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A Priori Determination of Track Modulus Based on Elastic Solutions

  • Railroad Engineering
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Abstract

The standard approach for modeling railway tracks idealizes the rails as two infinite beams, each supported over a separate continuous spring foundation. The foundation is characterized by a track modulus that embodies all components and materials underlying each rail as well as any cross-rail interaction. Track modulus is considered a basic parameter governing the field performance of tracks. Therefore, a priori determination of track modulus is needed in design of traditional railways, as well as in evaluating the performance-potential of non-traditional track solutions. In this study, a new method was suggested for a priori track modulus determination based on elastic solutions. Specifically sought were closed-form analytical formulations that could be representative and tractable. In this connection, a 3-D track model was developed, wherein: rail-pads were considered as linear springs, sleepers as finite beams, and all underlying soil-like materials as a homogenous half-space. Ultimately, track modulus was determined by linking calculations in the 3-D model and the standard model. This was done by requiring equal maximal displacement as well as identical load distribution along the rail under the weight of a single railcar axle. The method was illustrated considering a wide set of values for the different model parameters. The calculated results are comparable in magnitudes and exhibit similar sensitivities to the input parameters as reported in field studies or as derived from elaborate numerical schemes.

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References

  • Adegoke CW, Chang CS, Selig ET (1979) Study of analytical models for track support systems. Transportation Research Record 733:2–20

    Google Scholar 

  • AREMA Manual (2006) AREMA manual. American Railway Engineering and Maintenance-of-Way Association, Lanham, MD, USA

    Google Scholar 

  • Barden L (1962) Distribution of contact pressure under foundations. Géotechnique 12(3):181–198, DOI: https://doi.org/10.1680/geot.1962.12.3.181

    Google Scholar 

  • Biot MA (1937) Bending of infinite beam on elastic foundation. Journal of Applied Mechanics 4(1):1–7

    Google Scholar 

  • Bose T, Levenberg E, Zania V (2018) Analyzing track responses to train braking. Proceedings of the Institution of Mechanical Engineers Part F: Journal of Rail and Rapid Transit, 232(7):1984–1993, DOI: https://doi.org/10.1177/0954409718761242

    Google Scholar 

  • Burmister DM (1945a) The general theory of stresses and displacements in layered systems I. Journal of Applied Physics 16(2):89–94, DOI: https://doi.org/10.1063/1.1707558

    Google Scholar 

  • Burmister DM (1945b) The general theory of stresses and displacementsin layered soil systems II. Journal of Applied Physics 16(3):126–127, DOI: https://doi.org/10.1063/1.1707562

    Google Scholar 

  • Burmister DM (1945c) The general theory of stresses and displacements in layered soil systems III. Journal of Applied Physics 16(5):296–302, DOI: https://doi.org/10.1063/1.1707590

    Google Scholar 

  • Cai Z, Raymond GP, Bathurst RJ (1994) Estimate of static track modulus using elastic foundation models. Transportation Research Record 1470:65–72

    Google Scholar 

  • Chang CS, Adegoke CW, Selig ET (1980) GEOTRACK model for railroad track performance. Journal of the Geotechnical Engineering Division 106(11):1201–1218

    Google Scholar 

  • Daloglu AT, Vallabhan CVG (2000) Values of k for slab on winkler foundation. Journal of Geotechnical and Geoenvironmental Engineering 126(5):463–471, DOI: https://doi.org/10.1061/(ASCE)1090-0241(2000)126:5(463)

    Google Scholar 

  • Ebersöhn W, Trevizo MC, Selig ET (1993) Effect of low track modulus on track performance. In: Proceedings of the Fifth International Heavy Haul Conference, International Heavy Haul Association, Beijing, China

    Google Scholar 

  • Galin LA (1943) On the winkler-zimmermann hypothesis for beams. Journal of Applied Mathematics and Mechanics 7(4):293–300

    MathSciNet  MATH  Google Scholar 

  • Hay WW (1982) Railroad engineering. John Wiley & Sons, Inc., Hoboken, NJ, USA

    Google Scholar 

  • Hemsley JA (2000) Design applications of raft foundations. Thomas Telford Ltd., London, UK

    Google Scholar 

  • Ioannides AM (2006) Concrete pavement analysis: The first eighty years. International Journal of Pavement Engineering 7(4):233–249, DOI: https://doi.org/10.1080/10298430600798481

    Google Scholar 

  • Kerr AD (2000) On the determination of the rail support modulus k. International Journal of Solids and Structures 37:4335–4351, DOI: https://doi.org/10.1016/S0020-7683(99)00151-1

    MATH  Google Scholar 

  • Kerr AD (2003) Fundamentals of railway track engineering. Simmons-Boardman Books Inc., Omaha, NE, USA

    Google Scholar 

  • Khazanovich L, Tayabji SD, Darter MI (2001) Backcalculation of layer parameters for LTPP test sections, volume I: Slab on elastic solid and slab on dense-liquid foundation analysis of rigid pavements. Technical Report No. FHWA-RD-00-086, FHWA, McLean, VA, USA

    Google Scholar 

  • Klar A, Vorster TEB, Soga K, Mair RJ (2005) Soil-pipe interaction due to tunnelling: Comparison between winkler and elastic continuum solutions. Géotechnique 55(6):461–466, DOI: https://doi.org/10.1680/geot.2005.55.6.461

    Google Scholar 

  • Lee KM, Hou XY, Ge XW, Tang Y (2001) An analytical solution for a jointed shield-driven tunnel lining. International Journal of Numerical and Analytical Methods in Geomechanics 25:365–390, DOI: https://doi.org/10.1002/nag.134

    MATH  Google Scholar 

  • Lu S, Arnold R, Farritor S, Fateh M, Carr G (2008) On the relationship between load and deflection in railroad track structure. Proceedings of the AREMA 2008 annual conference, September 21-24, Salt Lake City, UT, USA

    Google Scholar 

  • Mair RJ (2008) Tunnelling and geotechnics: New horizons. Geotechnique 58(9):695–736, DOI: https://doi.org/10.1680/geot.2008.58.9.695

    Google Scholar 

  • Mishra D, Sharma S, Shrestha A, Li D, Basye C (2016) GEOTRACK-2015: An upgraded software tool for railroad track analysis. Joint rail conference american society of mechanical engineers, March 23-26, San Jose, CA, USA

    Google Scholar 

  • Nafari SF, Gül M, Cheng JR (2017) Quantifying live bending moments in rail using train-mounted vertical track deflection measurements and track modulus estimations. Journal of Civil Structural Health Monitoring 7(5):637–643, DOI: https://doi.org/10.1007/s13349-017-0248-1

    Google Scholar 

  • Narayanan RM, Jakub JW, Li D, Elias SE (2004) Railroad track modulus estimation using ground penetrating radar measurements. NDT & E International 37(2):141–151, DOI: https://doi.org/10.1016/j.ndteint.2003.05.003

    Google Scholar 

  • Newton SG, Clark RA (1979) An investigation into the dynamic effects on the track of wheelflats on railway vehicles. Journal of MechanicalEngineering Science 21(4):287–297, DOI: https://doi.org/10.1243/JMES_JOUR_1979_021_046_02

    Google Scholar 

  • Norman C, Farritor S, Arnold R, Elias SEG, Fateh M (2004) Design of a system to measure track modulus from a moving railcar. In: Proceedings of International Conference of Railway Engineering 2004. Engineering Technics Press, London, UK

    Google Scholar 

  • Poulos HG, Davis EH (1974) Elastic solutions for soil and rock mechanics. John Wiley & Sons, Hoboken, NJ, USA

    Google Scholar 

  • Prause RH, Kennedy JC (1977) Parametric study of track response. Report DOT-TSC-FRA-77-75, US Department of Transportation, Washington DC, USA

    Google Scholar 

  • Rajani B, Zhan C, Kuraoka S (1996) Pipe-soil interaction analysis of jointed water mains. Canadian Geotechnical Journal 33(3):393–404, DOI: https://doi.org/10.1139/t96-061

    Google Scholar 

  • Read D, Chrismer S, Ebersöhn W, Selig ET (1994) Track modulus measurements at the pueblo soft subgrade site. Transportation Research Record 1470:55–64

    Google Scholar 

  • Roghani A, Hendry MT (2017) Quantifying the impact of subgrade stiffness on track quality and the development of geometry defects. Journal of Transportation Engineering Part A: Systems 143(7):1–10, DOI: https://doi.org/10.1061/JTEPBS.0000043

    Google Scholar 

  • Sadeghi J, Barati P (2010) Improvements of conventional methods in railway track analysis and design. Canadian Journal of Civil Engineering 37(5):675–683, DOI: https://doi.org/10.1139/L10-010

    Google Scholar 

  • Schapery RA (1965) A method of viscoelastic stress analysis using elastic solutions. Journal of the Franklin Institute 279(4):268–289

    MathSciNet  MATH  Google Scholar 

  • Selig ET, Li D (1994) Track modulus: Its meaning and factors influencingit. Transportation Research Record 1470:47–54

    Google Scholar 

  • Selig ET, Waters JM (1994) Track geotechnology and substructure management. Thomas Telford Ltd., London, UK

    Google Scholar 

  • Setiadji BH, Fwa TF (2009) Examining k-E relationship of pavement subgrade based on load-deflection consideration. Journal of Transportation Engineering 135(3):140–148, DOI: https://doi.org/10.1061/(ASCE)0733-947X(2009)135:3(140)

    Google Scholar 

  • Steinbrenner WA (1936) A rational method for determination of the vertical normal stresses under foundations. Proceedings of the first international conference on soil mechanics and foundation engineering, June 22–26, Cambridge, MA, USA

    Google Scholar 

  • Stewart HE (1985) Measurement and prediction of vertical track modulus. Transportation Research Record 1022:65–71

    Google Scholar 

  • Terzaghi K (1955) Evaluation of coefficients of subgrade reaction. Geotechnique 5(4):41–50

    Google Scholar 

  • Tzanakakis K (2013) The effect of track stiffness on track performance. In: The railway track and its long term behaviour, STTT 2, Springer-Verlag, Berlin, Germany, 79–87

    Google Scholar 

  • Vesic AB (1961) Beams on elastic subgrade and the winkler's hypothesis. Proceedings of the 5th international conference on soil mechanics and foundation engineering, July 17–22, Paris, France

    Google Scholar 

  • Vesic AS, Saxena SK (1969) Analysis of structural behavior of road test rigid pavements. Highway Research Record 291:156–158

    Google Scholar 

  • Westergaard HM (1948) New formulas for stresses in concrete pavements of airfields. Transactions of the American Society of Civil Engineers, 113:425–439

    Google Scholar 

  • Winkler E (1867) Die Lehre von der Elastizitätund Festigkeit, mit Besonderer Rücksicht auf ihre Anwendung in der Technik. H. Dominicus, Prague, Czech Republic

    MATH  Google Scholar 

  • Wood AMM (1975) The circular tunnel in elastic ground. Géotechnique 25(1):115–127, DOI: 10.1680/geot.1975.25.1.115

    Google Scholar 

  • Zakeri JA, Abbasi R (2012) Field investigation on variation of rail support modulus in ballasted railway tracks. Latin American Journal of Solids and Structures 9(6):643–656, DOI: https://doi.org/10.1590/S1679-78252012000600002

    Google Scholar 

  • Zakeri JA, Sadeghi J (2007) Field Investigation on load distribution and deflections of railway track sleepers. Journal of Mechanical Science and Technology 21:1948–1956, DOI: https://doi.org/10.1007/BF03177452

    Google Scholar 

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Acknowledgements

The support from Innovation Fund Denmark is gratefully acknowledged. This study is part of ‘Roads2Rails: Innovative and cost-effective asphalt based railway construction system’ (Grand Solutions 5156-00006B).

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  1. Dept. of Civil Engineering, Technical University of Denmark, Nordvej, Kgs. Lyngby 2800, Denmark

    Tulika Bose & Eyal Levenberg

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  1. Tulika Bose
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  2. Eyal Levenberg
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Correspondence to Tulika Bose.

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Bose, T., Levenberg, E. A Priori Determination of Track Modulus Based on Elastic Solutions. KSCE J Civ Eng 24, 2939–2948 (2020). https://doi.org/10.1007/s12205-020-5372-5

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  • DOI: https://doi.org/10.1007/s12205-020-5372-5

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