Skip to main content
SpringerLink
Log in

Stability of Timoshenko beams with frequency and initial stress dependent nonlocal parameters

  • Original Research Article
  • Published:
Archives of Civil and Mechanical Engineering Aims and scope Submit manuscript

Abstract

This paper presents an analysis of the stability of Timoshenko beams which uses Eringen’s nonlocal elasticity theory. A numerical algorithm based on the exact solution for the free vibration of segmental Timoshenko beams was formulated. The algorithm enables one to calculate, with any degree of accuracy, the critical load levels in the beams on the macro and nanoscale. The beams were subjected to conservative and nonconservative static loads. The levels of critical loads in the beams were analysed assuming a functional dependence of the nonlocal parameters on the vibrational frequency and the state of stress.

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. S.P. Timoshenko, On the transverse vibrations of bars of uniform cross-section, London, Edinburgh, and Dublin, Philos. Mag. J. Sci. 43 (253) (1922) 125–131.

    Google Scholar 

  2. E.T. Kruszewski, National Advisory Committee for Aeronautics, 1909. Effects of Transverse Shear and Rotary Inertia on the Natural Frequencies of a Uniform Beam, 1949.

    Google Scholar 

  3. T. Huang, The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, J. Appl. Mech. 28 (4) (1961) 579–584.

    MathSciNet  MATH  Google Scholar 

  4. G. Cowper, The shear coefficient in Timoshenko’s beam theory, J. Appl. Mech. 33 (2) (1966) 335–340.

    MATH  Google Scholar 

  5. H. Saito, K. Otomi, Vibration and stability of elastically supported beams carrying an attached mass under axial and tangential loads, J. Sound Vib. 62 (2) (1979) 257–266.

    Google Scholar 

  6. A. Kounadis, J.T. Katsikadelis, Shear and rotatory inertia effect on Beck’s column, J. Sound Vib. 49 (2) (1976) 171–178.

    Google Scholar 

  7. T. Irie, G. Yamada, I. Takahashi, Vibration and stability of non-uniform Timoshenko beam subjected to a follower force, J. Sound Vib. 70 (4) (1980) 503–512.

    MATH  Google Scholar 

  8. V. Sundaramaiah, G. Venkateswara Rao, Stability of short Beck and Leipholz columns on elastic foundation, AIAA J. 21 (7) (1983) 1053–1054.

    Google Scholar 

  9. K. Sato, On the governing equations for vibration and stability of a Timoshenko beam: Hamilton’s principle, J. Sound Vib. 145 (2) (1991) 338–340.

    MathSciNet  Google Scholar 

  10. P. Ruta, The application of Chebyshev polynomials to the solution of the nonprismatic Timoshenko beam vibration problem, J. Sound Vib. 296 (2006) 243–263.

    MATH  Google Scholar 

  11. A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci. 10 (1972) 233–248. https://doi.org/10.1016/0020-7225(72)90039-0.

    MathSciNet  MATH  Google Scholar 

  12. A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves, Int. J. Eng. Sci. 10 (1972) 425–435. https://doi.org/10.1016/0020-7225(72)90050-X.

    MATH  Google Scholar 

  13. A.C. Eringen, On nonlocal fluid mechanics, Int. J. Eng. Sci. 10 (1972) 561–575., https://doi.org/10.1016/0020-7225(72)90098-5.

    MATH  Google Scholar 

  14. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703–4710. https://doi.org/10.1063/1.332803.

    Google Scholar 

  15. A.C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002.

    MATH  Google Scholar 

  16. R.D. Mindlin, Microstructure in linear elasticity, Arch. Ration. Mech. Anal. 10 (1964) 51–57.

    MATH  Google Scholar 

  17. R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Int. J. Solid Struct. 1 (1965) 417–438. https://doi.org/10.1016/0020-7683(65)90006-5.

    Google Scholar 

  18. B.S. Altan, E.C. Aifantis, On some aspects in the special theory of gradient elasticity, J. Mech. Behav. Mater. 8 (1997) 231–282.

    Google Scholar 

  19. N.A. Fleck, J.W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solid. 49 (2001) 2245–2271. https://doi.org/10.1016/S0022-5096(01)00049-7.

    MATH  Google Scholar 

  20. L. Li, Y.L. Hu, Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, Int. J. Eng. Sci. 107 (2016) 77–97.

    MathSciNet  MATH  Google Scholar 

  21. R.D. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal. 11 (1962) 415–448.

    MathSciNet  MATH  Google Scholar 

  22. R. Toupin, Elastic materials with couple-stresses, Arch. Ration. Mech. Anal. 11 (1962) 385–414.

    MathSciNet  MATH  Google Scholar 

  23. A.R. Hadjesfandiari, G.F. Dargush, Couple stress theory for solids, Int. J. Solids Struct. 48 (2010) 2496–2510.

    Google Scholar 

  24. H.M. Ma, X.L. Gao, J.N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, J. Mech. Phys. Solids 56 (2008) 3379–3391.

    MathSciNet  MATH  Google Scholar 

  25. M. Asghari, M.H. Kahrobaiyan, M.T. Ahmadian, A nonlinear Timoshenko beam formulation based on the modified couple stress theory, Int. J. Eng. Sci. 48 (2010) 1749–1761.

    MathSciNet  MATH  Google Scholar 

  26. A.C. Eringen, Microcontinuum Field Theories: Foundations and Solids, Springer, 1999.

    MATH  Google Scholar 

  27. C.M. Wang, Y.Y. Zhang, X.Q. He, Vibration of nonlocal Timoshenko beams, Nanotechnology 18 (2007) 105401.

    Google Scholar 

  28. M.A. Aydogdu, General nonlocal beam theory: its application to nanobeam bending buckling and vibration, Physica E 41 (2009) 1651–1655.

    Google Scholar 

  29. H.T. Thai, A nonlocal beam theory for bending, buckling, and vibration of nanobeams, Int. J. Eng. Sci. 52 (2012) 56–64. https://doi.org/10.1016/J.IJENGSCI.2011.11.011.

    MathSciNet  MATH  Google Scholar 

  30. Q. Wang, K.M. Liew, Application of nonlocal continuum mechanics to static analysis of micro and nanostructures, Phys. Lett. A363 (2007) 236–242.

    Google Scholar 

  31. C.M. Wang, S. Kitipornchai, C.W. Lim, M. Eisenberg, Beam bending solutions based on non-local Timoshenko beam theory, J. Eng. Mech. 134 (2008) 475–481.

    Google Scholar 

  32. S.P. Xu, An operational calculus-based approach to a general bending theory of nonlocal elastic beams, Eur. J. Mech. A/ Solids 46 (2014) 54–59.

    MathSciNet  MATH  Google Scholar 

  33. K. Torabi, A. Jafarzadeh Jazi, E. Zafari, Exact closed form solution for the analysis of the transverse vibration modes of a Timoshenko beam with multiple concentrated masses, Appl. Math. Comput. 238 (2014) 342–357.

    MathSciNet  MATH  Google Scholar 

  34. V. Senthilkumar, S.C. Pradhan, G. Prathap, Buckling analysis of carbon nanotube based on nonlocal Timoshenko beam theory using differential transform method, Adv. Sci. Lett. 3 (2010) 415–421.

    Google Scholar 

  35. F. Ebrahimi, P. Nasirzadeh, A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method, J. Theor. Appl. Mech. 53 (2015) 1041–1052.

    Google Scholar 

  36. M. Janghorban, A. Zare, Freevibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method, Physica E 43 (2011) 1602–1604.

    Google Scholar 

  37. T. Murmu, S.C. Pradhan, Buckling analysis of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity and Timoshenko beam theory and using DQM, Physica E 41 (2009) 1232–1239.

    Google Scholar 

  38. S.A.M. Ghannadpour, B. Mohammadi, Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory using Chebyshev polynomials, Adv. Mater. Res. 123–125 (2010) 619–622.

    Google Scholar 

  39. B. Mohammadi, S.A.M. Ghannadpour, Energy approach vibration analysis of nonlocal Timoshenko beam theory, Proc. Eng. 10 (2011) 1766–1771.

    Google Scholar 

  40. L. Behera, S. Chakraverty, Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials, Appl. Nanosci. 4 (2014) 347–358.

    Google Scholar 

  41. C.M.C. Roque, J.M. Ferreira, J.N. Reddy, Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method, Int. J. Eng. Sci. 49 (2011) 976–984.

    MATH  Google Scholar 

  42. M. Hemmatnezhad, R. Ansari, Finite element formulation for the free vibration analysis of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory, J. Theor. Appl. Phys. 7 (2013) 6.

    Google Scholar 

  43. P. Kasirajan, R. Amirtham, J.N. Reddy, Surface and non-local effects for non-linear analysis of Timoshenko beams, Int. J. Non-Linear Mech. 76 (2015) 100–111.

    Google Scholar 

  44. M. Alves, P. Ribeiro, Non-linear modes of vibration of Timoshenko nanobeams under electrostatic actuation, Int. J. Mech. Sci. 130 (2017) 188–202.

    Google Scholar 

  45. J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci. 45 (2007) 288–307.

    MATH  Google Scholar 

  46. P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, Application of nonlocal beam models for carbon nanotubes, Int. J. Solids Struct. 44 (2007) 5289–5300.

    MATH  Google Scholar 

  47. H. Heireche, A. Tounsi, A. Benzair, M. Maachou, E.A. Adda Bedia, Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity, Physica E 40 (8) (2008) 2791–2799.

    Google Scholar 

  48. C.M. Wang, Y.Y. Zhang, S. Sai Sudha Ramesh, Kitipornchai, Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory, J. Phys. D: Appl. Phys. 39 (2006) 3904–3909.

    Google Scholar 

  49. R. Ansari, M. Faghih Shojaei, V. Mohammadi, R. Gholami, H. Rouhi, Buckling and postbuckling of single-walled carbon nanotubes based on a nonlocal Timoshenko beam model, ZAMM J. Appl. Math. Mech./Z. Angew. Math. Mech. 95 (2015) 939–951. https://doi.org/10.1002/zamm.201300017.

    MathSciNet  MATH  Google Scholar 

  50. G.F. Wang, X.Q. Feng, Timoshenko beam model for buckling and vibration of nanowires with surface effects, J. Phys. D: Appl. Phys. 42 (2009) 155411.

    Google Scholar 

  51. S. Sahmani, R. Ansari, Nonlocal beam models for buckling of nanobeams using state-space method regarding different boundary conditions, J. Mech. Sci. Technol. 25 (2011) 2365–2375.

    Google Scholar 

  52. R. Ansari, V. Mohammadi, M. Faghih Shojaei, R. Gholami, S. Sahmani, Postbuckling analysis of Timoshenko nanobeams including surface stress effect, Int. J. Eng. Sci. 75 (2014) 1–10.

    Google Scholar 

  53. J.D. Aristizabal-Ochoa, Large deflection and postbuckling behavior of Timoshenko beam-columns with semi-rigid connections including shear and axial effects, Eng. Struct. 29 (2007) 991–1003.

    Google Scholar 

  54. Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, J. Appl. Phys. 98 (2005) 124301.

    Google Scholar 

  55. J. Yoon, C.Q. Ru, A. Mioduchwski, Timoshenko-beam effects on transverse wave propagation in carbon nanotubes, Composites B 35 (2004) 87–93.

    Google Scholar 

  56. N. Challamel, J. Lerbet, C.M. Wang, Z. Zhang, Analytical length scale calibration of nonlocal continuum from a microstructured buckling model, Z. Angew. Math. Mech. 92 (5) (2013) 402–413. https://doi.org/10.1002/zamm.201200130.

    MathSciNet  MATH  Google Scholar 

  57. N. Challamel, Z. Zhang, C.M. Wang, Nonlocal equivalent continuum for the buckling and the vibrations of microstructured beams, J. Nanomech. Micromech. 5 (1) (2015) A4014004, https://doi.org/10.1061/(ASCE)NM.2153-5477.0000062.

    Google Scholar 

  58. W.H. Duan, C.M. Wang, Y.Y. Zhang, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, J. Appl. Phys. 101 (2007) 024305.

    Google Scholar 

  59. CM. Wang, Z. Zhang, N. Challamel, W.H. Duan, Calibration of Eringen’s small length scale coefficient for initially stressed vibrating nonlocal Euler Beams based on microstructured beam model, J. Phys. D: Appl. Phys. 46 (2013) 345501.

    Google Scholar 

  60. L.J. Sudak, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. Appl. Phys. 94 (2003) 7281.

    Google Scholar 

  61. W. Glabisz, Stability of non-prismatic rods subjected to non-conservative loads, Comput. Struct. 46 (1993) 479–486.

    MATH  Google Scholar 

  62. S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1963.

    Google Scholar 

  63. H.H.E. Leipholz, Stability Theory: An Introduction to the Stability Problems of Elastic Systems and Rigid Bodies, John Wiley and Sons, Stuttgard, 1987.

    MATH  Google Scholar 

  64. V.V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, Oxford, 1963.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wroclaw University of Science and Technology, Faculty of Civil Engineering, Wybrzeze Wyspianskiego 27, Wroclaw, 50-370, Poland

    W. Glabisz, K. Jarczewska & R. Holubowski

Authors
  1. W. Glabisz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Jarczewska
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Holubowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Jarczewska.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Glabisz, W., Jarczewska, K. & Holubowski, R. Stability of Timoshenko beams with frequency and initial stress dependent nonlocal parameters. Archiv.Civ.Mech.Eng 19, 1116–1126 (2019). https://doi.org/10.1016/j.acme.2019.06.003

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1016/j.acme.2019.06.003

Keywords

Search

Navigation