Free vibration of functionally graded beams resting on Winkler-Pasternak foundation

  • Mehmet Avcar
  • Waleed Khalid Mohammed Mohammed
Part of the following topical collections:
  1. Geo-Resources-Earth-Environmental Sciences


In the present study, the free vibration of functionally graded beams resting on two parameter elastic foundation was examined. The properties of the functionally graded materials were presumed to vary continuously along the thickness direction. The foundation medium was assumed to be linear, homogeneous, and isotropic, and it was modeled by the Winkler-Pasternak model with two parameters for describing the reaction of the elastic foundation on the beam. The functionally graded beam was modeled with classical beam theory. The governing equation including the effects of functionally graded material properties, Winkler-Pasternak elastic foundation was solved using separation of variables. The eigenvalues of yielding fundamental equation versus clamped-clamped, clamped-free, clamped-simply supported, and simply supported-simply supported boundary conditions were found. To corroborate the results, comparisons were carried out with available results for homogeneous and functionally graded beams. The effects of Winkler-Pasternak type elastic foundation and functionally graded material properties on the values of dimensionless frequency parameter of beams were discussed. Briefly, it was found that the dimensionless frequency parameters of beam change according to material properties, presence of elastic foundation, and boundary conditions; moreover, the separate effects of these quantities on each other are interesting.


Winkler-Pasternak foundation Functionally graded materials Beam Free vibration 


Funding information

The financial support of the Suleyman Demirel University Scientific Research Projects Unit (SDU-BAP) with Grant No. 4912-YL1-17 is gratefully acknowledged. The authors would like to thank institution.


  1. Akgöz B, Civalek O (2016) Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronaut 119:1–12CrossRefGoogle Scholar
  2. Al Rjoub YS, Hamad AG (2017) Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method. KSCE J Civ Eng 21:792–806CrossRefGoogle Scholar
  3. Atmane HA, Tounsi A, Bernard F (2017) Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations. Int J Mech Mater Des 13(1):71–84CrossRefGoogle Scholar
  4. Avcar M (2014) Free vibration analysis of beams considering different geometric characteristics and boundary conditions. Int J Mech App 4:94–100Google Scholar
  5. Avcar M (2015) Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam. Struct Eng Mech 55:871–884CrossRefGoogle Scholar
  6. Avcar M (2016) Effects of material non-homogeneity and two parameter elastic foundation on fundamental frequency parameters of Timoshenko beams. Acta Phys Pol A 130:375–378CrossRefGoogle Scholar
  7. Avcar M, Mohammed WKM (2017) Examination of the effects of Winkler foundation and functionally graded material properties on the frequency parameters of beam. J Eng Sci Des 5:573–580 (in Turkish)Google Scholar
  8. Avcar M, Saplioglu K (2015) An artificial neural network application for estimation of natural frequencies of beams. Int J Adv Comput Sci Appl 6:94–102Google Scholar
  9. Chakraverty S, Pradhan KK (2016) Vibration of functionally graded beams and plates, ​1st edn. Academic Press, Elsevier, OxfordGoogle Scholar
  10. Chen WJ, Li XP (2013) Size-dependent free vibration analysis of composite laminated Timoshenko beam based on new modified couple stress theory. Arch Appl Mech 83:431–444CrossRefGoogle Scholar
  11. Civalek Ö, Demir Ç (2016) A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 289:335–352Google Scholar
  12. Civalek Ö, Öztürk B (2010) Free vibration analysis of tapered beam-column with pinned ends embedded in Winkler-Pasternak elastic foundation. Geomech Eng 2:45–56CrossRefGoogle Scholar
  13. Coşkun SB, Öztürk B, Atay MT (2011) Transverse vibration analysis of Euler-Bernoulli beams using analytical approximate techniques. Adv Vib Anal Res, chapter 1. InTech, Vienna, pp 1–22Google Scholar
  14. De Silva CW (2000) Vibration: fundamentals and practice. CRC Press LLC, Baco RatonGoogle Scholar
  15. Duy HT, Van TN, Noh HC (2014) Eigen analysis of functionally graded beams with variable cross-section resting on elastic supports and elastic foundation. Struct Eng Mech 52:1033–1049CrossRefGoogle Scholar
  16. Eisenberger M (1994) Vibration frequencies for beams on variable one- and two-parameter elastic foundations. J Sound Vib 176:577–584CrossRefGoogle Scholar
  17. Hetenyi M (1946) Beams on elastic foundations. The University of Michigan Press, Ann ArborGoogle Scholar
  18. Kerr AD (1964) Elastic and viscoelastic foundation models. ASME J Appl Mech 31491–498Google Scholar
  19. Kieback B, Neubrand A, Riedel H (2003) Processing technique for functionally graded materials. Mater Sci Eng 362(1–2):81–106CrossRefGoogle Scholar
  20. Koizumi M (1997) FGM activities in Japan. Compos B Eng 28:1–4CrossRefGoogle Scholar
  21. Li XF (2008) A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–Bernoulli beams. J Sound Vibr 318:1210–1229CrossRefGoogle Scholar
  22. Matsunaga H (1999) Vibration and buckling of deep beam-columns on two parameter elastic foundations. J Sound Vibr 228:359–376CrossRefGoogle Scholar
  23. Mohammed WKM (2017) Free vibration of functionally graded beam resting on Pasternak foundation, Suleyman Demirel University, Graduate School of Natural and Applied Sciences. Department of Civil Engineering, Master of Science Thesis, 79p(in Turkish)Google Scholar
  24. Morfidis K, Avramidis IE (2002) Formulation of a generalized beam element on a two-parameter elastic foundation with semi-rigid connections and rigid offsets. Comput Struct 80:1919–1934 CrossRefGoogle Scholar
  25. Obara P (2014) Vibrations and stability of Bernoulli-Euler and Timoshenko beams on two-parameter elastic foundation. Arch Civ Eng 60:421–440Google Scholar
  26. Pasternak PL (1954) On a new method of analysis of an elastic foundation by means of two foundation constants. Gosudarstvennoe Izdatelstro Liberaturi po Stroitelstvui Arkhitekture(in Russian)Google Scholar
  27. Pradhan KK, Chakraverty S (2013) Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos B Eng 51:175–184CrossRefGoogle Scholar
  28. Rahbar-Ranji A, Shahbaztabar A (2017) Free vibration analysis of beams on a Pasternak foundation using legendre polynomials and Rayleight-Ritz method. Odes’kyi Politechnichnyi Universytet Pratsi 3(53):20–31CrossRefGoogle Scholar
  29. Rao SS (2007) Vibration of continuous systems. Wiley, New YorkGoogle Scholar
  30. Sahraee S, Saidi AR (2009) Free vibration and buckling analysis of functionally graded deep beam-columns on two-parameter elastic foundations using the differential quadrature method. Proc IME C J Mech Eng Sci 223:1273–1284CrossRefGoogle Scholar
  31. Sedighi HM, Shirazi KH (2014) Accurate investigation of lateral vibrations of a quintic nonlinear beam on an elastic foundation: using an exact formulation of the beam curvature. J Appl Mech Tech Phys 55:1066–1074CrossRefGoogle Scholar
  32. Selvadurai APS (1979) Elastic analysis of soil-foundation interaction. Dev Geotech Eng 17:7–9Google Scholar
  33. Sharafi P, Hadi MN, Teh LH (2012) Geometric design optimization for dynamic response problems of continuous reinforced concrete beams. J Comput Civ Eng 28:202–209CrossRefGoogle Scholar
  34. Sharafi P, Samali B, Mortazavi M, Ronagh H (2018) Interlocking system for enhancing the integrity of multi-story modular buildings. J Autom Constr 85:263–272CrossRefGoogle Scholar
  35. Shen HS (2009) Functionally graded materials: nonlinear analysis of plates and shells. CRC Press, Boca RatonCrossRefGoogle Scholar
  36. Sina SA, Navazi HM, Haddadpour H (2009) An analytical method for free vibration analysis of functionally graded beams. Mater Des 30:741–747CrossRefGoogle Scholar
  37. Wattanasakulpong N, Ungbhakorn V (2012) Free vibration analysis of functionally graded beams with general elastically end constraints by DTM. World J Mech 2:297–310CrossRefGoogle Scholar
  38. Wei-Ren C, Heng C (2018) Vibration analysis of functionally graded Timoshenko beams. Int J Struct Stab Dyn 18(01):1850007. CrossRefGoogle Scholar
  39. WinklerE(1867). Die Lehre von der Elasticitaet und Festigkeit, Prag. Dominicus 182 pGoogle Scholar
  40. Ying J, Lü CF, Chen WQ (2008) Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Compos Struct 84:209–219CrossRefGoogle Scholar
  41. Zahedinejad P (2016) Free vibration analysis of functionally graded beams resting on elastic foundation in thermal environment. Int J Struct Stabil Dynam 16:1550029CrossRefGoogle Scholar
  42. Zhong H, Li X, He Y (2005) Static flexural analysis of elliptic Reissner-Mindlin plates on a Pasternak foundation by the triangular differential quadrature method. Arch Appl Mech 74:679–691CrossRefGoogle Scholar

Copyright information

© Saudi Society for Geosciences 2018

Authors and Affiliations

  1. 1.Department of Civil Engineering, Faculty of EngineeringSuleyman Demirel UniversityIspartaTurkey
  2. 2.Graduate School of Natural and Applied SciencesSuleyman Demirel UniversityIspartaTurkey

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