Skip to main content
SpringerLink
Log in

Refined beam elements with only displacement variables and plate/shell capabilities

  • Published:
Meccanica Aims and scope Submit manuscript

An Erratum to this article was published on 01 February 2012

Abstract

This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases.

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. Bathe K (1996) Finite element procedure. Prentice Hall, New York

    Google Scholar 

  2. Carrera E (2002) Theories and finite elements for multilayered, anisotropic, composite plates and shells. Arch Comput Methods Eng 9(2):87–140

    Article  MathSciNet  MATH  Google Scholar 

  3. Carrera E (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng 10(3):216–296

    Article  MathSciNet  Google Scholar 

  4. Carrera E, Brischetto S (2008) Analysis of thickness locking in classical, refined and mixed multilayered plate theories. Compos Struct 82(4):549–562

    Article  Google Scholar 

  5. Carrera E, Brischetto S (2008) Analysis of thickness locking in classical, refined and mixed theories for layered shells. Compos Struct 85(1):83–90

    Article  Google Scholar 

  6. Carrera E, Giunta G (2010) Refined beam theories based on Carrera’s unified formulation. Int J Appl Mech 2(1):117–143

    Article  Google Scholar 

  7. Carrera E, Petrolo M (2011) On the effectiveness of higher-order terms in refined beam theories. J Appl Mech 78(2). doi:10.1115/1.4002207

  8. Carrera E, Giunta G, Nali P, Petrolo M (2010) Refined beam elements with arbitrary cross-section geometries. Comput Struct 88(5–6):283–293. doi:10.1016/j.compstruc.2009.11.002

    Article  Google Scholar 

  9. Carrera E, Petrolo M, Nali P (2011) Unified formulation applied to free vibrations finite element analysis of beams with arbitrary section. Shock Vib 18(3):485–502. doi:10.3233/SAV-2010-0528

    Google Scholar 

  10. Carrera E, Petrolo M, Varello A (2011) Advanced beam formulations for free vibration analysis of conventional and joined wings. J Aerosp Eng. doi:10.1061/(ASCE)AS.1943-5525.0000130

    Google Scholar 

  11. Carrera E, Petrolo M, Zappino E (2011) Performance of CUF approach to analyze the structural behavior of slender bodies. J Struct Eng. doi:10.1061/(ASCE)ST.1943-541X.0000402

    Google Scholar 

  12. Dinis P, Camotim D, Silvestre N (2006) GBT formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open cross-sections. Thin-Walled Struct 44:20–38

    Article  Google Scholar 

  13. Eisenberger M (2003) An exact high order beam element. Comput Struct 81:147–152

    Article  Google Scholar 

  14. El Fatmi R (2007) Non-uniform warping including the effects of torsion and shear forces. Part I: a general beam theory. Int J Solids Struct 44:5912–5929

    Article  MATH  Google Scholar 

  15. El Fatmi R (2007) Non-uniform warping including the effects of torsion and shear forces. Part II: analytical and numerical applications. Int J Solids Struct 44:5930–5952

    Article  MATH  Google Scholar 

  16. Euler L (1744) De curvis elasticis. Bousquet, Lausanne

    Google Scholar 

  17. Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections. Comput Mech 27:199–207

    Article  MATH  Google Scholar 

  18. Gruttmann F, Sauer R, Wagner W (1999) Shear stresses in prismatic beams with arbitrary cross—sections. Int J Numer Methods Eng 45:865–889

    Article  MATH  Google Scholar 

  19. Jönsson J (1999) Distortional theory of thin-walled beams. Thin-Walled Struct 33:269–303

    Article  Google Scholar 

  20. Kapania K, Raciti S (1989) Recent advances in analysis of laminated beams and plates, part I: shear effects and buckling. AIAA J 27(7):923–935

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Kapania K, Raciti S (1989) Recent advances in analysis of laminated beams and plates, part II: vibrations and wave propagation. AIAA J 27(7):935–946

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Novozhilov VV (1961) Theory of elasticity. Pergamon, Elmsford

    MATH  Google Scholar 

  23. Oñate E (2009) Structural analysis with the finite element method: linear statics, vol 1. Springer, Berlin

    Book  MATH  Google Scholar 

  24. Petrolito J (1995) Stiffness analysis of beams using a higher-order theory. Comput Struct 55(1):33–39

    Article  MATH  Google Scholar 

  25. Reddy JN (1997) On locking-free shear deformable beam finite elements. Comput Methods Appl Mech Eng 149:113–132

    Article  ADS  MATH  Google Scholar 

  26. Reddy JN (2004) Mechanics of laminated composite plates and shells. Theory and analysis, 2nd edn. CRC Press, Boca Raton

    Google Scholar 

  27. Rendek S, Baláž I (2004) Distortion of thin-walled beams. Thin-Walled Struct 42:255–277

    Article  Google Scholar 

  28. Saadé K, Espion B, Warzée G (2004) Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections. Thin-Walled Struct 42:857–881

    Article  Google Scholar 

  29. Silvestre N (2007) Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes. Thin-Walled Struct 45:185–198

    Article  Google Scholar 

  30. Timoshenko SP (1921) On the corrections for shear of the differential equation for transverse vibrations of prismatic bars. Philos Mag 41:744–746

    Google Scholar 

  31. Timoshenko SP (1922) On the transverse vibrations of bars of uniform cross section. Philos Mag 43:125–131

    Google Scholar 

  32. Tsai SW (1988) Composites design, 4th edn. Think Composites, Dayton

    Google Scholar 

  33. Vinayak RU, Prathap G, Naganarayana BP (1996) Beam elements based on a higher order theory—I. Formulation and analysis of performance. Comput Struct 58(4):775–789

    Article  MATH  Google Scholar 

  34. Wagner W, Gruttmann F (2002) A displacement method for the analysis of flexural shear stresses in thin—walled isotropic composite beams. Comput Struct 80:1843–1851

    Article  Google Scholar 

  35. Yu W, Hodges DH (2004) Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. J Appl Mech 71:15–23

    Article  MATH  Google Scholar 

  36. Yu W, Volovoi VV, Hodges DH, Hong X (2002) Validation of the variational asymptotic beam sectional analysis (VABS). AIAA J 40:2105–2113

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aeronautic and Space Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Erasmo Carrera & Marco Petrolo

Authors
  1. Erasmo Carrera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Petrolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erasmo Carrera.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s11012-012-9539-0.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carrera, E., Petrolo, M. Refined beam elements with only displacement variables and plate/shell capabilities. Meccanica 47, 537–556 (2012). https://doi.org/10.1007/s11012-011-9466-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-011-9466-5

Keywords

Search

Navigation