Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 199–227 | Cite as

Nonlinear flexural–torsional dynamic analysis of beams of variable doubly symmetric cross section—application to wind turbine towers

Original Paper
First Online:
Received:
Accepted:

Abstract

In this paper, a boundary element solution is developed for the nonlinear flexural–torsional dynamic analysis of beams of arbitrary doubly symmetric variable cross section, undergoing moderate large displacements, and twisting rotations under general boundary conditions, taking into account the effect of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions and to twisting and/or axial loading. Four boundary-value problems are formulated with respect to the transverse displacements, to the axial displacement, and to the angle of twist and solved using the Analog Equation Method, a Boundary Element Method (BEM) based technique. Application of the boundary element technique yields a system of nonlinear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled with algebraic equations. Numerical examples of great practical interest including wind turbine towers are worked out, while the influence of the nonlinear effects to the response of beams of variable cross section is investigated.

Keywords

Beams of variable cross section Nonlinear analysis Dynamic analysis Flexural–torsional analysis Boundary element method 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

This research has been cofinanced by the European Union (European Social Fund, ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), Research Funding Program THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.

References

  1. 1.
    Mohri, F., Azrar, L., Potier-Ferry, M.: Vibration analysis of buckled thin-walled beams with open sections. J. Sound Vib. 275, 434–446 (2004) CrossRefGoogle Scholar
  2. 2.
    Machado, S.P., Cortinez, V.H.: Free vibration of thin-walled composite beams with static initial stresses and deformations. Eng. Struct. 29, 372–382 (2007) CrossRefGoogle Scholar
  3. 3.
    Lopes Alonso, R., Ribeiro, P.: Flexural and torsional non-linear free vibrations of beams using a p-version finite element. Comput. Struct. 86, 189–1197 (2008) CrossRefGoogle Scholar
  4. 4.
    Sapountzakis, E.J., Dourakopoulos, J.A.: Nonlinear dynamic analysis of Timoshenko beams by BEM. Part I: theory and numerical implementation. Nonlinear Dyn. 58, 295–306 (2009) MATHCrossRefGoogle Scholar
  5. 5.
    Sapountzakis, E.J., Dourakopoulos, J.A.: Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation. Nonlinear Dyn. 58, 307–318 (2009) MATHCrossRefGoogle Scholar
  6. 6.
    Sapountzakis, E.J., Tsipiras, V.J.: Nonlinear nonuniform torsional vibrations of bars by the boundary element method. J. Sound Vib. 329, 1853–1874 (2010) CrossRefGoogle Scholar
  7. 7.
    Sapountzakis, E.J., Dikaros, I.C.: Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM. Int. J. Non-Linear Mech. 46, 782–794 (2011) CrossRefGoogle Scholar
  8. 8.
    Kitipornchai, S., Trahair, N.S.: Elastic behaviour of tapered monosymmetric I-beams. J. Struct. Div. 101(ST8), 1661–1678 (1975) Google Scholar
  9. 9.
    Kameswara Rao, C., Mirza, S.: Free torsional vibrations of tapered cantilever I-beams. J. Sound Vib. 124(3), 489–496 (1988) CrossRefGoogle Scholar
  10. 10.
    Bradford, M.A.: Elastic buckling of tapered monosymmetric I-beams. J. Struct. Eng. 114(5), 977–996 (1988) CrossRefGoogle Scholar
  11. 11.
    Eisenberger, M., Reich, Y.: Static, vibration and stability analysis of non-uniform beams. Comput. Struct. 31(4), 567–573 (1989) MATHCrossRefGoogle Scholar
  12. 12.
    Eisenberger, M.: Exact solution for general variable cross-section members. Comput. Struct. 41(4), 765–772 (1991) MATHCrossRefGoogle Scholar
  13. 13.
    Ronagh, H.R., Bradford, M.A., Attard, M.M.: Nonlinear analysis of thin-walled members of variable cross-section. Part I: theory. Comput. Struct. 77, 285–299 (2000) CrossRefGoogle Scholar
  14. 14.
    Ronagh, H.R., Bradford, M.A., Attard, M.M.: Nonlinear analysis of thin-walled members of variable cross-section. Part II: application. Comput. Struct. 77, 301–313 (2000) CrossRefGoogle Scholar
  15. 15.
    Sapountzakis, E.J., Mokos, V.G.: Nonuniform torsion of composite bars of variable thickness by BEM. Int. J. Solids Struct. 41, 1753–1771 (2004) MATHCrossRefGoogle Scholar
  16. 16.
    Abdel-Jaber, M.S., Al-Qaisia, A.A., Abdel-Jaber, M., Beale, R.G.: Nonlinear natural frequencies of an elastically restrained tapered beam. J. Sound Vib. 313, 772–783 (2008) CrossRefGoogle Scholar
  17. 17.
    Hoseini, S.H., Pirbodaghi, T., Ahmadian, M.T., Farrahi, G.H.: On the large amplitude free vibrations of tapered beams: an analytical approach. Mech. Res. Commun. 36, 892–897 (2009) MATHCrossRefGoogle Scholar
  18. 18.
    Prathap, G., Varadan, T.K.: Non-linear vibrations of tapered cantilevers. J. Sound Vib. 55(1), l–8 (1977) CrossRefGoogle Scholar
  19. 19.
    Prathap, G., Varadan, T.K.: The large amplitude vibration of tapered clamped beams. J. Sound Vib. 58(1), 87–94 (1978) MATHCrossRefGoogle Scholar
  20. 20.
    Nageswara Rao, B., Venkateswara Rao, G.: Large amplitude vibrations of a tapered cantilever beam. J. Sound Vib. 127(1), 173–178 (1988) CrossRefGoogle Scholar
  21. 21.
    Nageswara Rao, B., Venkateswara Rao, G.: Large-amplitude vibrations of free–free tapered beams. J. Sound Vib. 141(3), 511–515 (1990) CrossRefGoogle Scholar
  22. 22.
    Kanaka Raju, K., Shastry, B.P., Venkateswara Rao, G.: A finite element formulation for the large amplitude vibrations of tapered beams. J. Sound Vib. 47(4), 595–598 (1976) CrossRefGoogle Scholar
  23. 23.
    Shavezipur, M., Hashemi, S.M.: Free vibration of triply coupled centrifugally stiffened nonuniform beams, using a refined dynamic finite element method. Aerosp. Sci. Technol. 13, 59–70 (2009) CrossRefGoogle Scholar
  24. 24.
    Liao, M., Zhong, H.: Nonlinear vibration analysis of tapered Timoshenko beams. Chaos Solitons Fractals 36, 1267–1272 (2008) MATHCrossRefGoogle Scholar
  25. 25.
    Verma, M.K., Krishna Murthy, A.V.: Non-linear vibrations of non-uniform beams with concentrated masses. J. Sound Vib. 33(1), 1–12 (1974) MATHCrossRefGoogle Scholar
  26. 26.
    Bazoune, A., Khulief, Y.A., Stephen, N.G., Mohiuddin, M.A.: Dynamic response of spinning tapered Timoshenko beams using modal reduction. Finite Elem. Anal. Des. 37, 199–219 (2001) MATHCrossRefGoogle Scholar
  27. 27.
    Katsikadelis, J.T., Tsiatas, G.C.: Non-linear dynamic analysis of beams with variable stiffness. J. Sound Vib. 270, 847–863 (2004) CrossRefGoogle Scholar
  28. 28.
    Katsikadelis, J.T.: The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies. Theor. Appl. Mech. 27, 13–38 (2002) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Amsterdam (1989) MATHGoogle Scholar
  30. 30.
    Ramm, E., Hofmann, T.J.: In: Mehlhorn, G. (ed.) Stabtragwerke, Der Ingenieurbau (Beam Structures in Civil Engineering), Band Baustatik/Baudynamik. Ernst & Sohn, Berlin (1995) Google Scholar
  31. 31.
    Rothert, H., Gensichen, V.: Nichtlineare Stabstatik (Nonlinear Frame Analysis). Springer, Berlin (1987) CrossRefGoogle Scholar
  32. 32.
    Sapountzakis, E.J., Mokos, V.G.: Warping shear stresses in nonuniform torsion by BEM. Comput. Mech. 30, 131–142 (2003) MATHCrossRefGoogle Scholar
  33. 33.
    Sapountzakis, E.J.: Nonuniform torsion of multi-material composite bars by the boundary element method. Comput. Struct. 79, 2805–2816 (2001) CrossRefGoogle Scholar
  34. 34.
    Katsikadelis, J.T.: Boundary Elements: Theory and Applications. Elsevier, Amsterdam–London (2002) Google Scholar
  35. 35.
    Brigham, E.: Fast Fourier Transform and Its Applications. Prentice Hall, Englewood Cliffs (1988) Google Scholar
  36. 36.
    Lewandowski, R.: Free vibration of structures with cubic non-linearity-remarks on amplitude equation and Rayleigh quotient. Comput. Methods Appl. Mech. Eng. 192, 1681–1709 (2003) MATHCrossRefGoogle Scholar
  37. 37.
    FEMAP for Windows: Finite element modeling and post-processing software. Help System Index. Version 10 (2008) Google Scholar
  38. 38.
    Siemens PLM Software Inc: NX Nastran User’s Guide (2008) Google Scholar
  39. 39.
    Quiligan, A., O’Connor, A., Pakrashi, V.: Fragility analysis of steel and concrete wind turbine towers. Eng. Struct. 36, 270–282 (2012) CrossRefGoogle Scholar
  40. 40.
    Hansen, M.O.L.: Aerodynamics of Wind Turbines. Earthscan, London (2008) Google Scholar
  41. 41.
    Jonkman, J.M.: Dynamics modeling and loads analysis of an offshore floating wind turbine. Technical Report NREL/TP-500-41958 (2007) Google Scholar
  42. 42.
    CEN/TC250: Eurocode 1: actions on structures-general actions—part 1-4: wind actions. prEN 1991-1-4 (2004) Google Scholar
  43. 43.
    Deodatis, G.: Simulation of ergodic multivariate stochastic processes. J. Eng. Mech. 122(8), 778–787 (1996) CrossRefGoogle Scholar
  44. 44.
    Vassilopoulou, I., Gantes, C., Gkimousis, I.: Response of cable networks under wind loading. In: 7 th Greek National Steel Structures Conference, Volos (2011) (in Greek with English summary) Google Scholar
  45. 45.
    Fornberg, B.: Generation of finite difference formulas on arbitrarily spaced grids. Math. Comput. 50(184), 669–706 (1988) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institute of Structural Analysis and Antiseismic Research, School of Civil EngineeringNational Technical University of AthensAthensGreece

Personalised recommendations