Acta Mechanica Sinica

, Volume 34, Issue 4, pp 728–743 | Cite as

Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations

  • Seyed Farhad Hosseini
  • Ali HashemianEmail author
  • Behnam Moetakef-Imani
  • Saied Hadidimoud
Research Paper


In the present paper, the isogeometric analysis (IGA) of free-form planar curved beams is formulated based on the nonlinear Timoshenko beam theory to investigate the large deformation of beams with variable curvature. Based on the isoparametric concept, the shape functions of the field variables (displacement and rotation) in a finite element analysis are considered to be the same as the non-uniform rational basis spline (NURBS) basis functions defining the geometry. The validity of the presented formulation is tested in five case studies covering a wide range of engineering curved structures including from straight and constant curvature to variable curvature beams. The nonlinear deformation results obtained by the presented method are compared to well-established benchmark examples and also compared to the results of linear and nonlinear finite element analyses. As the nonlinear load-deflection behavior of Timoshenko beams is the main topic of this article, the results strongly show the applicability of the IGA method to the large deformation analysis of free-form curved beams. Finally, it is interesting to notice that, until very recently, the large deformations analysis of free-form Timoshenko curved beams has not been considered in IGA by researchers.


Curved beams Nonlinear Timoshenko beam theory Large deformation Isogeometric analysis NURBS curves 


  1. 1.
    Reddy, J.N.: An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, Oxford (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Kim, W., Reddy, J.N.: A comparative study of least-squares and the weak-form Galerkin finite element models for the nonlinear analysis of Timoshenko beams. J. Solid Mech. 2, 101–114 (2010)Google Scholar
  3. 3.
    Reddy, J.N., Singh, I.R.: Large deflections and large-amplitude free vibrations of straight and curved beams. Int. J. Numer. Methods Eng. 17, 829–852 (1981)CrossRefzbMATHGoogle Scholar
  4. 4.
    Pan, K.Q., Liu, J.Y.: Geometric nonlinear dynamic analysis of curved beams using curved beam element. Acta. Mech. Sin. 27, 1023–1033 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Reddy, J.N.: An Introduction to the Finite Element Method, 3rd edn. McGraw-Hill, New York (2004)Google Scholar
  6. 6.
    Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ganapathi, M., Patel, B.P., Saravanan, J., et al.: Shear flexible curved spline beam element for static analysis. Finite Elem. Anal. Des. 32, 181–202 (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hashemian, A., Imani, B.M.: A new quality appearance evaluation technique for automotive bodies including effect of flexible parts tolerances. Mech. Based Des. Struct. Mach. 1, 1–12 (2017)CrossRefGoogle Scholar
  9. 9.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Schmidt, R., Wüchner, R., Bletzinger, K.U.: Isogeometric analysis of trimmed NURBS geometries. Comput. Methods Appl. Mech. Eng. 241–244, 93–111 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Li, X., Zhang, J., Zheng, Y.: NURBS-based isogeometric analysis of beams and plates using high order shear deformation theory. Math. Probl. Eng. 2013, 159027 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Zhang, G., Alberdi, R., Khandelwal, K.: Analysis of three-dimensional curved beams using isogeometric approach. Eng. Struct. 117, 560–574 (2016)CrossRefGoogle Scholar
  13. 13.
    Hassani, B., Taheri, A.H., Moghaddam, N.Z.: An improved isogeometrical analysis approach to functionally graded plane elasticity problems. Appl. Math. Model. 37, 9242–9268 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Moosavi, M.R., Khelil, A.: Isogeometric meshless finite volume method in nonlinear elasticity. Acta Mech. 226, 123–135 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cazzani, A., Malagù, M., Turco, E., et al.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21, 182–209 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kalali, A.T., Hassani, B., Hadidi-Moud, S.: Elastic-plastic analysis of pressure vessels and rotating disks made of functionally graded materials using the isogeometric approach. J. Theor. Appl. Mech. 54, 113–125 (2016)CrossRefGoogle Scholar
  17. 17.
    Tsiptsis, I.N., Sapountzakis, E.J.: Isogeometric analysis for the dynamic problem of curved structures including warping effects. Mech. Based Des. Struct. Mach. 46, 66–84 (2018)CrossRefGoogle Scholar
  18. 18.
    Nguyen, T.N., Ngo, T.D., Nguyen-Xuan, H.: A novel three-variable shear deformation plate formulation: theory and Isogeometric implementation. Comput. Methods Appl. Mech. Eng. 326, 376–401 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nguyen, H.X., Atroshchenko, E., Nguyen-Xuan, H., et al.: Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory. Comput. Struct. 193, 110–127 (2017)CrossRefGoogle Scholar
  20. 20.
    Le-Manh, T., Huynh-Van, Q., Phan, T.D., et al.: Isogeometric nonlinear bending and buckling analysis of variable-thickness composite plate structures. Compos. Struct. 159, 818–826 (2017)CrossRefGoogle Scholar
  21. 21.
    Bazilevs, Y., Calo, V.M., Hughes, T.J.R., et al.: Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput. Mech. 43, 3–37 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yoon, M., Ha, S.H., Cho, S.: Isogeometric shape design optimization of heat conduction problems. Int. J. Heat Mass Transf. 62, 272–285 (2013)CrossRefGoogle Scholar
  23. 23.
    Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Comput. Methods Appl. Mech. Eng. 272, 290–320 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bouclier, R., Elguedj, T., Combescure, A.: Locking free isogeometric formulations of curved thick beams. Comput. Methods Appl. Mech. Eng. 245–246, 144–162 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane-curved beams. Math. Mech. Solids 21, 562–577 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hosseini, S.F., Moetakef-Imani, B., Hadidi-Moud, S., et al.: The effect of parameterization on isogeometric analysis of free-form curved beams. Acta Mech. 227, 1983–1998 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Luu, A.T., Kim, N.I., Lee, J.: Isogeometric vibration analysis of free-form Timoshenko curved beams. Meccanica 50, 169–187 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Bauer, A.M., Breitenberger, M., Philipp, B., et al.: Nonlinear isogeometric spatial Bernoulli beam. Comput. Methods Appl. Mech. Eng. 303, 101–127 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kitarovic, S.: Nonlinear Euler–Bernoulli beam kinematics in progressive collapse analysis based on the Smith’s approach. Mar. Struct. 39, 118–130 (2014)CrossRefGoogle Scholar
  30. 30.
    Babu, C.R., Prathap, G.: A linear thick curved beam element. Int. J. Numer. Methods Eng. 23, 1313–1328 (1986)CrossRefzbMATHGoogle Scholar
  31. 31.
    Day, R.A., Potts, D.M.: Curved Mindlin beam and axi-symmetric shell elements—a new approach. Int. J. Numer. Methods Eng. 30, 1263–1274 (1990)CrossRefzbMATHGoogle Scholar
  32. 32.
    Raveendranath, P., Singh, G., Venkateswara Rao, G.: A three-noded shear-flexible curved beam element based on coupled displacement field interpolations. Int. J. Numer. Methods Eng. 51, 85–101 (2001)CrossRefzbMATHGoogle Scholar
  33. 33.
    Imani, B.M., Hashemian, S.A.: NURBS-based profile reconstruction using constrained fitting techniques. J. Mech. 28, 407–412 (2012)CrossRefGoogle Scholar
  34. 34.
    Randrianarivony, M., Brunnett, G.: Approximation by NURBS curves with free knots. In: Proceedings of Vision, Modeling, and Visualization (VMV 2002), pp. 195–201, Erlangen, Germany (2002)Google Scholar
  35. 35.
    Adam, C., Hughes, T.J.R., Bouabdallah, S., et al.: Selective and reduced numerical integrations for NURBS-based isogeometric analysis. Comput. Methods Appl. Mech. Eng. 284, 732–761 (2015)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Lee, S.J., Park, K.S.: Vibrations of Timoshenko beams with isogeometric approach. Appl. Math. Model. 37, 9174–9190 (2013)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Bathe, K.J., Bolourchi, S.: Large displacement analysis of three-dimensional beam structures. Int. J. Numer. Methods Eng. 14, 961–986 (1979)CrossRefzbMATHGoogle Scholar
  38. 38.
    Mallett, R.H., Berke, L.: Automated method for the large deflection and instability analysis of three-dimensional truss and frame assemblies, Technical report: Air Force Flight Dynamics Laboratory, AFFDL 66-102 (1966)Google Scholar
  39. 39.
    Dupuis, G.A., Hibbitt, H.D., McNamara, S.F., et al.: Nonlinear material and geometric behavior of shell structures. Comput. Struct. 1, 223–239 (1971)CrossRefGoogle Scholar
  40. 40.
    Lo, S.H.: Geometrically nonlinear formulation of 3D finite strain beam element with large rotations. Comput. Struct. 44, 147–157 (1992)CrossRefzbMATHGoogle Scholar
  41. 41.
    Hashemian, A., Hosseini, S.F., Nabavi, S.N.: Kinematically smoothing trajectories by NURBS reparameterization—an innovative approach. Adv. Robot. 31, 1296–1312 (2017)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sun-Air Research InstituteFerdowsi University of MashhadMashhadIran
  2. 2.Department of Mechanical EngineeringFerdowsi University of MashhadMashhadIran
  3. 3.Department of Mechanical EngineeringHakim Sabzevari UniversitySabzevarIran
  4. 4.Centre for Manufacturing and Materials EngineeringCoventry UniversityCoventryUK

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