Skip to main content
SpringerLink
Log in

Hamiltonian structural analysis of curved beams with or without generalized two-parameter foundation

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The solution of curved Timoshenko beams with or without generalized two-parameter elastic foundation is presented. Beam can be subjected to any kind of loads and imposed external actions, distributed or concentrated along the beam. It can have external and internal restraints and any kind of internal kinematical or mechanical discontinuity. Moreover, the beam may have any spatial curved geometry, by dividing the entire structure into segments of constant curvature and constant elastic properties, each segment resting or not on elastic foundation. The foundation has six parameters like a generalized Winkler soil with the addition of other two parameters involving the link between settlements in transverse direction, as occurs for Pasternak soil model. The solution is obtained by the Hamiltonian structural analysis method, based on an energetic approach, solving a mixed canonical Hamiltonian system of twelve differential equations, leading to the fundamental matrix. The solution, numerically expressed or in closed form, is fast and simple to be implemented on a software, being the solving matrix always of order 12 for any kind of geometry and loads, without increasing in computational complexity. Numerical examples are given, comparing the results of the proposed procedure with the literature data, for rectilinear and curved beams with different soil properties.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arici M., Granata M.F: A general method for nonlinear analysis of bridge structures. Bridge Struct. 1(3), 223–244 (2005)

    Article  Google Scholar 

  2. Arici M., Granata M.F.: Analysis of curved incrementally launched box concrete bridges using Transfer Matrix Method. Bridge Struct. 3(3–4), 165–181 (2007)

    Article  Google Scholar 

  3. Hetenyi M.: Beams on Elastic Foundation. University of Michigan Press, Ann Arbor (1946)

    Google Scholar 

  4. Arici M.: Analogy for beam-foundation elastic systems. J. Struct. Eng. ASCE 111(8), 1691–1702 (1985)

    Article  Google Scholar 

  5. Selvadurai A.P.S.: Elastic Analysis of Soil–Foundation interaction. Elsevier, Amsterdam (1979)

    Google Scholar 

  6. Vlasov, V.Z., Leont’ev, U.N.: Beams, plates and shells on elastic foundations, (Translated from Russian), Israel Program for Scientific Translation, Jerusalem. (1966)

  7. Shirima L.M., Giger M.W.: Timoshenko beam element resting on two-parameter elastic foundation. J. Eng. Mech. 118(2), 280–295 (1992)

    Article  Google Scholar 

  8. Harr M.E., Davidson J.L., Da-Min Ho., Pombo L.E., Ramaswamy S.V., Rosner J.C.: Euler beams on a two parameter foundation model. J. Soil Mech. Found. Div. ASCE 95, 933–948 (1969)

    Google Scholar 

  9. Zhaohua F., Cook R.D.: Beam elements on two-parameter elastic foundations. J. Eng. Mech. 109(6), 1390–1402 (1983)

    Article  Google Scholar 

  10. Chiwanga M., Valsangkar A.J.: Generalized beam element on two-parameter elastic foundation. J. Struct. Eng. ASCE 114(6), 1414–1427 (1988)

    Article  Google Scholar 

  11. Razaqpur A.G., Shah K.R.: Exact analysis of beams on two-parameter elastic foundations. Int. J. Solids Struct. 27(4), 435–454 (1991)

    Article  MATH  Google Scholar 

  12. Onu G.: Shear effect in beam finite element on two-parameter elastic foundation. J. Struct. Eng. ASCE 126(9), 1104–1107 (2000)

    Article  Google Scholar 

  13. Failla G.: Closed-form solutions for Euler–Bernoulli arbitrary discontinuous beams. Arch. Appl. Mech. 81, 605–628 (2011)

    Article  MATH  Google Scholar 

  14. Failla G., Santini A.: A solution method for Euler–Bernoulli vibrating discontinuous beams. Mech. Res. Commun. 35(8), 517–529 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Caddemi S., Caliò I., Cannizzaro F.: Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports. Arch. Appl. Mech. 83, 559–577 (2013)

    Article  Google Scholar 

  16. Colajanni P., Falsone G., Recupero A.: Simplified Formulation of Solution for Beams on Winkler Foundation allowing Discontinuities due to Loads and Constraints. Int. J. Eng. Ed. 25(1), 75–83 (2009)

    Google Scholar 

  17. Yavari A., Sarkani S., Reddy J.N.: Generalized solutions of beams with jump discontinuities on elastic foundations. Arch. Appl. Mech. 71, 625–639 (2001)

    Article  MATH  Google Scholar 

  18. Volterra E.: Bending of a circular beam resting on an elastic foundation. J. Appl. Mech. ASME 19, 1–4 (1952)

    Google Scholar 

  19. Rodriguez D.A.: Three-dimensional bending of a ring on an elastic foundation. J. Appl. Mech. ASME 28, 461–463 (1961)

    Article  MATH  Google Scholar 

  20. Aköz A.Y., Kadioğlu F.: The mixed finite element solution of circular beam on elastic foundation. Comput. Struct. 60(4), 643–651 (1996)

    Article  MATH  Google Scholar 

  21. Banan M.R., Karami G., Farshad M.: Finite elment analysis of curved beams on elastic foundation. Comput. Struct. 32(1), 44–53 (1989)

    Article  Google Scholar 

  22. Dasgupta S., Sengupta D.: Horizontally curved isoparametric beam element with or without elastic foundation including effect of shear deformation. Comput. Struct. 29(6), 967–973 (1988)

    Article  MATH  Google Scholar 

  23. Arici M., Granata M.F.: Generalized curved beam on elastic foundation solved by transfer matrix method. Struct. Eng. Mech. 40(2), 279–295 (2011)

    Article  Google Scholar 

  24. Wright, R.N., Abdel Samad, S.R., Robinson, A.R.: BEF analogy for analysis of box girders. Proc. ASCE, 94, ST-7, 1719–1743 (1968)

  25. Maisel B.I.: Analysis of concrete box beams using small computer capacity. Can. J. Civil Eng. 12, 265–278 (1985)

    Article  Google Scholar 

  26. Kerr A.D.: The continuously supported rail subjected to an axial force and a moving load. Int. J. Mech. Sci. 14(1), 71–78 (1972)

    Article  Google Scholar 

  27. Mallik A.K., Chandra S., Singh A.B.: Steady-state response of an elastically supported infinite beam to a moving load. J. Sound Vib. 291, 1148–1169 (2006)

    Article  Google Scholar 

  28. Szechy K.: The Art of Tunnelling. Akademiai Kiado, Budapest (1973)

    Google Scholar 

  29. Arici M.: Reciprocal conjugate method for space curved bars. J. Struct. Eng. ASCE 115(3), 560–575 (1989)

    Article  Google Scholar 

  30. Haktanir V., Kiral E.: Statical analysis of elastically and continuously supported helicoidal structures by the transfer and stiffness matrix methods. Comput. Struct. 49(4), 663–677 (1993)

    Article  MATH  Google Scholar 

  31. Lanczos, C.: The ariational Principles of Mechanics. Math. Expositions No. 4, pp. 113–124, 161–170. Univ. Toronto Press, Toronto (1964)

  32. Myskis A.D.: Advanced Mathematics for Engineers, pp. 606–623. MIR Publisher, Moscow (1975)

    Google Scholar 

  33. Pipes L.A., Hovanessian S.A.: Matrix-Computer Methods in Engineering, pp. 101–123. Wiley, New York (1969)

    MATH  Google Scholar 

  34. Granata M.F., Margiotta P., Arici M.: A parametric study of curved incrementally launched bridges. Eng. Struct. 49, 373–384 (2013). doi:10.1016/j.engstruct.2012.11.007

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. DICAM, Università di Palermo, Palermo, Italy

    Marcello Arici, Michele Fabio Granata & Piercarlo Margiotta

Authors
  1. Marcello Arici
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michele Fabio Granata
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Piercarlo Margiotta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michele Fabio Granata.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arici, M., Granata, M.F. & Margiotta, P. Hamiltonian structural analysis of curved beams with or without generalized two-parameter foundation. Arch Appl Mech 83, 1695–1714 (2013). https://doi.org/10.1007/s00419-013-0772-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-013-0772-3

Keywords

Search

Navigation