004 (1) 1,7°C
-0,0008 (2) 41,0°C Axial Axial strain [mm/mm] (3) 52,6°C -0,0012 (4) 51,0°C (post-buckling) -0,0016 (5) 51,6°C (post-buckling)
-0,0020 110 115 120 125 130 135 140 Distance along railway track [m]
Figure 4.40: Axial strains in rail A along the railway track for different temperature increases along the buckling curve. H4V4H4 8m20mm
Figure 4.41: Contour plot of the axial strains at 훥T = 51,6°C (post-buckling). H4V4H4 8m20mm
For a perfectly straight track, the compression forces in the rails increase combined with zero elongation of the rail up to and including the upper critical buckling temperature. When the track has buckled, the rails have elongated in combination with the lateral displacements. For a track with a small misalignment, some minor, but limited elongations are expected to occur already before the track is actually buckling. The total elongation of the rails in the model can be derived from the course of the axial strains, by the integration of the strains over the total length L of the rail according to formula (4.3):
Model of a diamond crossing 86 LLN l(x) xx(x) + xx ( x + x) L =xx ( x) dx = dx = x (4.3) l 2 xx==00n=1 As the results of the axial strains are taken for a finite number of elements, the integration is approximated by the summation of the average values of the strains multiplied by the distance interval.
The results for these calculations are given in Table 4.3 for the different load configurations along the buckling curve. At a small initial temperature increase, the elongation of the rails is almost zero. At the upper critical buckling temperature, the total elongation of the rail is limited but already non-zero, because the lateral misalignment has grown slightly. Once the track has buckled, the elongation of the rail is increasing rapidly as the lateral displacements of the track are increasing at the defect.
Table 4.3: Total elongation of rail A for different temperature increases along the buckling curve. H4V4H4 8m20mm
Temperature increase above Total elongation of rail neutral temperature (1) 1,7°C 0,00 mm (2) 37,9°C 0,04 mm (3) 52,6°C 0,38 mm (4) 51,0°C 1,22 mm (5) 51,6°C 1,89 mm
4.3.3 Influence of different parameters on the buckling strength
4.3.3.1 Characteristic load-displacement curve of the lateral ballast resistance
In section 3.4.2.1, it was decided to model the lateral ballast resistances by applying the formulas for changing sleeper length proposed by Zacher. The characteristic load-displacement curves in Figure 3.7 were obtained. However, the lateral ballast resistances can also be based on the contribution factors proposed by the NMBS, which yielded the input data given in Figure 3.8. To investigate the influence of this choice, the results for both input data can be compared. The buckling curves corresponding to both approaches are given in Figure 4.42. The difference in critical buckling temperatures is approximately 1,5°C. This difference is marginal, but not completely neglectable. To assess the safety of the diamond crossing, an extra margin on the critical buckling temperatures calculated with the Zacher approach could be taken, to take into account the influence of the definition of the conversion formulas.
Model of a diamond crossing 87 55
50 C] ° 45
40 temperature[ H4V4H4 8m20mm - Lateral ballast resistance according to Zacher 35
H4V4H4 8m20mm - Lateral ballast resistance according to NMBS Temperature increase Temperatureincrease above neutral
30 0 5 10 15 20 25 30 35 40 45 Lateral displacement [mm]
Figure 4.42: Comparison between the buckling curves for the input data of the lateral ballast resistance according to Zacher and NMBS. H4V4H4 8m20mm
4.3.3.2 Influence of the check rails
It is assumed that the check rails in zone 2 in Figure 4.6 contribute to an increased lateral stiffness of the track panel and therefore improve the buckling strength. In section 4.3.2.8 it was also concluded that the axial forces in the rails reduce locally thanks to the influence of longitudinal forces in the check rails. To examine the influence of the presence of the check rails on the buckling behaviour, a finite element model was made without the check rails. The resulting buckling curve is given in Figure 4.43. It is immediately concluded that the difference in buckling strength is neglectable. A slight decrease in the upper critical buckling temperature is noticed and the buckling behaviour tends a little more to progressive buckling. However, the uncertainty on the parameters is certainly larger than the difference observed here, such that the influence of the removal of check rails can be disregarded.
55
50
C] ° 45
40 H4V4H4 8m20mm - Removal check rails
temperature[ H4V4H4 8m20mm - With check rails
35 Temperature increase Temperatureincrease above neutral 30 0 10 20 30 40 50 Lateral displacement [mm]
Figure 4.43: Comparison between the buckling curves for a H4V4H4 diamond crossing with initial misalignment 8m20mm, with and without the presence of check rails
Model of a diamond crossing 88 4.4 Parameter study
With a parameter study, the sensitivity of the critical buckling temperature increases to changes in the structural characteristics is investigated. The ballast resistances and the fastener characteristics are increased and decreased by a percentage change. The corresponding critical buckling temperature increases are then determined and compared.
4.4.1 Influence of track defect
In the previous sections, the results of the critical buckling temperatures of the H4V4H4 diamond crossing with implemented defects 8m8mm and 8m20mm were presented. It is concluded that a larger misalignment amplitude results in a much lower critical buckling temperature. This is in accordance with the conclusions found in the literature (see section 2.5.7).
The influence of the defect wavelength is already investigated by different authors. Landuyt (7) performed a detailed parameter study for varying wavelengths and he concluded that the critical buckling temperature increases significantly for increasing wavelength. In the case of a diamond crossing, implementing a defect with wavelengths of 10 m or 15 m in zone 3 would mean that the initial misalignment would be present in the frog and common crossing. As these are heavy blocks of steel, it is not expected that a major misalignment could occur in this element. In any case, configurations with a misalignment with a larger wavelength than 8 m will not yield lower critical buckling temperature increases. Therefore, the influence of this parameter is not further investigated in the context of this dissertation. For the same reason, a defect with the crest of the misalignment in the heart of the common crossing is not investigated. At this location, the sleepers have a minimal length of 270 cm and thus the smallest lateral resistance in the ballast bed. But again, the fabricated heavy steel pieces are not assumed to have a misalignment in track and the lateral stiffness of these fabricated pieces is much higher compared to standard rails.
In the following sections, the parameter study is performed on the finite element model of the H4V4H4 diamond crossing with the initial misalignment 8m20mm implemented in zone 3 of the crossing.
4.4.2 Influence of lateral and longitudinal ballast resistance
In section 2.5.3, it was already concluded that the lateral ballast resistance has a significant influence on the critical buckling temperatures of the railway track. The influence of the longitudinal ballast resistance is rather limited, with the upper critical buckling temperature being almost not affected.
Landuyt (7) assumed that the longitudinal and lateral ballast resistance vary simultaneously, because he expects that if the ballast quality changes, it will do in all directions. The simultaneous variation is indeed true for a change in ballast quality. However, the ballast resistance does not only depend on the quality, but also on the height of the ballast shoulder (29), the placement of anchor caps or retaining walls (see section 2.6.1.6), influence of nearby tracks, etc. It is assumed that these parameters mainly affect the later ballast resistance and only in a minor way the longitudinal ballast resistance. Therefore, the influence of the lateral and longitudinal ballast resistance are investigated independently of each other.
Model of a diamond crossing 89 4.4.2.1 Lateral ballast resistance
The input values for the lateral and longitudinal ballast resistances are increased and decreased with percentages ranging between -20% and +40%. Changing the characteristic load-displacement curves by means of a percentage change means that the resistance increases more in absolute value for a longer sleeper compared to the standard sleeper length of 270 cm. In case the ballast quality is changed, this is a good idealisation. However, if the resistance is changed by for example the installation of anchor caps, it is assumed that the resistance is increased with the same absolute value for all sleeper lengths. In this case, the study with a percental change is less representable for the ballast resistance increase of all sleepers. But as the lateral ballast resistance is mostly important in the defect area and the sleeper length variation is not large there, it may be expected that the influence is not radical.
A decrease in lateral ballast resistance will in most cases be linked to a decrease in ballast quality or local degradation of the ballast. Besides that, the decrease in ballast resistance can also be due to the effects described in section 2.6.1.4. The lateral ballast resistance can decrease with 20% or more under the effect of heavy rain, vibrations due to train loading and uplift of railway track. An increased resistance can be linked to a better quality/consolidated ballast, but also to the application of the different techniques described in section 2.6.1.6. The theoretical influence on the buckling strength of the latter applications will be discussed in more detail in section 4.5.4. In this paragraph, only the theoretical influence of a percental change in lateral ballast resistance is reported.
Figure 4.44 presents the results for the parameter study of the lateral ballast resistance. The influence on the critical buckling temperatures is significant, especially for the upper critical buckling temperature increase. For decreasing lateral ballast resistance, the load-displacement curves tend more and more to progressive buckling. This means that the lateral displacements before buckling already become large. For a decrease of 20% in resistance, the lateral misalignment grows with almost 4 mm at a temperature increase of 40°C.
65
60
55
C] ° 50
45 Lateral ballast resistance -20%
temperature[ Lateral ballast resistance -10% 40 H4V4H4 - 8m20mm Lateral ballast resistance +10%
Temperature increase Temperatureincrease above neutral 35 Lateral ballast resistance +20% Lateral ballast resistance +40% 30 0 5 10 15 20 25 30 35 40 45 50 Lateral displacement [mm]
Figure 4.44: Influence of increasing/decreasing lateral ballast resistance on the buckling curve. H4V4H4 8m20mm
Model of a diamond crossing 90 The courses of the upper and lower critical buckling temperatures according to each configuration are plotted in Figure 4.45. For smaller ballast resistances, the upper and lower critical buckling temperatures are closer to each other compared to larger resistances. This confirms the tendency to progressive buckling for low ballast quality. With an increase in lateral ballast resistance of 40%, the upper critical buckling temperature increases with almost 10°C compared to the base model. This corresponds to an increase of approximately 17% in critical temperature. It is concluded that the buckling strength of the track and the lateral ballast resistance do not have a one-to-one relationship.
Although dynamic buckling is not studied in this dissertation, the effect of the reduced ballast resistances that possibly arise under dynamic loads can be quantified with the performed parameter study. As already outlined before in chapter 2, the lateral resistance can decrease with more than 20% due to vibrations or uplift of track, which results in a decrease in the critical buckling temperatures with more than 4°C. The upper critical buckling temperature increase above the neutral temperature may become lower than 48°C. Together with the results from the different way of modeling of the frog (configuration 1) in Figure 4.36, the critical buckling temperature increase in case of dynamic vibrations and uplift can become equal to approximately 40°C or 41°C. This means the safety of the crossing becomes critical. Moreover, the lateral displacements at +40°C reach values of more than 4 mm, which may become critical for safe operation of track regarding derailments. It must be mentioned that the reduction in critical buckling temperature is quantified for the ballast conditions representing recently tamped track. Under train loads, the ballast bed will be consolidated and the lateral ballast resistance in unloaded conditions will increase significantly.
The results could be linked to more practical modifications to the dimensions of the sleepers. To become an increase of the lower critical temperature of approximately 5°C, the lateral ballast resistance needs to be increased with about 20 %. This means that the lengths of sleepers in the defect area (330 cm – 390 cm) need to be increased to lengths of about 390 cm to 450 cm, according to the formulas by Zacher (12). A change in cross-section from 150 mm x 260 mm to 200 mm x 300 mm results in the same effect.
4.4.2.2 Longitudinal ballast resistance
In section 2.5.3.2 of chapter 2, it was concluded that the longitudinal ballast resistance has only a small effect on the buckling strength of the track. The different load-displacements curves of the parameter study are not plotted here, as almost no difference is visible between them. The resulting variations in critical buckling temperature increases of the parameter study are presented in Figure 4.45. It is noticed that the upper critical buckling temperature experiences almost no change from increasing or decreasing longitudinal ballast resistance. The lower critical temperature undergoes a more visible change, in accordance with Figure 2.18 from chapter 2. Although, the temperature increase is changed with only 0,5°C between -20% and +20%. It is indeed concluded that the longitudinal ballast resistance has almost no influence on the buckling resistance of continuously welded railway track.
Model of a diamond crossing 91 4.4.3 Influence of fastener characteristics
4.4.3.1 Influence of the torsional fastener resistance
The characteristic curve of the torsional fastener resistance, presented in Figure 3.11, is scaled with percentage changes between -20% and +20%. The results of the calculations with these modified torsional fastener properties are given in Figure 4.45. Again, the individual buckling curves are not plotted, because the differences are small. As expected from the literature study presented in section 2.5.4.1, the buckling strength increases with increasing torsional fastener resistance. The gain in strength is however small.
Also a model with zero torsional fastener resistance is made, corresponding to a -100% decrease. This means that the rails are free to rotate in the connections with the sleepers. In this case the upper critical buckling temperature has decreased to 50°C, but the difference with the original model stays limited. It can be concluded that in this particular case the torsional fastener resistances do not have a significant contribution to the buckling strength of the track.
4.4.3.2 Influence of longitudinal fastener resistance
Finally, the influence of the longitudinal fastener resistance is plotted in the same Figure 4.45. It is noticed that the influence on the buckling strength is more significant compared to other parameters. For an increase of the limit resistance at 0,2 mm from 13 kN to 18,2 kN, the critical temperatures increase with more than 2°C. The longitudinal resistance of the rails in the fasteners manage the longitudinal force transfer. It was concluded in the previous results that in the defect area the rails are slipping in the fastening systems while the misalignment is growing. It is therefore logical that the buckling strength of the crossing increases if this slipping is restrained more.
On the other hand, if the rails are fixed to the sleepers the drop in axial forces (expanding rails) has to occur almost completely through longitudinal movement of the sleepers in the ballast. This means that local expansion of rails is more prevented, which is disadvantageous regarding the release of compression forces. Considering this, the obtained buckling strength should decrease for increasing longitudinal fastener resistance. This was concluded by Landuyt (7), discussed in section 2.5.4.2. As the contrary is observed in Figure 4.45, it can be concluded that the restraint in slipping (yielding higher resistance to lateral displacements) has a more important effect on the buckling behaviour in the diamond crossing. The opposing effect can however be slightly observed in the curve, as the slope becomes less steep for increasing resistances.
4.4.4 Summary and comparison of parameter study
Figure 4.45 gives a comparison between the influence of the investigated parameters. It is noticed that a comparison in percentage changes may not be completely reliable. The ballast resistance is for example more sensitive to variations than the properties of the fastening systems, which are produced in quality controlled environment. Wear of these elements may reduce the resistance during the lifetime of the fastener. Although an exact comparison between the parameters is difficult, the global sensitivity can be roughly compared in the obtained graphs.
Model of a diamond crossing 92 It is immediately concluded that the most important parameter influencing the lateral track buckling is the lateral resistance of the sleepers in the ballast bed. This is in accordance with the different literature about this subject, which was discussed in chapter 2. The parameter which has only a marginal influence on the buckling strength is the longitudinal ballast resistance. The second most influential parameter is the longitudinal fastener resistance, which can be explained by its influence on the longitudinal force transfer to the sleepers and control of permissivity.
Lateral Ballast Resistance: Tb,max Lateral Ballast Resistance: Tb,min
Longitudinal Ballast Resistance: Tb,max Longitudinal Ballast Resistance: Tb,min
Torsional Fastener Resistance: Tb,max Torsional Fastener Resistance: Tb,min
Longitudinal Fastener Resistance: Tb,max Longitudinal Fastener Resistance: Tb,min
62
61
60
59
C] 58 °
57
56
55
54
53
52
51
Temperature increase Temperatureincrease above neutraltemperautre [ 50
49
48
47
46 -100% -80% -60% -40% -20% +0% +20% +40%
Percentage change in parameter value
Figure 4.45: Summary of the parameter study for the H4V4H4 diamond crossing (track defect 8m20mm in zone 3)
Model of a diamond crossing 93 4.5 Improving the buckling resistance of a diamond crossing
Let’s for simplicity define the lower critical buckling temperature increase as the limit temperature increase for which the track becomes unsafe to operate (level 1 safety). With enough external energy applied to the track, there is indeed a risk that the track will find an equilibrium in a post-buckling state at this temperature. For a track defect with a wavelength of 8 m and a defect amplitude of 20 mm implemented in zone 3, the diamond crossing is consequently at risk of buckling at a temperature increase above the neutral temperature of 51°C. With a different way of modeling of the frogs and common crossing (see modification 1 in section 4.3.2.8), the safety limit could even be situated at 45,5°C. Taking into account the influence of dynamic loads and vibrations on the lateral ballast resistance, the critical temperatures could further decrease by more than 4°C.
With the maximal expected rail temperature increase of 40°C in practice, the strength reserve for risk at track instability becomes small. At this point, the lateral displacements of the track as well as the stress concentrations in bended rails become important. The lateral displacements could already amount to 2 to 4 mm at this temperature on top of the initial defect amplitude of 20 mm (20 mm = intervention limit; only corrective action in the foreseeable future). According to the limitations by NMBS (Table 2.4) the track would be put out of service, meaning economic losses. Therefore, there could be a desire to improve the buckling resistance of the diamond crossing. In this subsection, some solutions are provided to increase the buckling strength of the H4V4H4 diamond crossing. Of course, regular maintenance and inspections are necessary to limit the track imperfections, which reduce the critical buckling temperatures significantly. In the following paragraphs, the critical buckling temperatures are calculated for the track defect 8m20mm.
4.5.1 Influence of the ballast quality
The input characteristics of the lateral and longitudinal ballast resistance defined in section 3.4.2 correspond to low quality ballast or recently tamped track. This means that the calculation of the critical buckling temperatures is done for a configuration of track in its worst possible conditions. For moderate or good quality ballast and consolidated track, the peak resistance increases and the buckling temperatures will be higher. The same is true for the lateral displacements before buckling, that would be smaller. The parameter study in section 4.4.2.1 showed the possible gain in buckling strength for varying lateral ballast resistance. The abovementioned parameters are not under control of the designer. Regular maintenance of the track is of importance, but a situation with low ballast quality always needs to be taken into account.
4.5.2 Increasing the sleeper lengths and widths
As concluded in section 4.4, the lateral ballast resistance is the most influential factor in the buckling phenomenon. A larger ballast resistance will thus have a significant influence on the buckling temperatures. Besides the quality of the ballast, this structural parameter can also be improved by lengthening and widening of the sleepers in the critical central zone of the diamond crossing. This length of the sleepers is, in contrast to the quality of the ballast, completely under control of the designer.
Model of a diamond crossing 94 The critical zone for buckling reaches out between the locations of the check rails as illustrated in Figure 4.46, corresponding mostly to zones 3 and 4 in Figure 4.6. As the lateral displacements are growing in a wider region than the initial defect location, it is necessary to increase the ballast resistance in this broad zone.
Figure 4.46: Critical zone for buckling in the diamond crossing H4V4H4
If for example the sleeper length would not be changed at the location of the common crossing (zone 4), this would introduce a local weak zone. Preliminary testing with such a model showed that this leads to local instability of the common crossing, which is not desirable. This is illustrated in Figure 4.47 for a configuration in which the length of the sleepers in the critical zone is increased to 570 cm, except for the sleepers at the location of the common crossing. With the defect implemented outside the region of the common crossing, the latter faces yet instability indicating that this is no optimal solution.
Figure 4.47: Contour plot of the lateral displacements of the railway track for a configuration with larger sleepers except at the location of the common crossing. H4V4H4 8m20mm
The most extreme configuration to improve the lateral ballast resistance, is increasing the sleeper lengths in the critical zone up to 570 cm, which is the largest sleeper present in the geometry. The resulting rendered geometry is illustrated in Figure 4.48.
Figure 4.48: Geometry of the diamond crossing for a configuration with sleepers with length 570 cm in the critical zone.
Model of a diamond crossing 95 A more suitable modification exists in enlarging the sleepers up to a length of 450 cm, like illustrated in Figure 4.49. To further improve the resistance, the sleepers could be made wider, increasing the standard width from 260 mm to 300 mm. The lateral ballast resistance is increased significantly with this modification.
Figure 4.49: Geometry of the diamond crossing for a configuration with sleepers with length 450 cm in the critical zone.
Finally, also a solution with the sleepers of original length 270 cm, 300 cm, 330 cm and 360 cm being enlarged to 390 cm is proposed. Again, the width of the sleepers could be modified too, increasing it from 260 mm to 300 mm. These modifications are the least radical.
The results for the proposed modifications are given in Figure 4.50 and Table 4.4. The increase in upper critical buckling temperatures is quite significant. Already 3°C is gained with sleepers of 390 cm. Also the larger width leads to an additional increase of 2,6°C. For larger sleepers in the critical zone of the diamond crossing, the difference between the upper and lower critical temperature becomes larger. The buckling behaviour thus tends less to progressive buckling, which is an additional improvement. Also the lateral displacements at +40°C have decreased with reasonable values. The proposed modifications thus yield a better resistance against local instability. As a consequence of the reduced displacements, also the stress peaks in bended rails will be smaller.
70
C] °
60
50 H4V4H4 8m20mm Sleepers 570 cm Sleepers 450 cm 40 Sleepers 450 cm - width 300 mm Sleepers 390 cm
Sleepers 390 cm - width 300 mm Temperature increase Temperatureincrease above neutraltemperature [ Sleeper spacing = 550 mm 30 0 5 10 15 20 25 30 35 40 45 50 Lateral displacement [mm]
Figure 4.50: Influence of the different modifications in the sleeper geometry on the buckling curve. H4V4H4 8m20mm
Model of a diamond crossing 96 Table 4.4: Overview of the critical buckling temperature increases and lateral displacements at ≅+40°C for different model configurations. Solutions to improve the buckling strength of the H4V4H4 diamond crossing
Critical buckling temperature increase Lateral displacement Model configuration Upper critical 횫Tb,max Lower critical 횫Tb,min at ≅+40°C
H4V4H4 8m20mm - original geometry 52,6 °C 51,0 °C 2,5 mm H4V4H4 8m20mm – Sleepers with 66,2 °C ≅ 60 °C 0,8 mm length 570 cm in critical zone H4V4H4 8m20mm – Sleepers with 59,4 °C ≅ 55,7 °C 1,4 mm length 450 cm in critical zone H4V4H4 8m20mm – Sleepers with 62,0 °C ≅ 56,7 °C 1,1 mm length 450 cm in critical zone + width = 300 mm H4V4H4 8m20mm – Sleepers with 55,6 °C 53,2 °C 1,8 mm length 390 cm in critical zone H4V4H4 8m20mm – Sleepers with 58,0 °C 54,8 °C 1,5 mm length 390 cm in critical zone + width = 300 mm H4V4H4 8m20mm – Sleeper spacing 60,6 °C ≅ 57,8 °C 1,3 mm = 550 mm in zone 3 H4V4H4 8m20mm – Lateral ballast 57,0 °C 54,2 °C 1,6 mm resistance +20% H4V4H4 8m20mm – Lateral ballast 61,5 °C 57,1 °C 1,2 mm resistance +40%
4.5.3 Denser sleeper spacing
As was already noticed in section 4.2.2, the sleeper spacing in zone 3 of the diamond crossing is large. The spacing is 705 mm compared to values ranging between 435 mm and 610 mm in the other zones. A better resistance against buckling can thus be achieved by installing the sleepers with a smaller spacing, for example 550 mm. This configuration is illustrated in Figure 4.51 and is evaluated.
The results are again given in Figure 4.50 and Table 4.4. The improvement is significant, as the critical buckling temperatures have increased by about 8°C. The lateral displacement at +40°C is halved. With only a small modification in geometry from an economical point of view, the safety margin is thus increased significantly. This solution to improve the buckling resistance seems to be the best option.
Figure 4.51: Difference in sleeper spacing in zone 3 of H4V4H4. Left: spacing = 550 mm. Right: spacing = 705 mm.
Model of a diamond crossing 97 4.5.4 Installing anchor caps on the sleepers
In section 2.6.1.6, different applications were described that influence the lateral (and to a lesser extent the longitudinal) ballast resistance. These can be used to improve the buckling strength of the diamond crossing. Table 4.5 summarizes the expected increases in lateral ballast resistance for the mentioned applications. These applications could subsequently be linked to the results calculated in the parameter study in section 4.4.2.1. In Table 4.4, the critical buckling temperatures for an increase in lateral ballast resistance of 20% and 40% are given.
Table 4.5: Expected increase in lateral ballast resistance for different reinforcing applications
Application Increase in lateral ballast resistance
Safety/anchor cap on 1 sleeper out of 3 +22% Safety/anchor cap on 1 sleeper out of 2 +40% Safety/anchor cap on every sleeper +90% Rail type anchor system (Figure 2.31B) +40% Retaining wall along the railway track +34% Gluing ballast shoulders +230%
Applying an anchor cap on every sleeper out of 3 results in an increase of the upper critical buckling temperature of about 5°C. An anchor cap on every second sleeper increases the upper critical buckling temperature by 9°C. The same can be concluded for the application of a retaining wall along the track and the rail type anchor system in Figure 2.31B. It must be taken into account that these results are obtained for the model wherein all springs representing the lateral ballast resistances are adjusted. The applications mentioned here only need to be applied in the critical zone according to Figure 4.46. But because the sleepers are only moving in the zone close to the defect, the difference in obtained results will be small and the aforementioned values are a good reference.
The influence of the application of an anchor cap on every sleeper (+90%) and gluing of the ballast shoulder (+230%) is not calculated. But relying on the tendency obtained in Figure 4.45, an increase of the upper critical buckling temperature of at least 20°C can be expected.
4.6 Comparison between H4V4H4 and H3V3V3
The finite element models in the previous subsections were all based on the geometry of the H4V4H4 diamond crossing. The crossing rails are here intersecting with an angle of 0,124 radians (1:8) as illustrated in Figure 4.52A. The H3V3H3 diamond crossing approaches with a smaller angle, being 0,108 radians (1:9,2). To assess the difference in safety between both geometries, a finite element model of the H3V3H3 diamond crossing is made with the same principles applied before. The track defect 8m20mm is again applied in the model.
Model of a diamond crossing 98 A
B
Figure 4.52: Simplified line models of the H4V4H4 (A) and H3V3H3 (B) diamond crossings
The plan of the H3V3H3 diamond crossing is given in Figure A.6 in Annex 2. With the angle between the intersecting rails being smaller, comes the fact that the total length of the diamond crossing is longer. The distance between the frog/check rails and the common crossing (zone 3 in Figure 4.6) is therefore somewhat larger. The defect is implemented as close as possible to the common crossing, being most critical regarding the smaller lateral resistance. The rendered mesh of the finite element model is illustrated in Figure 4.53.
Figure 4.53: Rendered mesh of the H3V3H3 model in Siemens NX
Globally seen, the length of the sleepers at the location of the defect is therefore also smaller. As a consequence, the rail forces are transferred to the sleeper on a smaller width of track panel, meaning a larger concentration of axial forces. Taking this into account, it can be expected that these factors may contribute to a small decrease in buckling strength compared to the H4V4H4 geometry. However, an important difference for both configurations is the sleeper spacing in zone 3 where the track defect is implemented. For the H4V4H4 diamond crossing, the spacing was equal to 705 mm, while for the H3V3H3 this is only 635,3 mm. This factor may again contribute to an increased buckling strength.
The buckling curves for both geometries are compared in Figure 4.54. It is immediately noticed that the difference is neglectable. The upper critical buckling temperatures are more or less equal and the lower critical buckling temperature for the H3V3H3 is slightly higher. The load-displacement curve for the H3V3H3 geometry corresponds almost completely to progressive buckling behaviour, making it less safe than the H4V4H4 diamond crossing.
It can be concluded that the two factors described above, which differentiate the H3V3H3 diamond crossing from the H4V4H4, almost outweigh each other. The denser sleeper spacing compensates the detrimental effect of the smaller sleepers and more concentrated axial forces. Nevertheless, since the lower critical buckling temperature is higher for the H3V3H3, the risk at buckling is lower. But the lateral displacements before buckling become slightly higher in contrast.
Model of a diamond crossing 99 55
50 C] ° 45
40
temperature [ temperature H4V4H4 - 8m20mm
35 H3V3H3 - 8m20mm Temperature increase Temperatureincrease above neutral the
30 0 5 10 15 20 25 30 35 40 45 Lateral displacement [mm]
Figure 4.54: Comparison between the buckling curves for the diamond crossing H4V4H4 and H3V3H3, with implemented track defect 8m20mm in zone 3
4.7 Model with fishplated joints
Before the application of continuous welded rail in turnouts and crossings, fishplated joints were used to connect the turnout/crossing with the railway track. These joints, illustrated in Figure 4.55 and Figure 4.57, allow (limited) thermal expansion of the rails. Up until this day, there are plenty of turnouts and crossings present in Belgian railway lines connected with fishplated joints. According to Ferdinande from Infrabel (52), the H4V4H4 is however always welded in the railway track.
Figure 4.55: Drawing of a fishplated joint (8)
According to the provided drawings of the H4V4H4 diamond crossing however, a tolerance gap of 3 mm is provided between the heart of the common crossing and the rails, as illustrated in Figure 4.56. A reason for this can be found by the fact that the heart of the common crossing and the other rails cannot be welded together due to the limited place on site, which makes an aluminothermic welding impossible. The connection can be made by a fishplated joint. The transversal bolts are provided with a tolerance, which makes the longitudinal horizontal displacement of the rails unrestrained. At the moment the rails have expanded and the 3 mm gap is closed, contact is made and longitudinal forces are transferred through the connection.
Model of a diamond crossing 100
Figure 4.56: Detail of the 3 mm gap joint on plan [mm] Figure 4.57: Photograph of a fishplated joint between two rails
The main objective of this master dissertation is the investigation of the buckling behaviour of continuously welded crossings. However, it is worthwhile to study the behaviour of the diamond crossing with the (theoretical) implementation of the fishplated joint. A typical cross-section and the geometry of the fishplates are illustrated in Figure 4.58. The locations where the fishplated joints are implemented in the finite element model of the H4V4H4 diamond crossing are illustrated in Figure 4.59.
Figure 4.58: Cross-section and geometry of fishplates (£600) (53)
Figure 4.59: Locations were the fishplated joints are implemented in the model of the H4V4H4 diamond crossing
Model of a diamond crossing 101 4.7.1 Modeling the fishplated joint
To model this connection in the finite element model, the element ‘CGAP’ in NX Nastran is used (54), which defines a gap and friction element. It simulates an unidirectional point-to-point contact. The properties for this element are defined on the PGAP entry. This element allows to model the initial gap opening of u0 = 3 mm between both rails.
When the gap is closed, meaning that uAB− u u0 , an axial stiffness (Ka) is present, while an open gap can be assigned a small open stiffness (Kb). When the gap is open, the CGAP element has no transverse stiffness. For a closed gap, the element has an elastic stiffness in the transverse direction Kt until the lateral force exceeds the friction force and slip starts to occur.
The application of the GAP element poses difficulties in the convergence of the solution in non-linear analysis, particularly when friction is involved (55). Penalty values are introduced to avoid penetration and to enforce the sticking condition between the contact points. The difficulties arise when these penalty values are not properly chosen. The CGAP element changes its status (open or closed) when the load is applied. Since the solution method is sensitive to abrupt changes in stiffness during iterations, it could cause a divergent or oscillatory solution when the CGAP changes its status. The value of the closed stiffness Ka should therefore be chosen to be three orders of magnitude higher than the stiffness of the neighbouring grid points. A larger value of Ka may slow down the convergence or cause divergence, while a small value may result in an inaccuracy.
The CGAP element is implemented in the model by an individual line of length 4 mm between the two ends of rail. The element’s x-axis is defined by its end nodes (corresponding to GA-GB). The orientation of the y- and z-axes is determined by the orientation vector, which defines the x-y plane, comparable with the definition of the CBUSH elements in chapter 3.
Figure 4.60 gives the input window for the PGAP entry. The initial gap opening is set to 3 mm, corresponding to Figure 4.56. The preload of the element is 0 N. The axial stiffness is determined by the characteristics of steel for the neighbouring meshed elements, equal to E⋅A/L. The exact stiffness is not necessary, as the closed gap only needs to physically transfer forces and a stable solution is only required. The applied axial stiffnesses are visible in the given window. The axial stiffness for open gap is set to 1 N/mm. This small value physically means that there is not any restraint for longitudinal expansions of the rails, but it gives no numerical problems which could possibly be the case when a value of 0 is entered. The coefficient of friction is taken large enough to preserve force transfer. The maximal allowable penetration is set to 0,01 mm, which is small enough but does not give any problems related to convergence of the solution. The other parameters are kept at their default values.
Model of a diamond crossing 102
Figure 4.60: Input data for the PGAP entry in Siemens NX
First, a model was made with the CGAP element defined, without taken into account the presence of the fishplates. This means that any restraint to transverse movement of the rails comes from the defined friction. The results revealed that the rail ends start to slide relatively to each other at a certain point, because the friction resistance is exceeded. However, in reality the fishplates have bending stiffness and will prevent the transverse sliding of rail ends. To obtain the desired behaviour of the fishplated joint, these fishplates must be implemented in the finite element model.
The difficulty in modeling the fishplates lies in the fact that they may not transfer any longitudinal forces. The reason for this can be found in the small clearance between the bolts and the fishplate to allow longitudinal movements of the rails. As a consequence, the fishplates cannot be modeled fixed to the rails. A solution was found by modeling spring elements between rails and fishplates, as illustrated in Figure 4.61. The springs can be assigned a very high stiffness along their own axis (x-axis in Figure 4.61, transverse to the rail axis) and a very low stiffness along their second degree of freedom (y-axis in Figure 4.61, parallel to the rail axis). As such, transversal forces induced by rail movements are fully transferred to the fishplates, but no longitudinal ones. The mesh representing the fishplate is given a double cross-section of the typical £600 fishplate in Figure 4.58, because in reality a fishplate is installed on both sides of the rail as seen in Figure 4.55.
Figure 4.61: Schematical representation of the implementation of the CGAP element and modeling of the fishplate
Model of a diamond crossing 103 4.7.2 Results and comparison with continuously welded diamond crossing
The influence of the fishplated joints is first examined for the longitudinal displacements. The joints must be able to accommodate longitudinal expansions of rails. Figure 4.62 illustrates the longitudinal displacements of the rails and sleepers at a temperature increase of 40°C, which is the maximal expected rail temperature in reality. At this point, the rails are expanding towards the common crossing in the gap tolerances of the fishplated joints. The expansions are taking place over a wide zone in the diamond crossing. The gap of 3 mm is not yet completely closed at this temperature increase as a maximal longitudinal displacement of about 2,8 mm is observed in the plot.
Figure 4.62: Contour plot of the longitudinal displacements at 훥T = 40°C. H4V4H4 8m20mm with fishplated joints
At the same temperature increase of 40°C, the lateral displacements are given in Figure 4.63. It is observed that the initial misalignment has increased slightly, which is as expected because compression forces are still being built up in the rails. This is explained by the fact that the longitudinal fastener resistance and longitudinal ballast resistance of the sleepers do not allow a completely free expansion. The lateral displacement is however limited to maximal 1 mm, which is less than observed in the results of the continuously welded diamond crossing in section 4.3.2.
Figure 4.63: Contour plot of the lateral displacements at 훥T = 40°C. H4V4H4 8m20mm with fishplated joints
Model of a diamond crossing 104 At a temperature increase of approximately 60°C, the gaps in the joints are already completely closed and longitudinal forces are transferred through the joint. This means that the railway track is at risk of becoming unstable. The resulting load-displacement curve is plotted in Figure 4.64. The critical buckling temperature increases are higher compared to the results of the continuously welded diamond crossing. The lateral displacements before buckling, or the growth of the misalignment, are clearly smaller. This is because the rails could expand and the compression forces in the rails are lower. The resulting buckled shape of the railway track is illustrated in Figure 4.65.
70
60
50
C] ° 40
30
temperature[ 20 H4V4H4 8m20mm - Continously welded H4V4H4 8m20mm - Fishplated joints
10 Temperature increase Temperatureincrease above neutral
0 0 5 10 15 20 25 30 35 40 45 Lateral displacement [mm]
Figure 4.64: Comparison between the buckling curves for a continuously welded diamond crossing and a diamond crossing connected with fishplated joints. H4V4H4 8m20mm
Figure 4.65: Contour plot of the lateral displacements at 훥T = 53,8°C (post-buckling). H4V4H4 8m20mm with fishplated joints
In continuously welded railway track, snap-through could occur at a temperature lower than the upper critical buckling temperature. In the case of jointed track, this is however dependent on whether the gap between the rails is closed and longitudinal forces are transferred through the joint or not. Snap-through of the track at the lower
Model of a diamond crossing 105 critical buckling temperature increase can only occur if the rails are transferring axial forces through the joint at this temperature. In the case investigated here with a track defect 8m20mm, the rails are transferring axial forces already below the lower critical buckling temperature of 58,7°C. This means snap-through is possible if sufficient external energy is applied. For smaller defects, the buckling regime will be much larger. However, in these cases the lower critical buckling temperatures will also be higher than the value of 58,7°C, so that the gap in the fishplated joint is certainly closed. As such, the H4V4H4 diamond crossing with the fishplated joints implemented is thus also vulnerable for snap-through behaviour at temperatures lower than the upper critical buckling temperature. However, these temperatures are much higher than the expected maximal rail temperature, so that there is no risk of buckling in practice. It is moreover noticed that the mechanical strength of the fishplates and the bolts themselves will probably be more critical than the actual buckling behaviour of the crossing at these temperatures.
Finally, the course of the axial forces for the diamond crossing with fishplated joints is examined. Figure 4.66 illustrates the results for rail A for different temperature increases. When the gaps in the joints are not yet closed, the joints do not transfer longitudinal forces and the compression forces in the rails are cut down to zero towards the ‘free’ rail ends. At the moment the joints start to transfer forces at approximately 40°C, the compression forces are developing from the location of the joints towards the clamped ends. In the post-buckling equilibrium, the compression forces have dropped in the defect area, corresponding to the results outlined in section 4.3.2.8. It is noticed that the influence of the expanding check rails is also observed in these plots by means of the small drops in compressive force at the locations of these elements.
-1200
-1000
-800
-600
10°C Axial Axial [kN] force -400 40°C 59,6°C (post-buckling)
-200
0 0 50 100 150 200 250 Distance along railway track [m]
Figure 4.66: Course of the axial forces in rail A along the railway track for different temperature increases. H4V4H4 8m20mm with fishplated joints
Model of a diamond crossing 106 4.8 Conclusions
In this chapter, the buckling behaviour of a continuously welded diamond crossing was investigated. On the basis of the obtained results of the displacements and the course of the internal forces, the reliability of the finite element model was validated. Since no inexplicable faults or behaviours were observed, it is concluded that the finite element model behaves as expected and can be used to verify the buckling strength of the diamond crossing.
The obtained critical buckling temperatures are sufficiently high to ensure a safe operation during hot summer days. Static buckling will not occur below a temperature of 51°C for the base model. This critical buckling temperature was obtained for a model with an implemented sinusoidal defect curve with wavelength 8 m and amplitude 20 mm in the weakest zone of the diamond crossing. This misalignment yields a track configuration for which corrective actions must be undertaken in the foreseeable future, but for which the track is not being put out of service immediately.
Depending on the definition of some parameters like the ballast resistances and the fastener characteristics, the critical buckling temperature could drop to 48°C. Also the way the frog and common crossing are modeled in the software affects the critical temperatures with some degrees. Nevertheless, with a maximal expected temperature increase above the neutral rail temperature of 40°C, the safety margin in case of static buckling is still large enough. Taken into account the level 2 safety approach in Table 2.1, the allowable temperature is 5°C lower than the lower critical buckling temperature for a small buckling regime. Following this reglementation, the safety of the diamond crossing can become critical in particular cases. Moreover, taking into account the reduced lateral ballast resistance due to dynamic vibrations and uplift of track, the critical buckling temperatures can be further reduced with more than 4°C. Therefore, it could be desired to improve the buckling resistance of the diamond crossing.
In this chapter, different solutions were presented to increase the buckling strength of the diamond crossing H4V4H4. Not only the critical buckling temperatures can be increased, but also the lateral displacements and peak stresses before buckling reduced. From an economical and practical point view, the most suitable solutions are the applications of safety caps on the sleepers in the critical zone or reducing the sleeper spacing in the same zone from 705 mm to 550 mm.
Not only the critical buckling temperatures form a criterion to continuously weld the diamond crossing in track, but also the resulting peak stresses in the rails and the permissivity of the rails in the fastening systems. The latter are sufficiently small and do not surpass the limit value of 30 mm. A maximal peak stress of 120 MPa was observed in the rail of the diamond crossing at a temperature increase of 38°C. This peak value may reduce the fatigue life of the rail, but does not exceed the reference limit value of 172 MPa obtained from EN 1991-2. This limitation is determined for specific track configurations which are not applicable in this study, but it gives a good indication of the order of magnitude of the allowable stresses. The lateral displacement of the initial misalignment could already reach values up to 3 or 4 mm before buckling. Considering the limits for safe operation of the railway track, these displacements may already become too large.
Model of a diamond crossing 107 Chapter 5 Model of a diamond crossing with double slips
5.1 Introduction
In the previous chapter, the buckling behaviour of a diamond crossing was investigated. In this type of crossing, the trains cannot switch track and the model consists out of four ongoing and intersecting rails. In a diamond crossing with double slips (Dutch: Engelse Wissel), the junction rail of the diamond crossing is replaced by tongues. Figure 5.1 gives a photograph of a diamond crossing with double slips in Belgium. As the geometry of rails is changed, a different behaviour in rail forces during temperature increase is expected and the buckling resistance will consequently be different from the diamond crossing.
Figure 5.1: Photograph of a diamond crossing with double slips (56)
To make a finite element model of a diamond crossing with double slips, the geometry of a TJD EUH4 is used, for which the plan is given in Figure A.7 in Annex 2. The geometry of the simplified line model of the intersecting tracks is actually the same as the H4V4H4 (same intersection angle), which makes it appropriate to compare with the results of the diamond crossing in chapter 4.
Model of a diamond crossing with double slips 108 5.2 Geometry
The same principles as discussed in chapter 3 and 4 are applied to the imported geometry of the TJD EUH 4 in the finite element software Siemens NX. Figure 5.2 gives the plan view of the line-model in the software. The movable parts of the switchblades are not modeled, for the reasons explained before. In the central region of the crossing, the extra rails of the slips contribute to a stiff track panel.
Figure 5.2: Plan view of the geometry of the TJD EUH4 diamond crossing with double slips in Siemens NX
The basic structure of the TJD EUH4 is similar to the H4V4H4 diamond crossing. As a consequence, the same convention of the designation of the zones in Figure 4.6 can be applied to the TJD EUH4, as given in Figure 5.3. For the reasons already mentioned in section 4.2.2, the most critical zone for buckling is again zone 3, which is between the check rails/frog and common crossing. The stiffness of zone 4 in the TJD EUH 4 is even higher than for the diamond crossing, because the large concentration of rails is contributing to the lateral stiffness of the track panel.
Zone 1 Zone 2 Zone 3 Zone 4
Figure 5.3: Designation of the different zones considered in the TJD EUH4
The track defect is thus implemented in zone 3 and applied to the four rails. This allows comparison between the diamond crossing H4V4H4 and the TJD EUH4. Again a misalignment with wavelength 8 m perfectly fits in this zone between the frog and common crossing. The buckling behaviour is investigated for two different defect amplitudes of 8 mm and 20 mm respectively. Note that parts of the movable switchblades are also present in zone 3, but the connections of these blades do not transfer significant forces to the sleepers. As a consequence they do not contribute to a significant increase in lateral stiffness of the track panel and their influence on the buckling behaviour can be disregarded. In addition, it must be mentioned that part of the junction rails that change in tongues have fixed connections to the sleepers in the defect zone 3, as clarified in Figure 5.4. The misalignment is not applied on these small sections of rails. If the misalignments were also applied to these elements, the influence on the buckling behaviour would be neglectable as the axial forces in the rail ends are low. It is however expected that their stiffness contributes to a slightly larger resistance to the growth of the lateral displacements.
Model of a diamond crossing with double slips 109
Zone 3
Figure 5.4: Part of junction rails (orange) that lie in the defect zone, but have not applied an initial misalignment
The TJD EUH4 is like the H4V4H4 symmetric in two directions, such that the buckling strength does not depend on the location and orientation of the misalignment in the considered zone. The defect is in the TJD EUH4 applied in the upward direction in Figure 5.3.
Contrary to the diamond crossing H4V4H4, the rails of the TJD EUH4 have a 50E2 profile in accordance to the plan, being smaller than the common 60E1 rail. As a result, the rails will develop smaller thermal forces. But this profile yields a reduced second moment of area and consequently also a smaller lateral bending stiffness. The smaller thermal forces reduce the buckling risk, but the reduced lateral stiffness makes the track again more vulnerable to buckling. It has to be investigated which of the two characteristics affect the buckling temperatures the most. The 50E2 rails are only assigned to the actual geometry of the TJD EUH4. The rails in the track before and after the TJD EUH4 are assigned the common 60E1 rail profile. This is illustrated in Figure 5.7, in which the light blue rails represent a 50E2 rail section and the dark blue rails a 60E1 profile. As the 60E1 profiles provide higher thermal forces outside the implemented track defect, this configuration will yield the most conservative situation.
Another difference in geometry with the H4V4H4 diamond crossing is found in the curvature of the rails. In the H4V4H4 model, the four intersecting rails are perfectly straight (except for the initial misalignment). In the TJD EUH4 diamond crossing with double slips, the two outer rails are curved.
At the location of the switchblades, the rail foots of the stock rails are abraded to allow the tongues to align with the rails. This is visible in the extract of the plan of the TJD EUH4 in Figure 5.5. This is not modeled in the finite element model, as the nominal rail section 50E2 is applied in the entire rail. As mentioned earlier, it is expected that this geometry may have an influence on the stress concentrations in the rail. These are local effects, which do not influence the buckling load of the track. The modified cross-section will however influence the second moment of area of the rail and the development of thermal forces. The lateral bending stiffness of the rail is decreased, while the nominal force in the rail does not differ much due to the influence of the nominal cross-sections nearby. It can thus be expected that the critical buckling load will be lower in case of the abraded rails, especially because these are located in the zone where the initial misalignment is implemented. However, here the Vossloh fastening systems are replaced by the connections with the gliding chairs, as seen in the extract of the plan and illustrated in the photograph of Figure 5.6. It is assumed that the latter provide a higher lateral and torsional resistance compared to the characteristics of the common Vossloh fastener. This again improves the local buckling resistance of the track. Therefore, it is assumed that no large differences will be obtained in the results when the abrasion of the rails are omitted from the finite element model, compared to the actual behaviour of the TJD EUH4 in reality.
Model of a diamond crossing with double slips 110
Figure 5.5: Extract of the plan of the TJD EUH4, showing the abraded rail foot of the stock rail at the location of the tongue (57)
Figure 5.6: Photograph illustrating the connection of the stock rail and movable part of the switchblade with gliding chairs in a classical turnout (58)
According to the plan, most of the sleepers in the TJD EUH4 have a width of 300 mm, as illustrated in Figure 5.7. Therefore, the input data for the ballast resistances need to be adjusted according to this width. The input data for these sleepers were illustrated in Figure 3.9, as explained in the corresponding section. To obtain a more conservative solution, also a model is made where the ballast resistances are not adjusted to the width of the sleeper. In this latter model, all the springs representing the lateral ballast resistances are assigned the input data given in Figure 3.7, according to sleepers with a standard width of 260 mm. This makes comparison with the diamond crossing H4V4H4 easier.
The last difference being observed in the TJD EUH4 track model is visible in zone 4 in Figure 5.3. Here, the track panel is wider compared to the diamond crossing and the sleepers have a length of 330 cm. In the diamond crossing, the sleepers at the heart of the crossing have a length of only 270 cm. The three-dimensional view of the rendered mesh of the TJD EUH4 model in Siemens NX is given in Figure 5.8.
Model of a diamond crossing with double slips 111
60E1 Sleeper 150x300mm 50E2 Sleeper 150x260mm
Figure 5.7: Meshed geometry of the TJD EUH4 in Siemens NX
Figure 5.8: Three-dimensional view of the meshed geometry of the TDJ EUH4 in Siemens NX
5.3 Results and safety assessment
5.3.1 Comparison between defect configurations
To make an outright comparison between the TJD EUH4 and the diamond crossing H4V4H4, a defect 8m8mm was also applied in plain track. The resulting buckling curves for the different defect configurations in the TJD EUH4, together with the results of the diamond crossing H4V4H4, are given in Figure 5.9. In these results, the finite element model is used without the adjustments of the lateral ballast resistances according to the sleeper widths of 300 mm. The characteristic load-displacement curves for standard sleepers with width 260 mm are thus applied. The corresponding critical buckling temperature increases are listed in Table 5.1.
Regarding the defect location and defect amplitude, the same conclusions can be made as were done earlier for the diamond crossing. For a defect amplitude of 20 mm in the crossing, the lower critical buckling temperature increase is approximately 56°C. This means that the TJD EUH4 diamond crossing with double slips is never at risk of buckling, because in reality the maximal expected rail temperature increase is only 40°C for track with a neutral temperature of 20°C. But similar to the diamond crossing, the lateral displacements and axial forces may become important at this temperature increase.
Model of a diamond crossing with double slips 112 It is noticed that the critical buckling temperature increases are about 5°C to 10°C higher compared to the H4V4H4 diamond crossing for the misalignments implemented in zone 3. It can thus be concluded that the TJD EUH4 is safer for thermal buckling than the H4V4H4 geometry. A detailed investigation of the parameters which yield this difference in buckling strength is given in chapter 6.
When the resulting load-displacement curves of both crossings with the defect implemented in plain track are compared, a small difference is observed. It could be expected that the difference in critical temperatures should be zero, because the TJD EUH4 and the H4V4H4 have the same geometry. However, the exact configuration of the rails is different (transition zone 60E1 and 50E2 rails), such that the axial forces will vary slightly in the plain track close to the crossing. This explains the observed marginal difference in the buckling curves for both track models.
140 TJD EUH4 - Track defect 8m8mm in plain track 120 TJD EUH4 - Track defect 8m8mm
100 in zone 3 of crossing C] ° TJD EUH4 - Track defect 80 8m20mm in zone 3 of crossing
60 H4V4H4 - Track defect 8m8mm
in plain track temperature[ 40 H4V4H4 - Track defect 8m8mm in zone 3 of crossing 20 Temperature increase Temperatureincrease above neutral H4V4H4 - Track defect 8m20mm in zone 3 of crossing 0 0 10 20 30 40 50 60
Lateral displacement [mm]
Figure 5.9: Load-displacement curves for different defect configurations in the TJD EUH4 diamond crossing with double slips. Comparison with the results for the H4V4H4 diamond crossing
Table 5.1: Critical buckling temperature increases for different defect configurations in the TJD EUH4 diamond crossing with double slips
Critical buckling temperature increase Model configuration Upper critical buckling Lower critical buckling
temp 횫Tb,max temp 횫Tb,min TJD EUH4 – track defect 8m8mm in plain track 125 °C 72,8 °C TJD EUH4 – track defect 8m8mm in zone 3 of 97,2 °C 69 °C crossing TJD EUH4 – track defect 8m20mm in zone 3 of crossing (ballast resistances according to 58,9 °C 56,0 °C sleepers with width 260 mm) TJD EUH4 – track defect 8m20mm in zone 3 of crossing (ballast resistances adjusted to sleepers 61,2 °C 57,6 °C with width 300 mm)
Model of a diamond crossing with double slips 113 In the plan geometry of the TJD EUH4, the sleepers have a width of 300 mm. The influence on the buckling behaviour when the ballast resistances are adjusted to the width of the sleepers is illustrated in Figure 5.10. The values of the critical buckling temperature increases are also listed in Table 5.1. As expected, the buckling strength increases with the implementation of the increased lateral ballast resistances. The gain in buckling safety is rather small, with an increase of the upper critical buckling temperature of about 2,3 °C. But the difference is certainly not neglectable.
65
60
55
C] °
50
temperature[ 45 TJD EUH4 - 8m20mm - Ballast resistances 260 mm
40 TJD EUH4 - 8m20mm - Ballast resistances adjusted to 300 mm Temperature increase Temperatureincrease above neutral
35 0 10 20 30 40 50
Lateral displacement [mm]
Figure 5.10: Comparison between the buckling curves for model configurations of the TJD EUH4, with and without adjustment of the lateral ballast characteristics to the width of the sleeper
In the following sections, the results are discussed for the TJD EUH4 with implemented track defect 8m20mm. The model configuration in which the ballast resistances are not adjusted to the sleeper’s width is used. This allows a better comparison with the H4V4H4 diamond crossing, which is constructed completely with sleepers with a width of 260 mm. The global conclusions are however exactly the same for the model configuration with adjusted resistances, only the absolute values can vary slightly. The overview of results mainly serves as validation of the reliability of the finite element model.
5.3.2 Lateral displacements
Similar to the H4V4H4 diamond crossing in chapter 4, the results are taken at certain points along the buckling curve corresponding to different temperature increases. These points for the TJD EUH4 are taken as illustrated in Figure 5.11. These include two pre-buckling temperatures, the upper critical buckling temperature and two post- buckling temperatures. The results at +40°C allow to assess the safety of the railway track at the maximal expected temperature increase in the rails in service. The results are evaluated along the course of the railway track. Figure 5.12 allows to link the geometry of the TJD EUH4 to the corresponding convention in linear distance. The initial track defect with wavelength 8 m is situated between 118 m and 126 m.
Model of a diamond crossing with double slips 114 70
60
50 (3) +58,9°C (5) +58,6°C
C] (4) +56,0°C °
40 (2) +40,0°C
30
temperature[ 20
10
(1) +1,66°C Temperature increase increase neutral above Temperature 0 0 10 20 30 40 50 60 70 Lateral displacement [mm]
Figure 5.11: Designation of the points along the buckling curve where the results are taken. TJD EUH4 8m20mm
Distance along railway track [m]
Distance along railway track [m]
Figure 5.12: Plot of the distance measured along the railway track applied to the TJD EUH4 model (contour plot of the lateral displacements)
Figure 5.13 gives the lateral displacements of the track for different temperature increases. Again, the lateral fastener resistance is high enough to ensure that the displacements of rails and sleepers are equal. At a temperature increase above the neutral temperature of 40°C, the maximal lateral displacement of the misalignment is about 1,3 mm, which is smaller than the obtained lateral displacement of 2,5 mm in the H4V4V4 diamond crossing. The influence zone of the lateral misalignments broadens with increasing temperature and increasing displacements, in accordance with the results of the diamond crossing.
Model of a diamond crossing with double slips 115 70
60 (1) 1,67°C (2) 40,0°C 50 (3) 58,9°C 40 (4) 56,0°C (post-buckling) (5) 58,6°C (post-buckling) 30
20
10 Lateral displacement Lateral displacement [mm]
0
-10 110 115 120 125 130 135 140 Distance along railway track [m]
Figure 5.13: Lateral displacements along the railway track for different temperature increases along the buckling curve. TJD EUH4 8m20mm
5.3.3 Longitudinal displacements and axial strains
To illustrate the results for the longitudinal displacements, the designation given in Figure 5.14 is applied. In Figure 5.15, the contour plot of the longitudinal displacements along the global x-axis after buckling is given. It is noticed that the rails and sleepers are deforming relatively to each other, which means the longitudinal fastener resistance has reached its limit value. The rails are thus sliding in the fastening system, behaviour that was already noticed for the H4V4H4 diamond crossing in chapter 4. The relative displacements between sleeper and rail are thus non-zero.
Rail H
Rail G
Figure 5.14: Designation of the naming convention of the rails in the diamond crossing with double slips
Model of a diamond crossing with double slips 116
Figure 5.15: Contour plot of the longitudinal displacements at 훥T = 58,6 °C (post-buckling), according to the global x- axis. TJD EUH4 8m20mm
5.3.3.1 Rail C
Figure 5.16 gives the longitudinal displacements of Rail C, which is a junction rail that evolves in a free-expanding tongue. The displacements are taken along the local longitudinal axis of the rail. The influence of the free- expanding end of this rail is seen in the graphs, which show longitudinal expansions of about 1 to 2 mm at a distance of 132 m. These expansions are limited and do not form any limitation in practice, because the tongues are relatively short in this track model. Between the distances of approximately 115 m and 128 m, the same course in longitudinal displacements as for the H4V4H4 diamond crossing in Figure 4.17 is observed. The same conclusions about the behaviour of the railway track in the defect zone discussed in section 4.3.2.3 apply thus here.
2,0
(1) 1,67°C 1,5 (2) 40,0°C (3) 58,9°C 1,0 (4) 56,0°C (post-buckling)
0,5 (5) 58,6°C (post-buckling)
0,0
-0,5 Longitudinal Longitudinal displacement [mm] -1,0
-1,5 50 60 70 80 90 100 110 120 130
Distance along railway track [m]
Figure 5.16: Longitudinal displacements of rail C along the railway track for different temperature increases. TJD EUH4 8m20mm
Model of a diamond crossing with double slips 117 For distances between 0 m and about 111 m, the rail is gradually expanding. This is can be explained by the fact that the rails have profiles 60E1 up to this point and 50E2 rail sections are applied between 111 m and 145 m. The sudden change in cross-section yields different nominal axial forces. The variation in axial forces could not occur suddenly and a transition zone is needed. As the 50E2 rail profile yields smaller thermal forces, the 60E1 rails must expand to allow a drop in nominal force. In the same way, the 50E2 rails must shorten to gradually build up the normal forces to higher values.
This behaviour is illustrated in Figure 5.17 by means of the axial strains in rail C. Positive strains, corresponding to expanding rails, are noticed up to a distance of 111 m after which the strain suddenly changes to negative values, corresponding to rails that shorten. This is completely in accordance with the explanation given for the longitudinal displacements. In this plot, the strains are increasing near the end of the rail, corresponding to the expansion of the ‘free’ end of the tongue.
2E-04
(2) 40,0°C
1E-04 Zero-strain
0E+00
Axial Axial strain [mm/mm] -1E-04
-2E-04 50 60 70 80 90 100 110 120 130 Distance along railway track [m]
Figure 5.17: Axial strains in rail C along the railway track at a temperature increase 훥T = 40°C. TJD EUH4 8m20mm
The relative longitudinal displacements between rail C and the sleeper underneath are plotted in Figure 5.18. The initial misalignment is growing in the defect zone by slipping of the rails in the fastening system. At a temperature increase of 40 °C, this is limited to 0,1 mm. It is concluded that the permissivity of the fastening system does not form any limitation for the application of a continuously welded diamond crossing with double slips.
For post-buckling configurations, the relative displacements grow fast and reach values above 4 mm. The post- buckling configuration is however a situation one typically wants to avoid for in service track. From the plot it can be concluded that the end of rail C is expanding by slipping in the fastening system, because the relative longitudinal displacements are also increasing at this location. The expansion does not occur through substantial longitudinal movement of the sleepers in the ballast.
Model of a diamond crossing with double slips 118 4,5 (1) 1,67°C (2) 40,0°C 3,5 (3) 58,9°C (4) 56,0°C (post-buckling) 2,5 (5) 58,6°C (post-buckling)
1,5
0,5 Relative longitudinal Relative longitudinal displacement [mm]
-0,5 90 95 100 105 110 115 120 125 130 Distance along railway track [m]
Figure 5.18: Relative longitudinal displacements of rail C and sleepers underneath along the railway track at different temperature increases. TJD EUH4 8m20mm
Figure 5.19 finally gives the variation in axial strains in rail C for the different temperature increases along the buckling curve. At the locations of the track instability, the absolute values of the strains are increasing in correspondence with the observed behaviour discussed in section 4.3.2.9. In this way, the conclusions made for the diamond crossing can also be applied here. The integration of the axial strains along the length of the rail yields the total expansions/shortening. For rail C, this is non-zero, as one end of the rail is not fixed and is able to expand.
2E-03
(1) 1,67°C 2E-03 (2) 40,0°C
1E-03 (3) 58,9°C (4) 56,0°C (post-buckling) 5E-04 (5) 58,6°C (post-buckling)
0E+00
Axial Axial strain [mm/mm] -5E-04
-1E-03
-2E-03 80 85 90 95 100 105 110 115 120 125 130 Distance along railway track [m]
Figure 5.19: Axial strains in rail C along the railway track for different temperature increases along the buckling curve. TJD EUH4 8m20mm
Model of a diamond crossing with double slips 119 5.3.3.2 Rail A
For the straight rail C, the longitudinal displacements could be defined along the rail’s axis. For rail A and B, the rail’s axis is curved and the definition of one fixed local axis along the entire length of the rail is not possible. The Siemens NX software does not allow to determine the results along a varying coordinate system along the rail’s path. In the defect region, where the longitudinal displacements are non-zero, the rail’s axis is however almost parallel to the global x-axis. This means that the influence of the lateral displacements on the longitudinal ones is rather limited. The results of the longitudinal displacements of rail A along the global x-axis are given in Figure 5.20. It is remarked that still a minor influence of the lateral displacements is present in the course of the displacements. The global behaviour of the longitudinal displacements is however visible and is in accordance with the results presented in Figure 4.17 for the diamond crossing. It is concluded that the rails are expanding towards the central region of the crossing, where the misalignment is growing laterally. The expanding rails are related to the drop in axial compressive forces, discussed in the following sections.
2,5
(1) 1,67°C 2,0 (2) 40,0°C 1,5 (3) 58,9°C (4) 56,0°C (post-buckling) 1,0 (5) 58,6°C (post-buckling)
0,5
0,0 Longitudinal Longitudinal displacement [mm] -0,5
-1,0 50 70 90 110 130 150 170 190 210 Distance along railway track [m]
Figure 5.20: Longitudinal displacements of rail A along the railway track for different temperature increases along the buckling track. TJD EUH4 8m20mm
Again, the expansion of the 60E1 rails towards the transition point with the 50E2 rails is visible in the plot, similar to the results for rail C. This behaviour is confirmed from the course of the axial strains in Figure 5.21, which shows the transition from gradual increasing positive values to negative ones at 111 m and 145 m. This indeed corresponds to expanding 60E1 rails and contracting 50E2 rails. At the defect location, the absolute values of the strains are increasing along the path of the buckling curve, in accordance with the previous results.
Model of a diamond crossing with double slips 120 2E-03 (1) 1,67°C 2E-03 (2) 40,0°C (3) 58,9°C 1E-03 (4) 56,0°C (post-buckling)
5E-04 (5) 58,6°C (post-buckling)
0E+00
Axial Axial strain [mm/mm] -5E-04
-1E-03
-2E-03 100 105 110 115 120 125 130 135 140 145 150 Distance along railway track [m]
Figure 5.21: Axial strains in rail A along the railway track for different temperature increases along the buckling curve. TJD EUH4 8m20mm
As both ends of rail A are clamped, the resulting total expansion/shortening after integration of the course of the axial strains should be close to zero. Using formulation (4.3), the calculated expansions of rail A for the different temperature increases are presented in Table 5.2. For small temperature increases, the total elongation is zero. For increasing temperatures, the total elongation is slightly increasing because the misalignment is growing. In pre- buckling configurations, the elongations are however smaller compared to the diamond crossing H4V4H4 (Table 4.3). This is in accordance with the difference in buckling curves observed in Figure 5.9, which shows that the lateral displacements are smaller for the TJD EUH4 before buckling. The elongations are indeed proportional to the growth of the misalignment in the rails.
Table 5.2: Total elongation of rail A for different temperature increases along the buckling curve. TJD EUH4 8m20mm
Temperature increase above Total elongation of rail neutral temperature (1) 1,7 °C 0,00 mm (2) 40,0 °C 0,05 mm (3) 58,9 °C 0,21 mm (4) 56,0 °C 0,63 mm (5) 58,6 °C 1,53 mm
5.3.4 Forces in the lateral ballast springs
Figure 5.22 illustrates the course of the forces in the springs representing the lateral ballast resistance. In case of the TJD EUH4, the track defect is present along sleepers with length 330 cm, 360 cm and 390 cm. The limit resistance forces for the lateral displacements in the ballast were given in Table 3.1. It is noticed that in the post-
Model of a diamond crossing with double slips 121 buckling configurations, the limit forces are reached in a broad range in the defect area. At a temperature increase of 40°C above the neutral rail temperature, the lateral ballast resistance does not yet reach any limit value anywhere. This means that that the ballast is able to withstand extra lateral forces everywhere in the defect area. This is certainly one of the reasons that the TJD EUH4 has a higher buckling strength, as it was noticed that in the H4V4V4 diamond crossing some sleepers already reached their limit resistance in the ballast at this temperature. Note that the force reserve is even higher in the finite element model in which the ballast resistances are adjusted according to the width of the sleeper. This immediately explains the higher buckling strength of the latter.
10
8 (1) 1,7°C 6 (2) 40,0°C 4 (3) 58,9°C
2 (5) 58,6°C (post-buckling)
0
Force[kN] -2
-4
-6
-8
-10 100 105 110 115 120 125 130 135 140 145 150 Distance along railway track [m]
Figure 5.22: Forces in the springs representing the lateral ballast resistance along the railway track for different temperature increases. TJD EUH4 8m20mm
5.3.5 Axial forces
In section 5.3.3, it was already concluded that the transition from rails with profiles 60E1 to 50E2 causes a variation in longitudinal axial forces in the diamond crossing with double slips. Both profiles have a cross-sectional area of respectively 76,70 cm² and 63,65 cm², which indeed corresponds to different thermal forces according to formula (4.1). The calculated axial forces for different temperatures increases are given in Table 5.3.
Table 5.3: Comparison axial forces calculated with (4.1) for 60E1 and 50E2 rail profiles
Axial force per rail [kN] Temperature increase above neutral temperature 60E1 50E2
1,7 °C -30,25 -25,07 40,0 °C -728,93 -604,20 58,9 °C -1073,35 -889,68 56,0 °C -1020,50 -845,87 58,6 °C -1067,88 -885,75
Model of a diamond crossing with double slips 122 Similar as in chapter 4, the axial forces are first compared by means of an equivalent total force along the railway track, obtained by making the summation of the individual axial forces in each of the rails designated in Figure 5.14. This allows the examination of the global behaviour of the axial forces for increasing temperatures, during the buckling of the railway track.
Figure 5.23 and Figure 5.24 illustrate the obtained results for the different considered temperature increases along the path of the load-displacement curve. A first observation are the small local drops in axial forces for all curves at the distances of approximately 116 m and 140 m. As already explained in section 4.3.2.8 of chapter 4, this has to do with the presence of the check rails. The added forces of the latter allow the rails to expand locally and consequently the axial forces drop. The influence of the check rails on the course of the axial forces is illustrated in Figure 5.25. It is noticed that with the addition of the check rails, the course of the total axial force increases locally. This is the consequence of the added rail section and accompanying added thermal force.
When the misalignment in the defect area is growing, a global drop in axial forces in the buckling influence zone is expected, as was the case for the diamond crossing in chapter 4. However, this will be more difficult to distinguish in the plots for the TJD EUH4, because of the variation in axial forces due to the changing rail profile. In the part of track with 60E1 rails the nominal thermal forces are higher than for the part with 50E2 rails and a transition zone between both is present. So also without imperfection, a drop in axial forces is visible in the course.
In addition to this phenomenon comes the fact that eight rails are locally present in the same section in the central region of the TJD EUH4. Outside this zone, the railway track contains four rails. Rail G and H in Figure 5.14 are non- clamped rails. This means that the nominal force calculated with formula (4.1) will not be reached, as these rails can expand somehow. Rails C and E, respectively D and F, intersect locally and contribute to increased cross- sections, resulting in higher total forces. The different zones referred to are illustrated in Figure 5.26.
-4500
-4000
-3500
-3000
-2500
-2000 (1) 1,7°C Axial Axial [kN] force -1500 (2) 40,0°C
-1000 (3) 58,9°C -500
0 0 50 100 150 200 250 Distance along railway track [m]
Figure 5.23: Total axial forces after summation of the axial forces of the individual rails, along the railway track for different temperature increases (pre-buckling). TJD EUH4 8m20mm
Model of a diamond crossing with double slips 123 -4400
-4200
-4000
-3800 Axial Axial [kN] force (3) 58,9°C
-3600 (4) 56,0°C (post-buckling)
(5) 58,6°C (post-buckling)
-3400 0 50 100 150 200 250 Distance along railway track [m]
Figure 5.24: Total axial forces after summation of the axial forces of the individual rails, along the railway track for different temperature increases (post-buckling). TJD EUH4 8m20mm
-4500 (5) 58,6°C (Without check rails) -4300 (5) 58,6°C (check rails added)
-4100
-3900
-3700 Axial Axial [kN] force
-3500
-3300 50 70 90 110 130 150 170 190
Distance along railway track [m]
Figure 5.25: Comparison between the summation of the axial forces of the rails along the railway track for a calculation with and without the axial forces in the check rails. TJD EUH4 8m20mm
4 rails 60E1 4 rails 50E2 8 rails 50E2 4 rails 50E2 4 rails 60E1
Figure 5.26: Different zones of rail section in the TJD EUH4 diamond crossing with double slips
Model of a diamond crossing with double slips 124 These considerations are illustrated in the course of the rail forces in Figure 5.27. The course of the total axial force is compared to the calculated nominal values. In plain track, the total force is equal to the expected nominal force for the 60E1 cross-section (corresponding to 4 rails). From a distance of about 50 m, the compression forces start to decrease towards the central region. It is noticed that the nominal force for 4 50E2 rails is never reached, because of the influence of the section with 8 rails. The maximal compression forces in rail G and H are added to the calculated nominal force in the 50E2 rails, yielding a value of -2550 kN. It is seen that this comes close to the observed forces in the finite element model. The remaining difference of approximately 50 kN comes from the intersecting rails C, D, E and F, of which the resulting force is hard to predict. Due to the additional compressive forces in the heart of the crossing, development of creep has to be expected in the junction rails and outer curved rails.
-3000 (2) 40,0°C -2900
-2800 Nominal force 60E1 -2700 (4 rails)
-2600 Nominal force 50E2
Axial Axial [kN] force (4 rails) -2500
-2400 Nominal force 50E2 (4 rails + rail G and H) -2300 0 50 100 150 200 250 Distance along railway track [m]
Figure 5.27: Comparison between the observed total axial force and nominal thermal forces for different configurations, at 훥T = 40,0°C. TJD EUH4 8m20mm
In the post-buckling configurations, being at +56,0°C and +58,6°C, the compression forces in the defect area should drop below the nominal values of the 50E2 sections. For the summation of the forces in the 4 rails, the absolute value of the total nominal force should drop below 3543 kN at a temperature increase of 56,8°C according to the values in Table 5.3. This is clearly the case in Figure 5.24, where the compression force at the location of the defect has dropped to -3430 kN. From the comparison between the global course of the axial forces at the upper critical buckling temperature and the post-buckling temperatures, it can already be noticed that the compression forces are decreasing. Finally, also the buckling influence zone is broadening, as expected.
Subsequently, the course of the individual rail forces is examined and can be compared to an average rail force, obtained by dividing the total axial force from previous results by four. The results at two temperature increases, respectively +40°C (pre-buckling) and +58,6°C (post-buckling), are given in Figure 5.30, Figure 5.31, Figure 5.32 and Figure 5.33 The post-buckling configurations are added to determine the effect of growing lateral misalignment on the course of the axial forces. This is however a configuration that one typically wants to avoid for in-service track.
Model of a diamond crossing with double slips 125 The results in the rails G and H are for simplicity disregarded from these plots. The order of magnitude compared to the other rail forces can be observed from Figure 5.28. The influence of rails G and H on the forces in rails A and B is comparable to the observed behaviour at the location of the check rails, by means of a local drop in axial forces because the rails can locally expand. This however leads to creep as already outlined.
-35
-30
-25
-20 Average Rail Force -15 Rail A
Axial Axial [kN] force Rail B -10 Rail G -5 Rail H
0 60 80 100 120 140 160 180 Distance along railway track [m]
Figure 5.28: Comparison between the course of the axial forces of the individual rails (A, B, G and H) and the average rail force, pre-buckling (+1,67°C). TJD EUH4 8m 20mm
The axial forces in rails C, D, E and F are going to zero at their free ends. The course of the forces in the defect area is in accordance with the axial strains from paragraph 5.3.3 and comparable with the results for the diamond crossing in section 4.3.2.8. The reason for the difference in the axial forces of rail A and B in Figure 5.32 can be found by looking at the geometry of the track. The curvature of rail B is in the same direction as the implemented track defect, so that the defect actually increases the curvature, as illustrated in Figure 5.29. For rail A instead, the misalignment opposes the curvature. Due to the effect of radial breathing, as explained in section 2.3.6, the axial forces are able to decrease more in rail B. An opposing effect is taking place for rail A. This confirms the observed differences in the plot of the axial forces in Figure 5.32.
Figure 5.29: Illustration of the direction of the implemented track defect
The fluctuating course of axial forces in rails C and D in the defect region can be explained by the difference in expanding and shortening rails, as was already outlined in chapter 4. The reason for the alternating expansion/shortening between both rails is found by looking at their different orientation. The same behaviour was also observed in the H4V4H4 diamond crossing in the defect region.
Model of a diamond crossing with double slips 126 The increased magnitudes of the compressive forces of rails C/D and rails E/F above the average force line in Figure 5.31, is explained by the additional compressive forces induced by the expansions of rails G and H. Creep can thus develop in these rails.
-750
-710
-670
-630 Axial force [kN] force Axial Average Rail Force Rail A -590 Rail B
-550 60 70 80 90 100 110 120 130 140 150 160 170 180 190
Distance along railway track [m]
Figure 5.30: Comparison between the course of the axial forces of the individual rails (A and B) and the average rail force, pre-buckling (+40,0°C). TJD EUH4 8m 20mm
-750
-710
-670
-630 Average Rail Force
Axial force [kN] force Axial -590 Rail C Rail D -550 Rail E Rail F -510
80 90 100 110 120 130 140 150 160 170 Distance along railway track [m]
Figure 5.31: Comparison between the course of the axial forces of the individual rails (C, D, E and F) and the average rail force, pre-buckling (+40,0°C). TJD EUH4 8m 20mm
Model of a diamond crossing with double slips 127 -1100
-1060
-1020
-980
-940
-900 Axial Axial [kN] force -860 Average Rail Force Rail A -820 Rail B -780
60 70 80 90 100 110 120 130 140 150 160 170 180 190
Distance along railway track [m]
Figure 5.32: Comparison between the course of the axial forces of the individual rails (A and B) and the average rail force, post-buckling (+58,6°C). TJD EUH4 8m 20mm
-1060
-1020
-980
-940
-900
-860 Average Rail Force
Axial force [kN] force Axial -820 Rail C Rail D -780 Rail E
-740 Rail F
-700 80 90 100 110 120 130 140 150 160 170
Distance along railway track [m]
Figure 5.33: Comparison between the course of the axial forces of the individual rails (C, D, E and F) and the average rail force, post-buckling (+58,6°C). TJD EUH4 8m 20mm
Finally, the extreme absolute values of the rail forces at a temperature increase of 40°C are examined. The corresponding rail stress is of importance to evaluate the fatigue life the rail. The maximal compression force is occurring in rail D, with a value of 670 kN. This corresponds to a stress of 105 MPa in the 50E2 rail, which is larger than the peak stress observed in the diamond crossing H4V4H4. It does not exceed the reference limit value of 172 MPa obtained in section 2.8.2. As already explained in section 4.3.2.8 for the diamond crossing, the maximal peak
Model of a diamond crossing with double slips 128 compression stress in the rail does not only depend on the nominal axial stress, but also the influence of rail bending in the defect area should be taken into account. From the results of the diamond crossing, the expected order of magnitude of this extra compression stress in the outermost fiber of the rail lies in the order of magnitude of 30 MPa. As the lateral displacements before buckling are smaller in the TJD EUH4 compared to the H4V4H4 model, it can be expected that this extra bending stress is smaller for the diamond crossing with double slips. However, the rails have a 50E2 section, such that the second moment of area is smaller and the additional bending stresses can be larger. Similar as for the diamond crossing, the stress in the most compressed fiber can be calculated from the bending moments in the rail at +40°C. The course of the bending moments Mzz are not explicitly plotted here. In the defect region, a maximal bending moment of 1,45 kNm is observed at a temperature increase of 40°C. Similar to expression (4.2) for the diamond crossing, the additional compressive stress in the outermost fiber can be calculated using expression (5.1):
My 1,45 106 Nmm 70,0 mm (5.1) =zz max = = 24,85 MPa bending 44 Izz 408,4 10 mm The cross-sectional characteristics of the 50E2 rail, given in Figure A.2 in Annex 1, are used in this expression. The additional compressive stress is smaller, but comparable in magnitude than was the case for the diamond crossing. As also discussed in section 4.3.2.8, this stress is important to assess the fatigue life of the rails.
At the location of the movable part of the switchblade, the rail foot of the stock rail is abraded. This is located in the region of the implemented track defect, where the stock rail is subjected to peak compressive stresses. This property is not taken into account in the finite element model. It is not expected that this has any significant influence on the buckling resistance of the track, but it has a local influence on the distribution of stresses in the rail section. Due to the abrasion of the rail foot, a concentration of stresses of three times the nominal stress can be expected here (22). It is noted that this concentration of stresses is important to consider for the fatigue of the rail. The resulting stresses will not yield to fracture of the rail, as the yield strength of the steel profiles have a minimal yield strength of at least 680 MPa to 880 MPa (59).
It must be noted that a slight difference in rail forces has to be expected for the finite element model in which the lateral ballast resistances are adjusted to the sleeper width of 300 mm. As the buckling strength is increased, the lateral displacement of the track in the defect area will decrease for the same temperature increase. And since the drop in axial forces is proportional to the elongations of the rail and consequently also to the lateral displacements, the absolute value of the rail forces will be slightly higher for the adjusted model. It is however expected that the difference is negligible.
5.4 Parameter study
In chapter 4, a parameter study was performed for the finite element model of the diamond crossing. It was concluded that the lateral ballast resistance and the longitudinal fastener resistance had the most important influence on the buckling strength of the track model. For the diamond crossing with double slips, the effect of
Model of a diamond crossing with double slips 129 these two parameters is verified in this section. The variation of the parameters is performed on the configuration with the track defect 8m20mm implemented in zone 3 of the crossing.
5.4.1 Influence of the misalignment amplitude
The effect of the misalignment amplitude was already investigated in Figure 5.9. It was concluded that larger initial misalignments result in a significant reduction in the critical buckling temperatures. This in accordance with the results for the diamond crossing and the literature.
5.4.2 Influence of the lateral ballast resistance
The results in Figure 5.34 immediately show that the lateral ballast resistance is the most influential parameter for the buckling strength of track, in accordance with the previous results. A decrease in ballast quality of 20 % yields a reduction in the critical buckling temperature of 4 to 5°C. For decreasing ballast resistance, the observed behaviour tends more to progressive buckling, which was also observed in Figure 4.45 for the results of the diamond crossing H4V4H4. The influence of decreased ballast resistance due to dynamic vibrations and uplift of track can again be quantified by means of this parameter study. A reduction in critical buckling temperatures of more than 4°C can be expected, but the track is never at risk of buckling at the maximal expected temperature increase of 40°C above the neutral temperature of the rail.
By artificially increasing the lateral resistance (for example by increasing the shoulder height, installation of safety caps, application of a retaining wall, etc.), the critical buckling temperatures increase by 5 to 10°C. Not only the critical buckling temperatures are affected, but also the pre-buckling lateral displacements and the stress peaks will be reduced. The applications to improve the buckling strength of the diamond crossing discussed in section 4.5 can also be applied to the track geometry of the TJD EUH4. The same results are expected.
5.4.3 Influence of the longitudinal fastener resistance
The results for a variation in the stiffness of the longitudinal fastener resistance are presented in Figure 5.34. The influence of this parameter is much less pronounced than is the case for the lateral ballast resistance. The critical buckling temperatures only vary with 1 to 2°C for significant changes in the stiffness of the fasteners. It can again be concluded that the buckling strength of track is mainly dependent on the lateral resistance of the sleepers in the ballast bed and the stiffnesses of the rails (second moments of area) of the track panel. The initial misalignment is actually the most influencing parameter, but this is of course not under control of the designer. The geometry of the rails determine the variation in critical buckling temperature between the different track models.
Model of a diamond crossing with double slips 130 Lateral Ballast Resistance: Tb,max Lateral Ballast Resistance: Tb,min Longitudinal Fastener Resistance: Tb,max Longitudinal Fastener Resistance: Tb,min
71 70 69
68 C]
° 67 66 65 64 63 62 61 60 59 58 57 56
55 Temperature increase Temperatureincrease above neutraltemperautre [ 54 53 52 51 -40% -30% -20% -10% +0% +10% +20% +30% +40% Percentage change in parameter
Figure 5.34: Summary of the parameter study for the TJD EUH4 diamond crossing with double slips (track defect 8m20mm in zone 3)
5.5 Model with fishplated joints
5.5.1 Principles
In the newly installed diamond crossings with double slips, all connections are continuously welded. However, the old types of the TJD EUH4 being present in the Belgian railway lines have several connections with fishplated joints. These joints are constructed as illustrated in Figure 5.35, in which the dashes represent the in total 24 fishplated joints. These connections were among others necessary because of the limited place to weld the individual parts together. A comparison between the buckling behaviour of this old type and the continuously welded TJD EUH4 can be made.
Model of a diamond crossing with double slips 131
Figure 5.35: Geometry of an old TJD EUH4: dashes = connection with fishplates, dots = welds
To create a finite element model of the TJD EUH4 with fishplated joints, the same principles as explained in section 4.7 were applied. At the location of the joints, a CGAP element is introduced with the same characteristics as in the aforementioned section. In this finite element model, the movable parts of the switchblades were introduced, as visible in Figure 5.36. These are modeled with a varying cross-section towards the tip of the blade and are not connected with the sleepers by fasteners. As a consequence, the influence on the structural behaviour should be limited as the blades will expand freely under temperature increases. In the same figure, also the lines representing the fishplates can be noticed. The track defect with wavelength 8 m and amplitude 20 mm is implemented in the track model in the same way as for the continuously welded TJD EUH4. Here, the lateral ballast resistances are adjusted to the sleeper width.
Figure 5.36: Extract of the line model of the TJD EUH4 with fishplated joints in Siemens NX, with modeling of the entire switchblades
5.5.2 Results
To validate the model and especially the modeling of the fishplated joints, the longitudinal displacements of the rails at a small temperature increase of 10°C are checked in Figure 5.37. It is observed that the rails are expanding towards and between the fishplated joints, which is as expected. Also the movable parts of the tongues are exhibiting expansions at their free ends, as expected. These are however limited, due to the short tongues. The model thus has the desired behaviour. The other characteristics like the axial forces for different temperature increases are not further discussed here. These are comparable with the model explained in section 4.7.
For this model of the TJD EUH4, the joints are already transferring axial forces at a lower temperature increase than was noticed for the model of the H4V4H4 diamond created in section 4.7. This is explained by the more complex installation of the 24 joints, which do not allow the same large expansions towards one point as was observed in Figure 4.62 for the H4V4H4 diamond crossing.
Model of a diamond crossing with double slips 132
Figure 5.37: Contour plot of the longitudinal displacements at a temperature increase of 10°C. TJD EUH4 8m20mm with fishplated joints
The influence on the buckling strength can be observed in Figure 5.38. As expected, the critical buckling temperature increases are significantly higher compared to a continuously welded diamond crossing with double slips. The increase in critical temperatures amounts to more than 10°C. Also the lateral displacements for smaller temperature increases are significantly smaller thanks to the allowance for rail expansions. At an increase of 40°C, the maximal extra misalignment is equal to 0,25 mm, being much smaller than the value for the continuously welded track. This value is comparable with the lateral displacement observed in the model created in section 4.7. So, it can be concluded that the TJD EUH4 with fishplated joints has indeed a larger safety against lateral track buckling than the continuously welded type.
80
70
60
C] 50 °
40
30 temperature [ temperature TJD EUH4 - 8m20mm - Ballast resistances 260 mm 20 TJD EUH4 - 8m20mm - Ballast resistances adjusted to 300 mm
Temperature increase Temperatureincrease above neutral the 10 TJD EUH4 - 8m20mm - Model with fishplated joints
0 0 10 20 30 40 50 Lateral displacement [mm]
Figure 5.38: Comparison between the buckling curves for the model of a continuously welded diamond crossing with double slips and the model with fishplated joints. TJD EUH4 8m20mm
Model of a diamond crossing with double slips 133 5.6 Conclusions
In this chapter, the buckling behaviour of a diamond crossing with double slips was investigated. A finite element model was created of the TJD EUH4, using the same principles of the finite element model of the H4V4H4 diamond crossing. In the model of the diamond crossing with double slips, the movable parts of the switchblades are omitted, because these do not transfer any significant longitudinal forces to the sleeper and are free to expand. Similar results for the buckling curves, the lateral and longitudinal displacements, the axial forces, etc. were obtained as was the case for the diamond crossing in chapter 4. The observed courses of the plots and behaviours of the model were as expected, such that the model can be assumed to be valid and reliable.
The critical buckling temperatures of the TJD EUH4 model are 6 to 8°C higher compared to the H4V4H4 model with the same defect 8m20mm applied to the weakest zone of the crossing. It is concluded that the diamond crossing with double slips has a higher resistance against lateral instabilities than the common diamond crossing. The lateral displacements of an initial misalignment with amplitude 20 mm are limited to 1,3 mm at a pre-buckling temperature increase of 40°C. The increased lateral stiffness of the track of the TJD EUH4 model is contributed to the increased stiffness of the track panel in the heart of the central crossing, the denser sleeper spacing and the smaller 50E2 rail sections.
The courses of the relative displacements of the rails in the fastening systems and the axial forces in the rails have been evaluated. No limitations for the application of a continuously welded TJD EUH4 have been noticed. The resulting relative displacements in the fasteners are much smaller than the limit value of 30 mm and the peak nominal stresses are not larger than 105 MPa. One must however be aware of the concentration of stresses in the stock rail, which is abraded at the rail foot. This is not taken into account in the finite element model.
Model of a diamond crossing with double slips 134 Chapter 6 Comparison between diamond crossing and diamond crossing with double slips
6.1 Introduction
In chapter 4 and chapter 5, the buckling behaviours of respectively a diamond crossing and a diamond crossing with double slips were investigated. The simplified line-geometry of the H4V4H4 (diamond crossing) and TJD EUH4 (diamond crossing with double slips) are exactly the same. However, the critical buckling temperatures are different for the same defect implemented at the same location within the track geometry. The upper critical buckling temperature increase from the TJD EUH4 is almost 9°C higher than for the H4V4H4. In this chapter, the parameters and characteristics which increase the buckling strength of the diamond crossing with double slips compared to the common diamond crossing are investigated.
6.2 Differences in track geometry and structural parameters
First, the differences between both track models are listed in this section. Based on these differences, the models could be rebuilt towards each other, investigating how the increased buckling strength of the diamond crossing with double slips is gained.
6.2.1 Track geometry
The simplified line model of both tracks is the same, which is illustrated in Figure 6.1. The basic geometry consists out of two railway tracks intersecting at an angle of 0,124 radians. However, the rail geometry is different for both types of crossings. The diamond crossing H4V4H4 has four intersecting rails, while the TJD EUH4 has two curved outer rails with four interrupted junction rails and additional ‘slips’. The difference is illustrated in Figure 6.2.
Figure 6.1: Simplified line model of the diamond crossing H4V4H4 and TJD EUH4 (47)
Comparison between diamond crossing and diamond crossing with double slips 135
A
B
Figure 6.2: (A) Rail geometry of the diamond crossing H4V4H4. (B) Rail geometry of the diamond crossing with double slips TJD EUH4
The influence on the buckling strength was already shortly outlined in chapter 5. The intersecting junction rails and slips add extra lateral stiffness to the common diamond crossing. But these in turn also result in larger longitudinal forces being present in the track panel. The curved outer rails deal with the phenomenon of radial breathing. For the misalignment growing in the same direction as the curvature, the compression forces in the rail drop faster and as a result the buckling strength is increased. It must be noticed that a curvature in the opposite direction as the growing misalignment should have the opposite effect.
6.2.2 Rail types
The central zone of the TJD EUH4 consists of 50E2 rails, while the common diamond crossing is built up completely by 60E1 rails. The 50E1 profiles yield smaller longitudinal forces for a certain temperature increase, which will make the track less vulnerable for buckling. Contrary to the smaller compression forces being present in the railway track, the second moment of area of the rail is decreased and this in turn decreases the buckling strength. The combined influence of both parameters needs therefore to be investigated.
6.2.3 Sleeper geometry
The layout of the sleepers for both types of crossings is different. A comparison between the geometry of the H4V4H4 and the TJD EUH4 is made in Figure 6.3. In the central, most narrow zone of the crossings, the sleepers of the TJD EUH4 are longer compared to the sleepers of the H4V4H4. In zone 3 of the crossing (see Figure 4.6 and Figure 5.3) the spacing is different, with the spacing of the TJD EUH4 being more dense. This was already discussed in chapter 5. Finally, most of the sleepers of the TJD EUH4 have a width of 300 mm instead of the standard dimension of 260 mm. Concluding, the increased length, the denser spacing and the increased width of the sleepers result in a larger lateral ballast resistance for the TJD EUH4. This increases the buckling strength significantly.
Figure 6.3: Comparison between the lay-out of the sleepers of the H4V4H4 (blue) and the TJD EUH4 (red).
Comparison between diamond crossing and diamond crossing with double slips 136 6.3 Investigation of the differences in buckling strength
To investigate the influence of the abovementioned differences in the H4V4H4 and TJD EUH4 railway track, both models were gradually built towards each other by changing the layout or parameters one by one. The modifications are applied to the models with the implemented track defect 8m20mm in zone 3 of the crossing. Subsequently, the buckling curves corresponding to the gradual change in the finite element model can be compared.
6.3.1 Model configurations
6.3.1.1 Changing the width of the sleepers in the TJD EUH 4 (configuration 1)
In a first step, the width of the sleepers in the TJD EUH4 model were all set to 260 mm. The smaller width yields a lower lateral and longitudinal ballast resistance.
6.3.1.2 Changing the rail cross-section in the TJD EUH4 (configuration 2)
The rails in the TJD EUH4 model have a 50E2 profile, while the H4V4H4 model is completely constructed with 60E1 profiles. In the second step, the rails in the TJD EUH4 are modified to a 60E1 cross-section together with the reduced width of the sleepers.
6.3.1.3 Reducing the stiffness and rail forces in the TJD EUH4 by removing rails (configuration 3)
The central zone of the TJD EUH4 has a larger lateral stiffness compared to the same region in the H4V4H4, due to the presence of four extra rails. Although this zone is outside the applied defect area, the larger stiffness certainly has its influence on the buckling behaviour as the deformed buckling shape extends into the central zone for increasing temperatures. To reduce the influence of this extra stiffness and extra longitudinal forces, rails G and H (see Figure 5.14) are removed from the model of configuration 2. The resulting model is illustrated in Figure 6.4.
Figure 6.4: Line model of TJD EUH4 with removal of the slips (configuration 3)
6.3.1.4 Applying the sleeper geometry of the TJD EUH4 to the H4V4H4 (configuration 4)
As was clear from Figure 6.3, the sleeper geometry of the TJD EUH4 gives a larger lateral and longitudinal ballast resistance compared to the sleeper layout of the H4V4H4. Therefore, a model is made in which the exact sleeper geometry of the TJD EUH4 is applied to the rail geometry of the H4V4H4 diamond crossing. The line model of this configuration is illustrated in Figure 6.5. The sleepers are assigned a standard cross-section of 150 mm x 260 mm.
Comparison between diamond crossing and diamond crossing with double slips 137
Figure 6.5: Line model of the H4V4H4 diamond crossing with the sleeper layout of the TJD EUH4
6.3.2 Results and comparison in buckling curves
The modified configurations were calculated in Siemens NX. The resulting load-displacement curves are plotted and compared in Figure 6.6. The difference in buckling behaviour for the sleepers with a smaller cross-section in the TJD EUH4 diamond crossing with double slips was already discussed in chapter 5. This is the buckling curve for configuration 1.
In configuration 2, the rails with 50E2 profiles in the TJD EUH4 (configuration 1) were changed to 60E1 profile. The resulting buckling curve lies below the previous configuration. The upper critical buckling temperature has decreased by about 2°C. It is thus concluded that the increased longitudinal forces have a greater influence on the buckling strength, compared with the larger second moment of area that would have increased the critical buckling temperatures.
In the third configuration, the two inner ‘slips’ (rail G and H) were removed from the finite element model. As these are just short rails, it is logical that the influence on the buckling curve is only minor. The results are nearly completely identical to the curve from configuration 2. The only visible difference is that configuration 2 tends slightly more to progressive buckling compared to configuration 3. The reason behind this will probably be the increased longitudinal forces with the presence of rails G and H, which yield higher compression forces in the rails in the defect area. The slightly decreased lower critical buckling temperature for configuration 3 can be explained by the lower lateral stiffness in the central region. As the misalignment grows and the buckling influence zone broadens, this decreased stiffness could yield smaller temperatures at which an equilibrium in post-buckling state is reached.
The last configuration converts both models as close as possible to each other. The sleeper geometry and rail sections are completely the same. The only difference that remains when configuration 3 and 4 are compared, is the changed geometry of rails. The buckling curve of this fourth configuration lies above the curve for configuration 3. The upper critical buckling temperature is about 3°C higher. It must thus be concluded that the pure rail geometry of the diamond crossing has a higher strength against lateral track buckling than the diamond crossing with double slips. However, the difference is rather marginal and the critical buckling temperature is strongly dependent on different parameters, because they have an uncertainty in the applied magnitudes (for example the resistance of the ballast). It is thus concluded that this difference is negligible.
Comparison between diamond crossing and diamond crossing with double slips 138
65
C] ° 60
55
50 TJD EUH4 - 8m20mm (Adjusted to sleeper width 300mm) Configuration 1 45 Configuration 2 Configuration 3 40 Configuration 4
Temperature increase Temperatureincrease above neutraltemperature [ H4V4H4 - 8m20mm
35 0 5 10 15 20 25 30 35 40 45 50 Lateral displacement [mm]
Figure 6.6: Comparison between the load-displacement curves for the different considered configurations
From these results it can be concluded that the difference in buckling strength of the H4V4H4 and TJD EUH4 track models is mainly due to the difference in sleeper layout on plan and the application of different rail sections. Nevertheless, both models are safe to operate in service as the critical buckling temperatures are sufficiently higher than the maximal expected rail temperature. Furthermore, a better understanding in the parameters which influence the critical buckling temperatures has been obtained by performing this configuration study.
6.4 Comparison between parameter studies
In section 4.4 and 5.4, a parameter study was performed for the track models of the diamond crossing H4V4H4 and the diamond crossing with double slips TJD EUH4 respectively. The same conclusions with regard to the influence of the individual parameters on the buckling behaviour of the track were made. A comparison could be made between both results, to investigate which track model has a higher sensitivity to variations in the parameters. Therefore, the variation of the critical buckling temperatures with respect to the base value are determined and compared.
6.4.1 Sensitivity to a change in lateral ballast resistance
A comparison between the results of the parameter studies for the lateral ballast resistances for the H4V4H4 diamond crossing and TJD EUH4 diamond crossing with double slips is made in Figure 6.7. It is observed that the TJD EUH4 is more sensitive to variations in lateral resistance compared to the H4V4H4 track model. The reason for
Comparison between diamond crossing and diamond crossing with double slips 139 this can probably be found in the denser sleeper spacing and the larger sleepers applied in the defect zone for the TJD EUH4. As more sleepers are resisting the lateral movement of track, it is expected that a percentage change in the lateral ballast resistance has a more significant effect on the buckling strength of the rails. The difference in geometry of the rails between both track models will moreover also induce this difference in sensitivity.
Tb,max - H4V4H4 8m20mm Tb,min - H4V4H4 8m20mm Tb,max - TJD EUH4 8m20mm Tb,min - TJD EUH4 8m20mm 12 11
10
C] ° 9 8 7 6 5 4 3 2 1 0 -1 -2 -3
Change Change critical in buckling temperature increase [ -4 -5 -6 -20% -10% +0% +10% +20% +30% +40% Percentage change in parameter
Figure 6.7: Comparison between the change in critical buckling temperatures for the H4V4H4 diamond crossing and the TJD EUH4 diamond crossing with double slips, subjected to a percentage change in the lateral ballast resistances
6.4.2 Sensitivity to a change in longitudinal fastener resistance
Figure 6.8 gives the comparison between the changes in critical buckling temperatures for varying longitudinal fastener resistances. Contrary to the results from previous paragraph, the TJD EUH4 is in this case less sensitive to changes in the longitudinal stiffness of the fastening systems compared to the H4V4H4 diamond crossing. The difference is however marginal. A possible reason can be found by looking at the longitudinal force transfer from the rails towards the sleepers. In the TJD EUH4 diamond crossing with double slips, the axial forces in the rails in the defect area are increased due to the presence of 8 rails in the heart of the crossing (as discussed in section 5.3.5). When the longitudinal fastener resistance is increased, the transfer of these increased axial forces is more pronounced, such that the rails are closer to their buckling resistance. As a consequence, the critical buckling temperature can increase less for the TJD EUH4 when the longitudinal fastener resistance increases. The denser sleeper spacing (and larger number of fasteners) is an additional reason for this difference.
Comparison between diamond crossing and diamond crossing with double slips 140 Tb,max - H4V4H4 8m20mm Tb,min - H4V4H4 8m20mm
Tb,max - TJD EUH4 8m20mm Tb,min - TJD EUH4 8m20mm
3,0
2,5
C] °
2,0
1,5
1,0
0,5
0,0
-0,5
-1,0
Change Change critical in buckling temperature increase [ -1,5
-2,0 -20% -10% +0% +10% +20% +30% +40% Percentage change in parameter
Figure 6.8: Comparison between the change in critical buckling temperatures for the H4V4H4 diamond crossing and the TJD EUH4 diamond crossing with double slips, subjected to a percentage change in the longitudinal fastener resistances
Comparison between diamond crossing and diamond crossing with double slips 141 Chapter 7 Model of a switch diamond
7.1 Introduction
In chapter 4 and chapter 5, finite element models of respectively a diamond crossing and a diamond crossing with double slips were investigated. Neither of these were found to evolve to a critical point for lateral instability in realistic climate conditions. The critical buckling temperatures lie far enough above the expected maximum rail temperature increase of 40°C, in case dynamic buckling is not taken into account. In this chapter, a last typical track model applied in the Belgian railways is investigated: a switch diamond. In this geometry, the intersecting junction rails are replaced by switchblades and the common crossing is replaced by a so called Z-block. The switch diamond is often implemented for crossings with small angles to allow safe passage of trains. The outer rails are still continuous and are kinked. A photograph of a curved switch diamond is given in Figure 7.1. The difference in buckling behaviour compared to the common diamond crossing is investigated in this chapter.
Figure 7.1: Photograph of a curved switch diamond (60)
Model of a switch diamond 142 7.2 Geometry
In Figure A.8 in Annex 2, the plan of the (XZX)1/9,2 switch diamond is given. This corresponds to a crossing with an angle of 0,108 radians. To compare the buckling behaviour with the results from the diamond crossing H4V4H4 in chapter 4, a model is made with the larger angle of 0,124 radians. This corresponds consequently to a (XZX)1/8 switch diamond. The sleeper geometry given in the plan of Figure A.8 is maintained for this model.
Similar to previous models, the movable parts of the switchblades are omitted from the model. The geometry of the rails is thus the same as for the H4V4V4 model, except for the omitted switchblades and the application of the Z-block in the crossing. The rails have a 60E1 profile and the Z-block is assigned a 170 mm x 170 mm cross-section. The sleepers have different cross-sections along the track, as illustrated in the plan view of the meshed geometry in Figure 7.2. The ballast resistances are adapted according to these cross-sections. The input values for these adjusted geometries were discussed in section 3.4.2.
Sleepers 150x300mm Sleepers 200x300mm Sleepers 200x420mm Sleepers 150x260mm
Z-block (170x170mm)
Figure 7.2: Meshed geometry of the (XZX)1/8 switch diamond in Siemens NX
Since the rail geometry is the same as for the H4V4H4 diamond crossing, the different considered zones as illustrated in Figure 4.6 also apply here. Again the lateral stiffness in zones 2 and 4 are high because of the presence of the frog/check rails and the Z-block. At the location of the Z-block, longer and larger sleepers are applied that significantly increase the lateral resistance in this central zone. In zone 3, the rigidity of the track panel is reduced, because only two rails are connected to the sleepers in this region. This also means a reduction in buckling strength. But the concentration of axial forces is consequently also smaller, which in turn again improves the safety against buckling. It has to be investigated which of both factors is the most determining. Due to the presence of the Z- block, which has a larger cross-section than the 60E1 rails, the axial forces will increase in the rails nearby. This implies a negative effect on the buckling strength of the switch diamond.
The sleeper spacing in zone 3 of the crossing is smaller for the (XZX)1/8 compared to the H4V4H4 diamond crossing. This property should improve the buckling strength of the track. Again the most critical track defect 8m20mm is implemented in the same zone as for the diamond crossing, downwards in plan view. With the aforementioned reasons, it is assumed that this is indeed the most critical zone for lateral track buckling. Moreover, it allows comparison with the results for the diamond crossing.
Model of a switch diamond 143 7.3 Results and safety assessment
7.3.1 Buckling curve
In the finite element model, the ballast characteristics are assigned values according to the dimensions of the sleepers on plan. A second model was made, in which the sleepers all have the standard sleeper cross-sectional dimensions of 150 mm x 260 mm. This configuration comes close to the characteristics of the H4V4H4 diamond crossing, allowing an equivalent comparison between the buckling behaviour of both track geometries. Only the instability with an implemented track defect with wavelength 8 m and amplitude 20 mm is investigated. The resulting load-displacement curves are illustrated in Figure 7.3 and the corresponding critical buckling temperature increases in Table 7.1.
It is concluded that the (XZX)1/8 diamond crossing with switchblades has more safety against buckling compared to the diamond crossing. The upper critical buckling temperature increase is about 11°C higher. On the other hand, the lower critical buckling temperature, which could become critical for lateral instability if sufficient external energy is applied to the track, is 7°C larger. The diamond crossing H4V4H4 is more prone to progressive buckling, as the difference in upper and lower critical buckling temperature is small. The lateral displacements before buckling of the (XZX)1/8 switch diamond are limited to approximately 0,9 mm at a temperature increase of 40°C, which is clearly lower than observed for the diamond crossing. It is noticed that a track defect with amplitude 8 mm would lead to much larger critical buckling temperatures and smaller lateral displacements at +40°C, in accordance with the results found for the H4V4H4 diamond crossing. The upper and lower critical buckling temperature increases would certainly be higher than respectively 87°C and 67°C, the values found for the track defect 8m8mm in the diamond crossing. This can be concluded from the results found here for the defect 8m20mm.
70
60
50
C] ° 40
30 H4V4H4 - 8m20mm
temperature[ (XZX)1/8 - 8m20mm 20 (XZX)1/8 - 8m20mm - All sleepers 150x260mm
Temperature increase Temperatureincrease above neutral 10
0 0 10 20 30 40 50 60 70 80 Lateral displacement [mm]
Figure 7.3: Load displacement curves for different configurations in the (XZX)1/8 switch diamond. Comparison with the results for the H4V4H4 diamond crossing
Model of a switch diamond 144 The critical buckling temperatures of the model with standard sleeper dimensions are about 2,5°C lower compared to the results for the original geometry. This decrease is the logical result of the lower lateral ballast resistance. The difference in obtained safety is however limited and still significantly higher compared to the diamond crossing. The higher buckling strength of the switch diamond compared to the diamond crossing can be contributed to the denser sleeper spacing in the defect region and the presence of only 2 rails that transfer significant longitudinal forces to the sleepers in the critical zone.
Table 7.1: Critical buckling temperature increases for the (XZX)1/8 switch diamond
Critical buckling temperature increase Model configuration Upper critical buckling temp Lower critical buckling temp
횫Tb,max 횫Tb,min (XZX)1/8 – track defect 8m20mm in zone 3 63,4 °C 57,7 °C of crossing XZX)1/8 – track defect 8m20mm in zone 3 of crossing (ballast resistances according to 60,9 °C 55,4 °C standard sleeper dimensions)
The results presented in the following sections are based on the finite element model with the original sleeper geometries and corresponding ballast resistances. Similar to the previous chapters, the results are taken at certain loadsteps, corresponding to the designated points in Figure 7.4. The measured distances along the railway track are presented in Figure 7.5. The geometry is in fact the same as for the diamond crossing H4V4H4, so that the defect is again situated between 118,5 m and 126,5 m.
70
60 (3) +63,4°C (5) +61,2°C
50 (4) +57,7°C
C] ° 40 (2) +40,0°C
30
temperature[ 20
10
(1) +1,66°C Temperature increase Temperatureincrease above neutral 0 0 20 40 60 80 100 120 Lateral displacement [mm]
Figure 7.4: Designation of the points along the buckling curve where the results are taken. (XZX)1/8 8m20mm
Model of a switch diamond 145 Distance along railway track [m]
Distance along railway track [m]
Figure 7.5: Plot of the distance measured along the railway track applied to the (XZX)1/8 switch diamond (contour plot of the lateral displacements)
7.3.2 Lateral displacements
The lateral displacements for the different temperature increases along the buckling curve are presented in Figure 7.6. The results at the final loadstep (5: +61,2°C) are omitted from this plot as the displacements became too large to allow a clear comparison with the other results. The development of the lateral displacements shows the same behaviour as for the H4V4H4 and TJD EUH4 models. The conclusions made in chapter 4 and 5 can thus be applied here.
10
0
-10
-20
(1) 1,7°C -30 (2) 40,0°C
-40 (3) 63,4°C Lateral displacement Lateral displacement [mm]
(4) 57,7°C (post-buckling) -50
-60 110 115 120 125 130 135
Distance along railway track [m]
Figure 7.6: Lateral displacements along the railway track for different temperature increases. (XZX)1/8 8m20mm
Model of a switch diamond 146 As the calculation of the finite element model of the (XZX)1/8 switch diamond proceeded for larger displacements in the Nastran solver compared to the other models in previous chapters, the post-buckling displacements can be investigated in more detail. Figure 7.7 illustrates the development of lateral displacements in the post-buckling regime. In the last step of the calculation, the maximal misalignment reached more than 12 cm. At this latter configuration, the buckling shape is changing. The central region of the Z-block (127 m - 131 m) is becoming unstable as well and a new misalignment wave is developing.
The temperature increase at which this behaviour occurs is more than 60°C. This is much higher than the maximal temperature increases reached in the H4V4H4 or TJD EUH4 models. At this temperature, the lateral forces that are transferred to the ballast are consequently also higher than the forces which the latter models are facing in the post-buckling regime. In addition to this, the axial forces in the massive Z-block are larger than the common 60E1 rail section. As a consequence of these large transferred axial forces, a much wider region of sleepers has reached the limit displacements in the ballast and are ‘free’ to move. Figure 7.8 illustrates this conclusion, in which the lateral ballast springs between 115 m and 132 m have reached their limit resistance force. This explains the observed behaviour of changing buckling shape in the figure below. However, these considerations are of less importance for the designer of the track model, because the post-buckling configurations are situations which one typically wants to avoid.
40
20
0
-20
-40
-60
-80 (4) 57,7°C (post-buckling)
Lateral displacement Lateral displacement [mm] -100 (5) 61,2°C (post-buckling) -120
-140 110 115 120 125 130 135 140 Distance along railway track [m]
Figure 7.7: Lateral displacements along the railway track in post-buckling regime. (XZX)1/8 8m20mm
Model of a switch diamond 147 10 8 6 4 2 0 -2
-4 Force[kN] -6 -8 -10 -12 -14 100 105 110 115 120 125 130 135 140 145 150 Distance along railway track [m]
Figure 7.8: Forces in the springs representing the lateral ballast resistance along the railway track at the post-buckling temperature increase +61,2°C. (XZX)1/8 8m20mm
7.3.3 Longitudinal displacements and axial strains
The naming convention of the rails in the model of the (XZX)1/8 switch diamond is given in Figure 7.9. Rails C, D, E and F have no fixed boundary condition at one end, which represents the transition to the movable parts of the switchblades. These rails can thus expand in one direction and the same behaviour as for rail C in the diamond crossing with double slips in Figure 5.16 is noticed. The results are thus comparable and are not further discussed here.
Figure 7.9: Designation of the naming convention of the rails in the diamond crossing with switchblades
Rails A and B are kinked and the direction of the local rail axis changes consequently. Figure 7.10 gives the longitudinal displacements of rail A along the rail’s axis of the left part of Figure 7.9. This means that the course of the longitudinal displacements is influenced by the lateral displacements from a distance of 128 m on. This is also observed in the plots.
It is noticed that the rail is displaced towards the central region. The reason can be found in the presence of the Z- block. Due to its larger cross-section than the 60E1 rail section, the axial thermal forces are larger here, resulting in a variation in axial forces along the course of the track. This transition in forces occurs gradually, meaning that the 60E1 rails face larger forces than the nominal force. As a consequence, these rails need to be compressed, which can only occur through longitudinal displacements towards the Z-block region. This behaviour is similar to the transition from 60E1 towards 50E2 rail sections in the TJD EUH4 model in chapter 5. Also the transition from four to two rails in the switchblade area has the corresponding influence on the longitudinal displacements.
Model of a switch diamond 148 5,0 4,0 3,0 2,0 1,0 0,0 -1,0 -2,0 (1) 1,7°C -3,0 (2) 40,0°C -4,0 (3) 63,4°C -5,0 (4) 57,7°C (post-buckling) Longitudinal Longitudinal displacement [mm] -6,0 (5) 61,2°C (post-buckling) -7,0 -8,0 50 70 90 110 130 150 170 190 Distance along railway track [m]
Figure 7.10: Longitudinal displacements of rail A along the railway track for different temperature increases. (XZX)1/8 8m20mm
The axial strains in the rail at a temperature increase of 40°C, presented in Figure 7.11, confirms the observed structural behaviour. The Z-block is expanding (positive strains), while the 60E1 rails at both sides of the block are compressed (negative strain). In plain track, outside the diamond crossing, the rails are expanding slightly (positive strains). The reason for this is found by looking at the boundary conditions. The clamped supports at the ends of the model make sure that the total elongation of the rail should be approximately zero for pre-buckling temperatures. This confirms that elongations in plain track are needed to counteract the shortenings in the section of track close to the Z-block. The expansion is also the result of the expanding rails C/D and E/F.
3E-04 (2) 40,0°C Zero-strain 2E-04
1E-04
0E+00 Axial Axial strain [mm/mm]
-1E-04
-2E-04 40 60 80 100 120 140 160 180 200 Distance along railway track [m]
Figure 7.11: Course of the axial strains in rail A along the railway track at a temperature increase of 40°C (pre-buckling). (XZX)1/8 8m20mm
Model of a switch diamond 149 The longitudinal displacements in the defect area are mainly occurring through slip in the fastening system. Outside this zone, the longitudinal displacements observed in Figure 7.10 are generally occurring through displacements of the sleepers in the ballast with only a small relative displacement between sleeper and rail. It is noticed that the relative displacements of the rails in the fastening systems are again very limited, such that the limitation value of 30 mm will never be reached.
Finally, the axial strains for post-buckling configurations are plotted in Figure 7.12. The course in the defect area is similar to the results of other models. The total elongations of the rail, calculated from the course of the axial strains with formula (4.3), are given in Table 7.2. Again, this is in accordance with the results from the previous chapters.
3E-03
2E-03
1E-03
0E+00
-1E-03
Axial Axial strain [mm/mm] -2E-03 (3) 63,4°C (4) 57,7°C (post-buckling) -3E-03 (5) 61,2°C (post-buckling)
-4E-03 90 100 110 120 130 140 150 160
Distance along railway track [m]
Figure 7.12: Axial strains in rail A along the railway track for post-buckling temperature increases. (XZX)1/8 8m20mm
Table 7.2: Total elongation of rail A for different temperature increases along the buckling curve. (XZX)1/8 8m20mm
Temperature increase above Total elongation of rail neutral temperature (1) 1,7 °C 0,00 mm (2) 40,0 °C 0,01 mm (3) 63,4 °C 0,21 mm (4) 57,7 °C 1,73 mm (5) 61,2 °C 5,93 mm
Model of a switch diamond 150 7.3.4 Axial forces
In the previous section, the transition in rail forces was already outlined. Figure 7.13 illustrates the considered transition zones. The total axial forces will drop when the railway track evolves from four to two rails, but will increase again at the location of the z-block.
4 rails 60E1 2 rails 60E1 Z-block 2 rails 60E1 4 rails 60E1
Figure 7.13: Transition zones in axial rail forces in the switch diamond
The individual rail forces are again summed along the track, to obtain the plots in Figure 7.14. The course of the total axial forces corresponds to the expected transition. At a temperature increase of 40°C for example, the rail forces tend to go towards a nominal value which is half of the nominal value of four rails in the central zone. At the location of the Z-block however, the forces increase locally due to the larger nominal cross-section. In post-buckling configurations, the course of the total compression forces is clearly dropping in the defect area. The drop can be noticed when comparing the curves for the temperature increases of 57,7°C and 61,2°C. This is the consequence of the growing misalignment, allowing the rails to expand locally.
-5000 (1) 1,67°C -4500
-4000 (2) 40,0°C
-3500 (3) 63,4°C -3000
-2500 (4) 57,7°C (post- buckling) -2000 Axial Axial [kN] force (5) 61,2°C (post- -1500 buckling)
-1000
-500
0 0 50 100 150 200 250
Distance along railway track [m]
Figure 7.14: Total axial forces after summation of the axial forces of the individual rails along the railway track, for different temperature increases. (XZX)1/8 8m20mm
Model of a switch diamond 151 The influence of the larger cross-section of the Z-block on the nominal forces is examined in Figure 7.15. A second model is made in which the 170 mm x 170 mm cross-section of the Z-block is replaced by a common 60E1 rail section. It is seen that the presence of the Z-block locally increases the total axial forces.
-3000
-2800
-2600
-2400
-2200
-2000
Axial Axial [kN] force -1800
-1600 (XZX)1/8 - 8m20mm -1400 (XZX)1/8 - 8m20mm - Z-block as 60E1 rail -1200 0 50 100 150 200 250 Distance along railway track [m]
Figure 7.15: Comparison between the course of the axial forces for the models with and without the modeling of the Z- block, 훥T = 40°C. (XZX)1/8 8m20mm
In Figure 7.16, the axial forces in rail A and B in this second model are plotted. A first remark is made on the sudden increase in compression force at a distance of 105 m (and symmetrically at 151 m). This is exactly at the point where the two individual plain tracks evolve into the crossing, which consists out of one track panel for the four rails. This transition means a sudden increase in longitudinal ballast resistance as the sleeper’s length changes from 270 cm to 570 cm. Before this point, the rails and sleeper can move longitudinally in the ballast bed thanks to the smaller longitudinal ballast resistance. This offers an explanation for the decreasing axial forces in the region 40 – 105m, as the rails are expanding here.
At the transition point at 105 m, the sleepers suddenly face a much greater longitudinal ballast resistance. The longitudinal displacements are consequently restrained and the rails stop expanding and are compressed. A steep increase in compression forces is the result. The transition between expanding and compressing rails is also seen in Figure 7.11 in the plot of the axial strains.
Model of a switch diamond 152 -900
Rail A -850 Rail B
-800
-750 Axial Axial [kN] force
-700
-650 0 50 100 150 200 250 Distance along railway track [m]
Figure 7.16: Axial forces in rail A and B at a temperature increase of 40°C for the model with the Z-block replaced by 60E1 rails. (XZX)1/8 8m20mm
Rails C and D are expanding at their ‘free’ ends and the rail forces go to zero. At the heel of the switch, rail A and B are thus facing extra compression when the forces of the rail C and D are transferred to the sleeper. This is seen as the steep increase in compression forces in Figure 7.16. This is comparable to the behaviour of stock and switch rail in a turnout. Figure 7.17(a) gives the results obtained by Landuyt (7) for the axial forces in the stock rail of a turnout. The same fluctuation in normal forces is noticed. Due to the symmetry of the diamond crossing, the decrease in compression forces after the peak is not fully developed in Figure 7.16. At a distance of 128,5 m, both sides of the switch diamond ‘meet’ each other, explaining this behaviour.
Figure 7.17: Axial forces in stock and switch rails of P3350 for 훥T=40°C. Results obtained by Landuyt (7)
The model without the Z-block gave the best representation of the course of the axial forces. The base model of the (XZX)1/8 was however this with the Z-block modeled with a cross-section of 170 mm by 170 mm. The axial forces in the individual rails along the railway track of this model are plotted in Figure 7.18 (pre-buckling) and
Model of a switch diamond 153 Figure 7.19 (post-buckling). The axial forces are going to zero in rails C, D, E and F, comparable with the results in Figure 7.17(b).
Compared to the course in Figure 7.16, the influence of the increased cross-section of the Z-block is visible in rail A and B. The central drop in Figure 7.16 has disappeared in the plot in Figure 7.18 because of the added thermal force. In the post-buckling configuration, the compression forces are decreasing more in rail B. The explanation is found by looking at the effect of radial breathing. As can be noticed from Figure 7.7, at the location of the Z-block (125 m – 130 m), the misalignment is directed upwards, being in the same direction as the kink in the rail B. This kink is comparable to the effect of a curvature in the phenomenon of radial breathing. The growing misalignment in the same direction as the kink yields a faster decrease in compression forces. This behaviour is thus observed and confirmed in the plot of Figure 7.19.
A peak axial force of 900 kN in rail A and B is observed at the pre-buckling temperature increase of 40°C. For the 60E1 rail section, this corresponds to a compressive stress of 117 MPa. This peak stress is comparable with the obtained value for modification 1 of the diamond crossing in Figure 4.37. It is noticed that the magnitude is dependent on the dimensions of the modeled Z-block. The thermal forces are directly related to the cross-sectional surface area of this element. But also without the larger cross-section of the Z-block modeled, the peak compressive force amounts to 870 kN as observed in Figure 7.16. The reference limit value of the allowable compressive stress in the rail of 172 MPa, obtained in section 2.8.2, is not exceeded.
-1000
-800
-600
Average Rail Force Rail A -400
Axial Axial [kN] force Rail B Rail C Rail D -200 Rail E Rail F
0 0 50 100 150 200 250 Distance along railway track [m]
Figure 7.18: Comparison between the course of the axial forces of the individual rails and the average rail force, pre- buckling (+40,0°C). (XZX)1/8 8m20mm
Model of a switch diamond 154 -1400
-1200
-1000
-800
-600 Average Rail Force
Axial Axial [kN] force Rail A Rail B -400 Rail C Rail D -200 Rail E Rail F
0 0 50 100 150 200 250 Distance along railway track [m]
Figure 7.19: Comparison between the course of the axial forces of the individual rails and the average rail force, pre- buckling (+57,7°C). (XZX)1/8 8m20mm
7.4 Conclusions
The last model being investigated in this dissertation is the (XZX)1/8 switch diamond. The difference with the common diamond crossing is the interruption of the junction rails and application of movable switchblades. The outer rails are kinked, with the installation of Z-blocks, which have a larger cross-sectional area and yield higher nominal thermal forces. Again, the same principles as applied before to create a finite element model were applied and from the results it was concluded that the model was valid.
It is concluded that the switch diamond has a higher buckling resistance compared to the common diamond crossing. The upper and lower critical buckling temperature increases are respectively 63,4 °C and 57,7°C for an implemented track defect with an amplitude of 20 mm. With respect to the lower critical buckling temperature, this means that a strength reserve of almost 7°C is obtained in comparison with the H4V4H4 model. As the maximal expected rail temperature increase is only 40°C, it is concluded that there is no risk for severe lateral instabilities in the switch diamond. The permissivity of the fastening systems and the axial forces in the rails are comparable with the diamond crossing, such that these do not form a limitation for the installation of continuously welded switch diamonds.
Model of a switch diamond 155 Chapter 8 Conclusions and further research
8.1 Aim of the dissertation and validation of the results
The aim of this dissertation was the investigation of the buckling behaviour of different crossings applied in the Belgian railway lines with the aid of a finite element model in the software Siemens NX. The existing study by Landuyt (7) provided a reference dissertation for the application of a model of railway track in the finite element software. The goal of the obtained results was to provide a clear recommendation for the application of continuously welded crossings, considering the criterions for safe-operation of the track. These criterions include the critical buckling temperatures, but also the lateral displacements before buckling, the permissivity of the fastening systems and the stress concentrations in the rails. The buckling behaviour of the crossings has been investigated for thermal loads only. The influence of a direct implementation of train loads and dynamic behaviour in the finite element models has been disregarded. The influence of dynamic vibrations and uplift of track has been quantified through a reduction in the ballast resistances, which is a simplification. The railway track models that have been investigated are a diamond crossing, a diamond crossing with double slips and a switch diamond. The geometries of respectively H4V4H4, TJD EUH4 and (XZX)1/8 provided by Infrabel were applied in this study.
The results presented in the previous chapters served as a validation of the finite element model in the first place. The courses of the displacements, the axial forces, forces in the ballast bed, strains, etc. have been discussed and linked to each other. Since no inexplicable behaviour in the models was observed, the reliability of the finite element models is proven. By modifying the geometry of the models and the input characteristics of the resistances, a better understanding in the parameters affecting the buckling behaviour of the crossings is obtained. Finally, also models with fishplated joints were created to compare the behaviour with the continuously welded track models.
8.2 Safety assessment of the considered continuously welded crossings
8.2.1 Critical buckling temperatures
The critical buckling temperatures were determined for the H4V4H4 diamond crossing, the TJD EUH4 diamond crossing with double slips and the (XZX)1/8 switch diamond. Considering the lower critical buckling temperature as the criterion for safe-operation of track, the critical buckling temperature increases above the neutral temperatures were respectively 51°C, 57°C and 58°C for the abovementioned crossings. These are obtained for an implemented track defect with amplitude 20 mm and wavelength 8 m in the weakest zone of the track. The input
Conclusions and further research 156 data for the ballast resistances are corresponding to recently tamped track or low quality ballast, which is the most critical configuration for lateral track buckling. For a track with a neutral temperature of 20°C, the maximal expected temperature increase is 40°C. This means none of the investigated crossings is at risk of buckling during operation, for the base values of the implemented structural parameters.
However, taking into account the effects of dynamic vibrations and uplift of track due to passing trains through a reduction in lateral ballast resistance, the critical temperatures can decrease with more than 4°C. Furthermore, a different way of modeling of the frogs and common crossings can lead to further decrease of the buckling temperatures. This means that the critical buckling temperature increases of the diamond crossing may possibly be close to the limitation of 40°C in certain conditions. Following the level 2 safety approach by Esveld (17), the allowable temperature increase is obtained by diminishing the lower critical buckling temperature by 5°C in case of a buckling regime smaller than 5°C. It could thus be desired to increase the buckling strength of the diamond crossing.
The common diamond crossing has the smallest buckling resistance of the investigated models, which can mainly be contributed to the large sleeper spacing in the defect region and the smaller cross-sectional dimensions of the sleepers compared to the other models. The lateral ballast resistance is the most influencing parameter in the buckling strength of track, explaining the significant influence of the aforementioned aspects.
8.2.2 Permissivity of the rail fastening systems and lateral displacements before buckling
In any of the modeled tracks, the relative displacements between the rails and the sleepers are limited to about 0,1 mm at temperature increases of 40°C above the neutral temperature. Because the limit value of the permissivity is set to 30 mm, this parameter does not form a limitation for the application of continuously welded crossings. The lateral displacements before buckling may reach values up to 4 mm in some configurations of the diamond crossing. With an initial misalignment amplitude of 20 mm, which is above the intervention limit, the lateral displacement of the track can grow above the immediate action limit during a hot summer day. The diamond crossing would in this case be put out of service, meaning economic losses. The lateral displacements of the diamond crossing with double slips and the switch diamond are much more limited and reach values of 1,5 mm at most at +40°C.
8.2.3 Axial forces and stresses in the rails
The peak axial forces in each model have been determined at a temperature increase of approximately 40°C above the neutral temperature. The maximal peak compressive forces yielded axial stresses of 90 MPa to 120 MPa. In the defect region, additional bending stresses are present due to the growing misalignment. These additional stresses reach values up to 30 MPa in the most compressed fiber. The yield strength of the rails is much higher than these values, such that fracture of rails is not an issue. The peak stresses are important to assess the fatigue life of the rail, which is subjected to temperature cycles and consequently also alternating stress peaks. Referring to limitations applied in EN 1991-2 for bridge-rail interaction, which give a good indication of the order of magnitude of allowable stresses, it is concluded that the maximal compressive stresses in the crossings are not an issue for the application of continuously welded rails.
Conclusions and further research 157 8.2.4 Parameters influencing the buckling behaviour and applications to improve the buckling resistance
From the parameter studies performed in chapter 4 and chapter 5, it was concluded that the most influential parameter that affects the critical buckling temperatures is the lateral ballast resistance. The longitudinal ballast resistance and the torsional fastener resistance have almost no effect on the buckling strength. The longitudinal fastener resistance has a minor influence on the buckling temperatures. A change in the latter with 40%, affects the critical temperature with only 2°C.
Several modifications for the H4V4H4 diamond crossing were proposed to improve the buckling resistance. Increasing the sleeper length in the critical zone of the crossing can increase the buckling temperature by 6 to 9°C. The economical most interesting option is to decrease the sleeper spacing in the critical zone. Changing the spacing from 705 mm to 550 mm increases the critical buckling temperatures by more than 6°C. With the application of safety caps or retaining walls, the buckling temperature can also be increased by 3 to 6°C.
8.3 Critical inference on the obtained results
8.3.1 Reasons why the critical buckling temperatures could be lower in reality
In this dissertation, some assumptions were made in modeling the crossings in a finite element model. The results are thus depending on parameters which have an uncertainty in their application or parameters that have been disregarded. As a final conclusion, an overview is given of possible events or parameters that can further reduce the critical buckling temperatures of the considered track models.
8.3.1.1 Dynamic track buckling and uplift of track
The most critical factor that has been disregarded in this dissertation is the effect of dynamic forces on the buckling behaviour of the track. As was mentioned in section 2.4, the critical buckling temperatures for dynamic buckling could in particular cases be 6 to 7°C lower than calculated in a static buckling analysis. In this dissertation, this effect was taking into account by means of a reduction in lateral ballast resistance, yielding decreases of 4°C and more. A more detailed study by direct implementation of dynamic forces and uplift of track in the finite element model can however result in more critical conclusions. The lateral dynamic forces on the rails in the defect region have not been taken into account in this dissertation.
8.3.1.2 Alternating temperatures
During temperature cycles, the lateral displacements in the initial misalignment could accumulate, yielding a growing amplitude of the track imperfection. After a long period of hot summer days, the critical misalignment could thus have grown to a point where the buckling resistance has decreased significantly, without immediate corrective intervention being possible.
Conclusions and further research 158 8.3.1.3 Initial stress state of ballast characteristics and fasteners
Pio Pucillo performed a sensitivity analysis to the utilisation of raw experimental data obtained from in situ tests of the sleeper-ballast lateral resistance (61). He found that a high value of the initial stiffness of the lateral ballast resistance may lead to overestimation of the maximum buckling temperature. In this dissertation, the track defect was implemented together with stress-free springs characterising the ballast resistances. In reality however, the stiffness of the ballast resistance could already have decreased due to the initial lateral misalignment of track. As a consequence, the lateral ballast stiffness is possibly overestimated in the finite element model. The same is true for the fastener resistances. The critical buckling temperatures could be reduced when this initial deformation would be taken into account for the load-displacements characteristics of the ballast and fastener resistances. It must be taken into account that the typical ballast resistance tests are performed in a static way, such that the influence may not be significantly large.
8.3.2 Reasons why the obtained results could be too conservative
The applied configurations in the finite element model may also be too conservative. Some aspects are adduced which lead to increased critical buckling temperatures of the track models in reality.
8.3.2.1 Determination of the lateral ballast resistance
In the critical region of the crossings, the increased weight of the frogs, common crossings and Z-blocks were not included in the determination of the lateral ballast resistances of the sleepers underneath these elements. The most important contribution to the lateral resistance of the sleeper in the ballast bed comes from the self-weight of sleeper and track. It may thus be expected that the increased weight of these elements contribute to a non- negligible effect on the lateral resistance of the track in the ballast bed. The critical temperatures will consequently also be larger for these increased lateral ballast resistances.
Furthermore, it must be mentioned that the track model is assumed to be installed in the ballast bed without the presence of other railway tracks nearby. The characteristics of the ballast resistance are determined in the context of a single track. The application of the investigated crossings are however generally in bundles of railway track. As a consequence, the ballast bed is much more confined in all directions by the dense concentration of tracks. The stiffnesses of the characterising load-displacement curves will thus be higher. The modeled ballast resistances are thus conservative and subsequently also the obtained results for the critical buckling temperatures.
8.3.2.2 Overestimation of the initial imperfection in track
The critical buckling temperatures were determined for initial misalignments of 20 mm. This extreme value is based on the criterions for safe operation by NMBS (42). However, the assumption of a misalignment of 20 mm may be too conservative for the case of the considered diamond crossings. In reality, this will probably never occur in the track panel of the crossings.
Conclusions and further research 159 8.4 Further research
8.4.1 Bundle of different crossings/turnouts
In this dissertation, the buckling behaviour of the crossings was investigated with the assumption that these are installed within two sections of plain track. In reality, these crossings are present in large bundles of railway track, like it is for example the case in the setup of Kinkempois for which Figure 8.1 gives an extract. The axial forces in the rails, which triggers the lateral instability, can significantly differ in these configurations compared to the nominal plain track setups assumed in the finite element models. Especially in case turnouts are installed close to the crossings, the axial forces are increased in the stock rail over a relative large section of track. This is among others concluded by Landuyt (7). The influence on the critical buckling temperatures of the investigated crossings can be determined by creating a finite element model of the complete setup of the turnouts and crossings. Creating such a finite element model is however time-consuming and requires the use of external calculation computers, which could not be done in the timeframe of this dissertation.
Figure 8.1: Extract of the plan of the setup in Kinkempois (62)
8.4.2 3D modeling of the rails
In the context of this master dissertation, the rails were meshed in the finite element model with beam elements. Only the axial stresses could be obtained. In assessing the fatigue life of the rails, a more detailed stress distribution in the rails at the track defect is needed. To obtain the local stress concentration in a particular section of rail, a 3D model of the rail should be made. This however makes the finite element model much more complex and the time needed to calculate the model would also increase significantly. For these models, an extensive study of the different parameters effecting the buckling behaviour would be time-consuming. Therefore, it was not part of this master dissertation.
An additional remark could be made to the modeling of the TJD EUH4 diamond crossing with double slips. At the locations of the movable parts of the switchblades, the rail foot is abraded to allow alignment of stock rail and
Conclusions and further research 160 tongue. This property was not modeled in the finite element, with the reason explained in chapter 5. A more detailed study could be made with the reduced cross-section applied in the finite element model. Although it is expected that the influence on the critical buckling temperatures will be minimal, it can be interesting to investigate the course of the axial forces and the stresses with this modification. Together with the abraded rail foot, the characteristics of the gliding chairs should be modeled in more detail in order to obtain results in accordance with the expectations in reality.
8.4.3 Implementation of dynamic train loads
Landuyt (7) already suggested to investigate the influence of train loads on the buckling behaviour of track. As already outlined before, the critical buckling load of track could be up to 7°C lower in case dynamic buckling is analysed. This is particularly the case for curved rails, for which lateral train loads become important. In this dissertation the effect of dynamic vibrations and uplift of track has only be investigated through a reduction in lateral ballast resistance, quantified by Kish (32). Train loading can be implemented in the model by a static system, as for example done by Esveld (23) in the CWERRI software. However, in case vehicle induced buckling is investigated, a more advanced dynamic theory is needed. Also Esveld (23) concluded that the vibrations have a substantial influence on the lateral resistance of the track. In the context of this dissertation, the implementation of a dynamic train loading system in the finite element model was too complex.
8.4.4 Different implementation of the initial misalignments in a finite element model
As already outlined previously, the initial misalignment was implemented in the finite element model in a stress- free state of the rails. Also the initial states of the structural resistances are not adjusted according to the initial imperfection as discussed in section 8.3.1.3. It can be expected that these aspects may have an influence on the determined critical buckling temperatures. In section 3.7.1, a proposal was made to take these effects into account, which was however not possible in the Nastran solver. Further research is thus possible to model the described aspects in the calculation model.
The track defect was described in this dissertation by a symmetrical 1-cos(x) sinusoidal curve. However, also asymmetrical buckling shapes exists, that were not taken into account in this study. Further research could thus focus on the implementation of an asymmetrical initial misalignment and the determination of the influence on the critical buckling temperatures. Although Samavedam (20) concluded that the differences in critical temperatures between different buckling modes in plain track are rather small (see section 2.3.5), it may be worthful to investigate the influence in the case of continuously welded crossings.
8.4.5 Verification of the obtained results with tests on site
The obtained results in the finite element models could be validated by measurements on site. The strains and lateral displacements of the rails can be measured in time and linked to the corresponding rail temperature at each time interval. Especially hot summer periods are therefore desired. The difficulty in assessing these results is the fact that the initial misalignment determines the buckling behaviour of the track. Moreover, the misalignments were
Conclusions and further research 161 implemented in the finite element models in a stress-free state of the rail, which will never be the case in reality. One should be able to measure the initial misalignment in the track at the neutral rail temperature, and implement this exact misalignment in a finite element model. For example the detailed measurements of the EM130 measurement train (8) could be used to implement the exact geometry of the track into a finite element model.
Besides the initial misalignment of the track, also the lateral ballast resistance has a significant influence on the development of lateral displacements of the track and the strains in the rails. It is therefore also important to approach the real characteristics of the ballast resistance on site as much as possible for the implementation in a finite element model. Therefore, one could perform lateral ballast resistance tests on site (for example the uncut panel pull method described in section 2.6.1.1) to determine the characteristic load-displacement curve. This can be implemented in the finite element model. The results of the calculations could subsequently be compared to measurements on site. In this way, the modeled buckling behaviour of the crossings in a finite element model can be validated.
Conclusions and further research 162 References
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References 167 Annexes
Annex 1: Rail profiles
Figure A.1: Cross-section and characteristics of rail profile 60E1 (UIC60). Dimensions in mm (63)
Annexes 168
Figure A.2: Cross-section and characteristics of rail profile 50E2. Dimensions in mm (64)
Annexes 169
Figure A.3: Cross-section and characteristics of checkrail profile 33C1. Dimensions in mm (65)
Figure A.4: Cross-section of fishplates £600 Arcelor Mittal. Dimensions in mm (53)
Annexes 170 Annex 2: Plans of crossings/turnouts
Figure A.5: Plan of the H4V4H4 diamond crossing (47)
Annexes 171
Figure A.6: Plan of the H3V3H3 diamond crossing (66)
Annexes 172
Figure A.7: Plan of the TJD EUH4 diamond crossing with double slips (57)
Annexes 173
Figure A.8: Plan of the (XZX)1/9,2 switch diamond (67)
Annexes 174 Annex 3: Results – Diamond crossing
Figure A.9: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Lateral displacements at 훥T = 1,66°C (pre-buckling)
Figure A.10: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Lateral displacements at 훥T = 41,0°C (pre-buckling)
Annexes 175
Figure A.11: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Lateral displacements at 훥T = 52,6°C (upper critical buckling temperature increase)
Figure A.12: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Lateral displacements at 훥T = 51,6°C (post- buckling)
Annexes 176
Figure A.13: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Longitudinal displacements at 훥T = 1,66°C (pre- buckling)
Figure A.14: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Longitudinal displacements at 훥T = 41,0°C (pre- buckling)
Annexes 177
Figure A.15: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Longitudinal displacements at 훥T = 52,6°C (upper critical buckling temperature increase)
Figure A.16: Diamond crossing H4V4H4, defect 8m20mm in zone 3. Longitudinal displacements at 훥T = 51,6°C (post- buckling)
Annexes 178