Acta Mechanica

, Volume 227, Issue 4, pp 1157–1179 | Cite as

Analytical solution for the elastic bending of beams lying on a variable Winkler support

  • Diego Froio
  • Egidio RizziEmail author
Original Paper


The interaction between structures and supporting media is crucial for several engineering applications. Among existing models, the elastic one originally attributed to Emil Winkler is rather well-known to both researchers and engineers, with main reference to a constant foundation modulus. The present paper reviews first the state of the art on this model and then considers, analytically, the little-explored case of a space-dependent stiffness coefficient. The analytical solution of the ordinary differential equation describing the static deflection of a simply-supported Euler–Bernoulli elastic beam lying on a variable Winkler elastic foundation is sought. Specifically, the case of a nonlinear, minus four power variation of the stiffness coefficient is considered, allowing for closed-form representations. These are derived and examined in view of interpreting their parametric variations due to changes in the mechanical properties of the beam-foundation system. In the end, a consistent validation comparison between analytical solution and alternative reimplemented numerical treatments is produced.


Elastic Foundation Railway Track Elastic Support Uniform Beam Subgrade Reaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aydogan M.: Stiffness-matrix formulation of beams with shear effect on elastic foundation. J. Struct. Eng. ASCE 121(9), 1265–1270 (1995)CrossRefGoogle Scholar
  2. 2.
    Barden L.: Distribution of contact pressure under foundations. Geotechnique 12(3), 181–198 (1962)CrossRefGoogle Scholar
  3. 3.
    Beaufait F.W.: Numerical analysis of beams on elastic foundation. J. Mech. Div. Proc. ASCE 103(EM1), 205–209 (1977)Google Scholar
  4. 4.
    Biot M.A.: Bending of an infinite beam on an elastic foundation. J. Appl. Mech. Trans. ASME 59(203), 1–7 (1937)Google Scholar
  5. 5.
    Borák P., Marcián L.: Beams on elastic foundation using modified Betti’s theorem. Int. J. Mech. Sci. 88, 17–24 (2014)CrossRefGoogle Scholar
  6. 6.
    Bowles J.E.: Analytical and Computer Methods in Foundation Engineering. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  7. 7.
    Castro Jorge P., Simões F.M.F., Pinto da Costa A.: Dynamics of beams on non-uniform nonlinear foundations to moving loads. Comput. Struct. 148, 26–34 (2015)CrossRefGoogle Scholar
  8. 8.
    Castro Jorge P., Pinto da Costa A., Simões F.M.F.: Finite element dynamic analysis of finite beams on a bilinear foundation under a moving load. J. Sound Vib. 346, 328–344 (2015)CrossRefGoogle Scholar
  9. 9.
    Chen C.N.: Solution of beam on elastic foundation by DQEM. J. Eng. Mech. ASCE 124(12), 1381–1384 (1998)CrossRefGoogle Scholar
  10. 10.
    Clastornik J., Eisenberger M., Yankelevsky D.Z., Adin M.A.: Beams on variable Winkler foundation. J. Appl. Mech. Tran. ASME 53(4), 925–928 (1986)CrossRefzbMATHGoogle Scholar
  11. 11.
    Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill, New Delhi (1955)zbMATHGoogle Scholar
  12. 12.
    Dodge A.: Influence functions for beams on elastic foundations. J. Struct. Div. 90(ST4), 63–102 (1964)Google Scholar
  13. 13.
    Eisenberger M., Yankelevsky D.Z.: Exact stiffness matrix for beams on elastic foundation. Comput. Struct. 21(6), 1355–1359 (1985)CrossRefGoogle Scholar
  14. 14.
    Filonenko-Borodich M.M.: Some approximate theories of the elastic foundation (in Russian). Uchenyie Zapiski Moskovskogo Gosudarstvennogo Universiteta Mekhanica 46, 3–18 (1940)Google Scholar
  15. 15.
    Franklin J.N., Scott R.F.: Beam equation with variable foundation coefficient. J. Eng. Mech. Div. Proc. ASCE 105(5), 811–827 (1979)Google Scholar
  16. 16.
    Fraser D.C.: Beams on elastic foundations. A computer-oriented solution for beams with free ends. Civ. Eng. Trans. 11(1), 25–30 (1969)Google Scholar
  17. 17.
    Frydrýšek, K.: Beams on elastic foundation solved via probabilistic approach (SBRA Method). In: Bérenguer, C., Grall, A., Soares, C.G. (eds.) Advances in Safety, Reliability and Risk Management, pp. 1849–1854. Taylor and Francis, London (2012)Google Scholar
  18. 18.
    Guo Y., Weitsman Y.J.: Solution method for beams on nonuniform elastic foundations. J. Eng. Mech. Proc. ASCE 128(5), 592–594 (2002)CrossRefGoogle Scholar
  19. 19.
    Gazis D.C.: Analysis of finite beams on elastic foundations. J. Struct. Div. Proc. ASCE 84(ST4), 1–18 (1958)MathSciNetGoogle Scholar
  20. 20.
    Hayashi K.: Theorie des Trägers auf elastischer Unterlage und ihre Anwendung auf den Tiefbau nebst einer Tafel der Kreis- und Hyperbelfunktionen. Springer, Berlin (1921)CrossRefGoogle Scholar
  21. 21.
    Hendry A.W.: New method for the analysis of beams on elastic foundations. Civ. Eng. Public Works Rev. 53(621), 297–299 (1958)Google Scholar
  22. 22.
    Hertz H.: On the equilibrium of floating elastic plates. Wiedemann’s Ann. 22, 449–455 (1884)CrossRefGoogle Scholar
  23. 23.
    Hetényi M.: Beams on Elastic Foundation. The University of Michigan Press, Ann Arbor (1946)zbMATHGoogle Scholar
  24. 24.
    Hetényi M.: A general solution for the bending of beams on an elastic foundation of arbitrary continuity. J. Appl. Phys. 21(1), 55–58 (1950)CrossRefzbMATHGoogle Scholar
  25. 25.
    Hetényi M.: Beams and plates on elastic foundations and related problems. Appl. Mech. Rev. 19(2), 95–102 (1966)Google Scholar
  26. 26.
    Horibe T.: Boundary integral equation method analysis for beam-columns on elastic foundation. Trans. Jpn. Soc. Mech. Eng. Part A 62(601), 2067–2071 (1996)CrossRefGoogle Scholar
  27. 27.
    Hosur V., Bhavikatti S.S.: Influence lines for bending moments in beams on elastic foundations. Comput. Struct. 58(6), 1225–1231 (1996)CrossRefzbMATHGoogle Scholar
  28. 28.
    Iwiński, T.: Theory of Beams—The Application of the Laplace Transformation Method to Engineering Problems. Pergamon Press, Oxford (1967); (edited and translated by E.P. Bernat)Google Scholar
  29. 29.
    Iyengar K.T.S.R., Anantharamu S.: Finite beam-columns on elastic foundations. J. Eng. Mech. Div. Proc. ASCE 89(EM6), 139–160 (1963)Google Scholar
  30. 30.
    Iyengar K.T.S.R., Anantharamu S.: Influence lines for beams on elastic foundations. J. Struct. Div. Proc. ASCE 91(ST3), 45–56 (1965)Google Scholar
  31. 31.
    Jang T.S.: A new semi-analytical approach to large deflections of Bernoulli–Euler–v.Kárman beams on a linear elastic foundation: nonlinear analysis of infinite beams. Int. J. Mech. Sci. 66, 22–32 (2013)CrossRefGoogle Scholar
  32. 32.
    Jang T.S.: A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kárman beam on a nonlinear elastic foundation. Acta Mech. 225(7), 1967–1984 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Jones G.: Analysis of Beams on Elastic Foundations Using Finite Difference Theory. Thomas Telford Publishing, New York (1997)CrossRefGoogle Scholar
  34. 34.
    Kerr A.D.: Elastic and viscoelastic foundation models. J. Appl. Mech. Trans. ASME 31(3), 491–498 (1964)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kerr A.D.: A study of a new foundation model. Acta Mech. 1(2), 135–147 (1965)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Knothe, K., Tausendfreund, D.: Emil Oskar Winkler <1835–1888>. Begründer der Statik der Baukonstruktionen an der TH Berlin. Leben und Werk. In: K. Schwarz for the president of Berlin TU, eds., 1799–1999. Von der Bauakademie zur Technischen Universität Berlin. Geschichte und Zukunft, Ernst & Sohn, Berlin (2000)Google Scholar
  37. 37.
    Knothe K.: Fiedlerbriefe und Bibliographie Emil Winklers. Institut für Geschichte der Naturwissenschaften, München (2004)Google Scholar
  38. 38.
    Kurrer K.E., Ramm E.: The History of the Theory of Structures: From Arch Analysis to Computational Mechanics. Ernst & Sohn Verlag für Architektur und technische Wissenschaften, Berlin (2008)CrossRefGoogle Scholar
  39. 39.
    Lentini M.: Numerical solution of the beam equation with nonuniform foundation coefficient. J. Appl. Mech. Trans. ASME 46(4), 901–904 (1979)CrossRefzbMATHGoogle Scholar
  40. 40.
    Lee S.L., Wang T.M., Kao J.S.: Continuous beam-columns on elastic foundation. J. Eng. Mech. Div. Proc. ASCE 87(EM2), 55–70 (1961)Google Scholar
  41. 41.
    Levinton, Z.: Elastic foundations analyzed by the method of redundant reactions. Trans. ASCE 114(1), 40–52 (1949); (also in Journal of the Structural Division, Proceedings of the ASCE, 73(12), 1529–1541)Google Scholar
  42. 42.
    Malter H.: Numerical solutions for beams on elastic foundations. J. Struct. Div. Proc. ASCE 84(1562), 757–770 (1958)Google Scholar
  43. 43.
    Miyahara F., Ergatoudis J.G.: Matrix analysis of structure foundation interaction. J. Struct. Div. Proc. ASCE 102(ST1), 251–265 (1976)Google Scholar
  44. 44.
    Miranda C., Nair K.: Finite beams on elastic foundation. J. Struct. Div. Proc. ASCE 92(ST2), 131–142 (1966)Google Scholar
  45. 45.
    Murphy G.M.: Ordinary Differential Equations and Their Solutions. Van Nostrand, Princeton (1960)zbMATHGoogle Scholar
  46. 46.
    Pasternak P.L.: On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants (in Russian). Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow (1954)Google Scholar
  47. 47.
    Penzien J.: Discontinuity stresses in beams on elastic foundations. J. Struct. Div. Proc. ASCE 86(ST4), 1083–1095 (1960)Google Scholar
  48. 48.
    Pipes L.A.: Applied Mathematics for Engineers and Physicists. Mc-Graw Hill, New York (1946)zbMATHGoogle Scholar
  49. 49.
    Popov E.P.: Successive approximations for beams on an elastic foundations. Trans. ASCE 116(1), 1083–1095 (1951)Google Scholar
  50. 50.
    Ray K.C.: Influence lines for pressure distribution under a finite beam on elastic foundation. J. Am. Concret. Inst. 30(6), 729–740 (1958)Google Scholar
  51. 51.
    Reissner E.: A note on deflection of plates on a viscoelastic foundation. J. Appl. Mech. Trans. ASME 25(1), 144–145 (1958)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Schwedler, J.W.: Discussion on iron permanent way. In: Proceedings of the Institution of Civil Engineers, vol. 67, pp. 95–118. London, UK (1882)Google Scholar
  53. 53.
    Selvadurai A.P.S.: Elastic Analysis of Soil-Foundation Interaction. Elsevier, Amsterdam (1979)Google Scholar
  54. 54.
    Szuladzinski G.: Discrete models of beams on an elastic foundation. J. Eng. Mech. Div. Proc. ASCE 101(EM6), 839–853 (1975)Google Scholar
  55. 55.
    Terzaghi K.: Evaluation of coefficients of subgrade reaction. Geotechnique 5(4), 297–326 (1955)CrossRefGoogle Scholar
  56. 56.
    Timoshenko, S.P.: Method of analysis of statical and dynamical stresses in rail. In: Proceedings of the 2nd International Congress for Applied Mechanics, vol. 54, pp. 1–12. Zurich, Switzerland (1927)Google Scholar
  57. 57.
    Timoshenko S.P.: History of Strength of Materials. McGraw-Hill, New York (1953)Google Scholar
  58. 58.
    Ting B.Y.: Finite beams on elastic foundation with restraints. J. Struct. Div. Proc. ASCE 108(ST3), 611–621 (1982)Google Scholar
  59. 59.
    Tsiatas B.Y.: Nonlinear analysis of non-uniform beams on nonlinear elastic foundation. Acta Mech. 209(1), 141–152 (2010)CrossRefzbMATHGoogle Scholar
  60. 60.
    Usmani R.A.: A uniqueness theorem for a boundary value problem. Proc. Am. Math. Soc. 77(3), 329–335 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Vlasov, V.Z., Leontiev, U.N.: Beams, Plates, and Shells on Elastic Foundations (translated from Russian). Israel Program for Scientific Translations, Jerusalem (1966)Google Scholar
  62. 62.
    Winkler, E.: Die Lehre von der Elastizität und Festigkeit, mit Besonderer Rücksicht auf ihre Anwendung in der Technik, für Polytechnische Schulen, Bauakademien, Ingenieure, Maschinenbauer, Architekten, etc., Verlag H. Dominicus, Prague (1867)Google Scholar
  63. 63.
    Wolfram Research, Inc.: Mathematica, Version 9.0, User Guide. Champaign, IL (2012)Google Scholar
  64. 64.
    Yankelevsky D.Z., Eisenberger M.: Analysis of beam column on elastic foundation. Comput. Struct. 23(3), 351–356 (1986)CrossRefzbMATHGoogle Scholar
  65. 65.
    Yankelevsky D.Z., Eisenberger M., Adin M.A.: Analysis of beams on nonlinear Winkler foundation. Comput. Struct. 31(2), 287–292 (1989)CrossRefGoogle Scholar
  66. 66.
    Zimmermann H.: Die Berechnung des Eisenbahnoberbaues. Wiley Ernst and Sohn, Berlin (1888)Google Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Department of Engineering and Applied SciencesUniversity of BergamoDalmineItaly

Personalised recommendations