A new 2D beam finite element for nonlinear elastic analysis including warping and shear effects

  • João Paulo PasconEmail author
Technical Paper


In this paper, a finite-element formulation based on positional description is proposed to predict the large deformation behavior of short beams including shear deformation and cross section warping. A higher-order shear deformation beam theory is originally employed by means of two generalized vectors. A linear strain rate is also included in order to enrich the approximation of strains along the transverse direction. The formulation accounts for finite displacements, finite strains and plane stress conditions in elastic regime. Homogeneous materials, as well as functionally graded composite beams with varying properties along the height, are considered. In all, four beam formulations are described, combining the level of transverse enrichment and the cross section kinematics. A short cantilever beam problem with rectangular cross section is analyzed in detail to illustrate the differences among the four element formulations regarding displacements, strains and stresses. The numerical results confirm that the inclusion of both linear strain rate and cross section warping are needed to correctly predict the mechanical behavior of the cantilever beam, reproducing variable shear deformation across the height.


2D beam finite element Nonlinear elastic bending Cross section warping Shear deformation Functionally graded material 



The author appreciates all the essential support given by the following two departments, both from the University of São Paulo (USP): the Materials Engineering Department of the Lorena School of Engineering (DEMAR/EEL) for providing the necessary infrastructure to carry out the work and from the Structural Engineering Department of the São Carlos School of Engineering (SET/EESC) for allowing the remote access to their cluster.


  1. 1.
    Karamanli A (2018) Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory. Compos Struct 189:127–136CrossRefGoogle Scholar
  2. 2.
    Levinson M (1981) A new rectangular beam theory. J Sound Vib 74:81–87CrossRefGoogle Scholar
  3. 3.
    Reddy JN (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51:745–752CrossRefGoogle Scholar
  4. 4.
    Dash P, Singh BN (2012) Buckling and post-buckling of laminated composite plates. Mech Res Commun 46:1–7CrossRefGoogle Scholar
  5. 5.
    Shokrieh MM, Parkestani AN (2017) Post buckling analysis of shallow composite shells based on the third order shear deformation theory. Aerosp Sci Technol 66:332–341CrossRefGoogle Scholar
  6. 6.
    Emam SA (2011) Analysis of shear-deformable composite beams in postbuckling. Compos Struct 94:24–30CrossRefGoogle Scholar
  7. 7.
    She GL, Yuan FG, Ren YR (2017) Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory. Appl Math Model 47:340–357MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen Y, Jin G, Zhang C, Ye T, Xue Y (2018) Thermal vibration of FGM beams with general boundary conditions using a higher-order shear deformation theory. Compos B 153:376–386CrossRefGoogle Scholar
  9. 9.
    Benatta MA, Mechab I, Tounsi A, Bedia EAA (2008) Static analysis of functionally graded short beams including warping and shear deformation effects. Comput Mater Sci 44:765–773CrossRefGoogle Scholar
  10. 10.
    Nguyen-Xuan H, Tran LV, Thai CH, Kulasegaram S, Bordas SPA (2014) Isogeometric analysis of functionally graded plates using a refined plate theory. Compos Part B 64:222–234CrossRefGoogle Scholar
  11. 11.
    Canales FG, Mantari JL (2016) Buckling and free vibration of laminated beams with arbitrary boundary conditions using a refined HSDT. Compos B 100:136–145CrossRefGoogle Scholar
  12. 12.
    Kulkarni SA, Bajoria KM (2007) Large deformation analysis of piezolaminated smart structures using higher-order shear deformation theory. Smart Mater Struct 16:1506–1516CrossRefGoogle Scholar
  13. 13.
    Semnani AMD, Mostafaei H, Bahrami MN (2016) Free flexural vibration of geometrically imperfect functionally graded microbeams. Int J Eng Sci 105:56–79MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ghayesh MH (2018) Dynamics of functionally graded viscoelastic microbeams. Int J Eng Sci 124:115131MathSciNetCrossRefGoogle Scholar
  15. 15.
    Taati E (2018) On buckling and post-buckling behavior of functionally graded micro-beams in thermal environment. Int J Eng Sci 128:63–78MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ghayesh MH (2019) Viscoelastic dynamics of axially FG microbeams. Int J Eng Sci 135:75–85MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ghayesh MH, Farokhi H, Gholipour A, Tavallaeinejad M (2018) Nonlinear oscillations of functionally graded microplates. Int J Eng Sci 122:56–72CrossRefGoogle Scholar
  18. 18.
    She GL, Ren YR, Yan KM (2019) On snap-buckling of porous FG curved nanobeams. Acta Astronaut. CrossRefGoogle Scholar
  19. 19.
    Faleh NM, Ahmed RA, Fenjan RM (2018) On vibrations of porous FG nanoshells. Int J Eng Sci 133:1–14MathSciNetCrossRefGoogle Scholar
  20. 20.
    She GL, Yuan FG, Ren YR, Liu HB, Xiao WS (2018) Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos Struct 203:614–623CrossRefGoogle Scholar
  21. 21.
    She GL, Yuan FG, Karami B, Ren YR, Xiao WS (2019) On nonlinear bending behavior of FG porous curved nanotubes. Int J Eng Sci 135:58–74MathSciNetCrossRefGoogle Scholar
  22. 22.
    Srividhya S, Raghu P, Rajagopal A, Reddy JN (2018) Nonlocal nonlinear analysis of functionally graded plates using third-order shear deformation theory. Int J Eng Sci 125:1–22MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pascon JP (2016) Finite element analysis of flexible functionally graded beams with variable Poisson’s ratio. Eng Comput 33:2421–2447CrossRefGoogle Scholar
  24. 24.
    Coda HB, Paccola RR (2008) A positional FEM Formulation for geometrical non-linear analysis of shells. Latin Am J Solids Struct 5:205–223Google Scholar
  25. 25.
    Coda HB, Greco M (2004) A simple FEM formulation for large deflection 2D frame analysis based on position description. Comput Methods Appl Mech Eng 193:3541–3557CrossRefGoogle Scholar
  26. 26.
    Holzapfel GA (2004) Nonlinear Solid mechanics—a continuum approach for engineering. Wiley, ChichesterGoogle Scholar
  27. 27.
    Chi SH, Chung YL (2006) Mechanical behavior of functionally graded material plates under transverse load—part I: analysis. Int J Solids Struct 43:3657–3674CrossRefGoogle Scholar
  28. 28.
    Banic D, Bacciocchi M, Tornabene F, Ferreira AJM (2017) Influence of Winkler–Pasternak foundation on the vibrational behavior of plates and shells reinforced by agglomerated carbon nanotubes. Appl Sci 7:1–55CrossRefGoogle Scholar
  29. 29.
    Pai PF, Palazotto AN (1996) Large-deflection analysis of flexible beams. Int J Solids Struct 33:1335–1353CrossRefGoogle Scholar
  30. 30.
    Khabbaz RS, Manshadi BD, Abedian A (2009) Nonlinear analysis of FGM plates under pressure loads using the higher-order shear deformation theories. Compos Struct 89:333–344CrossRefGoogle Scholar
  31. 31.
    Arciniega RA, Reddy JN (2007) Large deformation analysis of functionally graded shells. Int J Solids Struct 44:2036–2052CrossRefGoogle Scholar
  32. 32.
    Kadoli R, Akhtar K, Ganesan N (2008) Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model 32:2509–2525CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Materials Engineering Department, Lorena School of EngineeringUniversity of São PauloLorenaBrazil

Personalised recommendations