Skip to main content
SpringerLink
Log in

An unconstrained dynamic approach for the Generalised Beam Theory

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

This paper presents a new approach for the cross-sectional analysis within the framework of the Generalised Beam Theory (GBT), and it is applicable to open, closed or partially closed cross-sections. This approach falls within a category of cross-sectional analysis available in the literature for which a suitable set of deformation modes, including conventional, extension and shear, is determined from dynamic analyses. The novelty of the proposed approach relies on the use of an unrestrained planar frame for the evaluation of the conventional and extension modes, therefore allowing the identification of both sets of modes from the solution of a planar eigenvalue problem, whose eigenmodes correspond to the sought conventional and extension modes. In the available dynamic GBT approach, the conventional modes are obtained from a planar dynamic analysis enforcing inextensibility to the members of the frame, and the extension modes are then evaluated from the conventional modes enforcing particular constraint conditions. Numerical examples are presented considering open unbranched, open branched and partially closed cross-sections to highlight the ease of use of the proposed approach and to discuss the contribution of the different modes to the structural response. The accuracy of the numerical results is validated against those calculated with a shell element model developed in Abaqus.

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. Vlasov V.Z.: Thin-Walled Elastic Beams. Monson, Jerusalem (1961)

    Google Scholar 

  2. Barr A.D.S.: An extension of the Hu-Washizu variational principle in linear elasticity for dynamic problems. J. Appl. Mech. 33, 465 (1966)

    Article  ADS  Google Scholar 

  3. Washizu K.: Variational Methods in Elasticity and Plasticity. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  4. Reissner E.: On a certain mixed variational theorem and a proposed application. Int. J. Numer. Methods Eng. 20, 1366–1368 (1984)

    Article  MATH  Google Scholar 

  5. Djoko J.K., Lamichhane B.P., Reddy B.D.: Conditions for equivalence between the Hu-Washizu and related formulations, and computational behavior in the incompressible limit. Comput. Methods Appl. Mech. Eng. 195, 4161–4178 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Yang J.S., Batra R.C.: Mixed variational principles in nonlinear piezoelectricity. Int. J. Non-Linear Mech. 30, 719–725 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Maurini C., dell’Isola F., Pouget J.: On models of layered piezoelectric beams for passive vibration control. J Phys. IV 115, 307–316 (2004)

    Google Scholar 

  8. Maurini C., Pouget J., dell’Isola F.: On a model of layered piezoelectric beams including transverse stress effect. Int. J. Solids Struct. 41, 4473–4502 (2004)

    Article  MATH  Google Scholar 

  9. Maurini C., Pouget J., dell’Isola F.: Extension of the Euler Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach. Comput. Struct. 84, 1438–1458 (2006)

    Article  Google Scholar 

  10. Sciarra G., dell’Isola F., Hutter K.: A solid-fluid mixture model allowing for solid dilatation under external pressure. Contin. Mech. Thermodyn. 13, 287–306 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Sciarra G., dell’Isola F., Coussy O.: Second gradient poromechanics. Int. J. Solids Struct. 44, 6607–6629 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Sciarra G., dell’Isola F., Ianiro N. et al.: A variational deduction of second gradient poroelasticity Part I: general theory. J. Mech. Mater. Struct. 3, 507–526 (2008)

    Article  Google Scholar 

  13. Altenbach H., Birsan M., Eremeyev V.A.: On a thermodynamic theory of rods with two temperature fields. Acta Mech. 223(8), 1583–1596 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Birsan M., Altenbach H., Sadowski T., Eremeyev V.A., Pietras D.: Deformation analysis of functionally graded beams by the direct approach. Compos. Part B Eng. 43(3), 1315–1328 (2012)

    Article  Google Scholar 

  15. Cazzani A., Atluri S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11, 229–251 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cazzani A., Lovadina C.: On some mixed finite element methods for plane membrane problems. Comput. Mech. 20, 560–572 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Garusi E., Tralli A., Cazzani A.: An unsymmetric stress formulation for Reissner-Mindlin plates: a simple and locking-free rectangular element. Int. J. Comput. Eng. Sci. 5, 589–618 (2004)

    Article  Google Scholar 

  18. Cazzani A., Garusi E., Tralli A., Atluri S.N.: A four-node hybrid assumed-strain finite element for laminated composite plates. Comput. Mater. Contin. 2, 23–38 (2005)

    MATH  Google Scholar 

  19. Schardt R.: Verallgemeinerte Technicsche Biegetheory. Springler, Berlin (1989)

    Book  Google Scholar 

  20. Schardt R.: Generalised Beam Theory: an adequate method for coupled stability problems. Thin-Walled Struct. 19, 161–180 (1994)

    Article  Google Scholar 

  21. Davies J.M., Leach P.: First-order Generalised Beam Theory. J. Constr. Steel Res. 31, 187–220 (1994)

    Article  Google Scholar 

  22. Leach P., Davies J.M.: An experimental verification of the Generalised Beam Theory applied to interactive buckling problems. Thin-Walled Struct. 25(1), 61–79 (1996)

    Article  Google Scholar 

  23. Jiang C., Davies J.M.: Design of thin-walled purlins for distortional buckling. Thin-Walled Struct. 29(1–4), 189–202 (1997)

    Article  Google Scholar 

  24. Silvestre N., Camotim D.: First-order generalised beam theory for arbitrary orthotropic materials. Thin-Walled Struct. 40, 755–789 (2002)

    Article  Google Scholar 

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

    Article  Google Scholar 

  26. Nedelcu M.: GBT formulation to analyse the behaviour of thin-walled members with variable cross-section. Thin-Walled Struct. 48, 629–638 (2010)

    Article  Google Scholar 

  27. Goncalves R., Dinis P.B., Camotim D.: GBT formulation to analyse the first-order and buckling behaviour of thin-walled members with arbitrary cross-sections. Thin-Walled Struct. 47, 583–600 (2009)

    Article  Google Scholar 

  28. Silvestre N., Camotim D.: Asymptotic-numerical method to analyse the postbuckling behaviour, imperfection-sensitivity, and mode interaction in frames. J. Eng. Mech. ASCE 131(6), 617–632 (2005)

    Article  Google Scholar 

  29. Basaglia C., Camotim D., Silvestre N.: GBT-based buckling analysis of thin-walled steel frames with arbitrary loading and support conditions. Int. J. Struct. Stab. Dyn. 10(3), 363–385 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Nedelcu M.: GBT-based buckling mode decomposition from finite element analysis of thin-walled members. Thin-Walled Struct. 54, 156–163 (2012)

    Article  Google Scholar 

  31. Silvestre N., Camotim D.: GBT buckling analysis of pultruded FRP lipped channel members. Comput. Struct. 81, 1889–1904 (2003)

    Article  Google Scholar 

  32. Silvestre N., Camotim D.: Second-order generalised beam theory for arbitrary orthotropic materials. Thin-Walled Struct. 40, 791–820 (2002)

    Article  Google Scholar 

  33. Silvestre N., Camotim D.: Local-plate and distortional postbuckling behaviour of cold-formed steel lipped channel columns with intermediate stiffeners. J. Struct. Eng. ASCE 132(4), 529–540 (2006)

    Article  Google Scholar 

  34. Adany S., Schafer B.W.: Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-Walled Struct. 44(5), 563–584 (2006)

    Article  Google Scholar 

  35. Adany S., Schafer B.W.: Buckling mode decomposition of single-branched open cross-section members via finite strip method: application and examples. Thin-Walled Struct. 44(5), 585–600 (2006)

    Article  Google Scholar 

  36. Casafont M., Marimon F., Pastor M.M.: Calculation of pure distortional elastic buckling loads of members subjected to compression via the finite element method. Thin-Walled Struct. 47(6–7), 701–729 (2009)

    Article  Google Scholar 

  37. Goncalves R., Ritto-Corrêa M., Camotim D.: A new approach to the calculation of cross-section deformation modes in the framework of generalised beam theory. Comput. Mech. 46(5), 759–781 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Silvestre N., Camotim D., Silva N.F.: Generalised beam theory revisited: from the kinematical assumptions to the deformation mode determination. Int. J. Struct. Stab. Dyn. 11(5), 969–997 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. De Miranda S., Gutierrez A., Miletta R., Ubertini F.: A generalized beam theory with shear deformation. Thin-Walled Struct. 67, 88–100 (2013)

    Article  Google Scholar 

  40. Ranzi G., Luongo A.: A new approach for thin-walled member analysis in the framework of GBT. Thin-Walled Struct. 49, 1404–1414 (2011)

    Article  Google Scholar 

  41. Piccardo G., Ranzi G., Luongo A.: A direct approach for the evaluation of the conventional modes within the GBT formulation. Thin-Walled Struct. 74, 133–145 (2014)

    Article  Google Scholar 

  42. Piccardo, G., Ranzi, G., Luongo, A.: A complete dynamic approach to the GBT cross-section analysis including extension and shear modes. Math. Mech. Solids (2013). doi:10.1177/1081286513493107

  43. Ranzi, G., Luongo, A.: An analytical approach for the cross-sectional analysis of GBT. In: Proceedings of the Institution of Civil Engineers: Structures and Buildings (2013). doi:10.1680/stbu.12.00057

  44. Jönsson J., Andreassen M.J.: Distortional eigenmodes and homogeneous solutions for semi-discretized thin-walled beams. Thin-Walled Struct. 49(6), 691–707 (2011)

    Article  Google Scholar 

  45. Andreassen M.J., Jönsson J.: Distortional solutions for loaded semi-discretized thin-walled beams. Thin-Walled Struct. 50, 116–127 (2012)

    Article  Google Scholar 

  46. Andreassen M.J., Jönsson J.: A distortional semi-discretized thin-walled beam element. Thin-Walled Struct. 62, 142–157 (2013)

    Article  Google Scholar 

  47. Jönsson J.: Distortional warping functions and shear distributions in thin-walled beams. Thin-Walled Struct. 33, 245–268 (1999)

    Article  Google Scholar 

  48. Jönsson J.: Determination of shear stresses, warping functions and section properties of thin-walled beams using finite elements. Comput. Struct. 68, 393–410 (1998)

    Article  MATH  Google Scholar 

  49. Cheung Y.K.: Finite Strip Method in Structural Analysis. Pergamon Press, Oxford (1976)

    MATH  Google Scholar 

  50. Cheung, Y.K., Fan, S.C., Wu, C.Q.: Spline finite strip in structural analysis. In: Proceedings of the International Conference on Finite Element Method, Shanghai, China (1982)

  51. Friedrich R.: Finite strip method: 30 years—a bibliography (1968–1998). Eng. Comput. 17(1), 92–111 (2000)

    Article  MATH  Google Scholar 

  52. Lau S.C.W., Hancock G.J.: Buckling of thin flat-walled structures by a spline finite strip method. Thin-Walled Struct. 4(4), 269–294 (1986)

    Article  Google Scholar 

  53. Mahendran M., Murray N.W.: Elastic buckling analysis of ideal thin-walled structures under combined loading using a finite strip method. Thin-Walled Struct. 4(5), 329–362 (1986)

    Article  Google Scholar 

  54. Young B.: Bifurcation analysis of thin-walled plain channel compression members. Finite Elem. Anal. Des. 41(2), 211–225 (2004)

    Article  Google Scholar 

  55. Adany S., Schafer B.W.: Buckling mode decomposition of single-branched open cross-section members via finite strip method: application and examples. Thin-Walled Struct. 44, 585–600 (2006)

    Article  Google Scholar 

  56. Vrcelj Z., Bradford M.A.: A simple method for the inclusion of external and internal supports in the spline finite strip method (SFSM) of buckling analysis. Comput. Struct. 86(6), 529–544 (2008)

    Article  Google Scholar 

  57. Eccher G., Rasmussen K.J.R., Zandonini R.: Geometric nonlinear isoparametric spline finite strip analysis of perforated thin-walled structures. Thin-Walled Struct. 47(2), 219–232 (2009)

    Article  Google Scholar 

  58. Pignataro M., Luongo A.: Asymmetric interactive buckling of thin-walled columns with initial imperfections. Thin-Walled Struct. 5(5), 365–386 (1987)

    Article  Google Scholar 

  59. Luongo A., Pignataro M.: Multiple interaction and localization phenomenon in postbuckling of compressed thin-walled members. AIAA J. 26(11), 1395–1402 (1988)

    Article  MATH  ADS  Google Scholar 

  60. Ren W., Fang S., Young B.: Finite-element simulation and design of cold-formed steel channels subjected to web crippling. J. Struct. Eng. 132(12), 1967–1975 (2006)

    Article  Google Scholar 

  61. Bakkera M.C.M., Peközb T.: The finite element method for thin-walled members—basic principles. Thin-Walled Struct. 41(2-3), 179–189 (2003)

    Article  Google Scholar 

  62. Chen J., Young B.: Cold-formed steel lipped channel columns at elevated temperatures. Eng. Struct. 29(10), 2445–2456 (2007)

    Article  Google Scholar 

  63. Luongo A., Pignataro M.: Multiple interaction and localization phenomena in the postbuckling of compressed thin-walled members. AIAA J. 26, 1395–1402 (1988)

    Article  MATH  ADS  Google Scholar 

  64. Luongo A.: On the amplitude modulation and localization phenomena in interactive buckling problems. Int. J. Solids Struct. 27(15), 1943–1954 (1991)

    Article  MATH  Google Scholar 

  65. Luongo A.: Mode localization by structural imperfections in one-dimensional continuous systems. J. Sound Vib. 155(2), 249–271 (1992)

    Article  MATH  ADS  Google Scholar 

  66. Luongo A.: Mode localization in dynamics and buckling of linear imperfect continuous structures. Non-linear Dyn. 25(1), 133–156 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  67. Luongo A., Rega G., Vestroni F.: On nonlinear dynamics of planar shear-indeformable beams. J. Appl. Mech. 108, 619–624 (1986)

    Article  Google Scholar 

  68. Pignataro M., Luongo A.: Interactive buckling of an elastically restrained truss structure. Thin Walled Struct. 19(2–4), 197–210 (1994)

    Article  Google Scholar 

  69. Di Egidio A., Luongo A., Vestroni F.: A nonlinear model for the dynamics of open cross-section thin-walled beams. Part I: formulation. Int. J. Nonlinear Mech. 38(7), 1067–1081 (2003)

    Article  MATH  Google Scholar 

  70. Di Egidio A., Luongo A., Vestroni F.: A nonlinear model for the dynamics of open cross-section thin-walled beams. Part II: forced motion. Int. J. Nonlinear Mech. 38(7), 1083–1091 (2003)

    Article  MATH  Google Scholar 

  71. Concalves R., Ritto-Correa M., Camotim D.: A new approach to the calculation of cross-section deformation modes in the framework of generalised beam theory. Comput. Mech. 46(5), 759–781 (2010)

    Article  MathSciNet  Google Scholar 

  72. Bathe K.J.: Finite Element Procedures. Prentice Hall, New Jersey (2006)

    Google Scholar 

  73. Cook R.D., Malkus D.S., Plesha M.E.: Concepts and Applications of Finite Element Analysis. Wiley, New York (2002)

    Google Scholar 

  74. Dassault Systèmes Simulia, ABAQUS user’s manual, version 6.8EF-2. Providence, RI, USA: Dassault Systèmes Simulia Corp.; 2008

Download references

Author information

Authors and Affiliations

  1. School of Civil Engineering, The University of Sydney, Sydney, Australia

    Gerard Taig & Gianluca Ranzi

  2. Dipartimento d’Ingegneria Civile, Edile-Architettura, ed Ambientale, Università dell’Aquila, L’Aquila, Italy

    Francesco D’Annibale

Authors
  1. Gerard Taig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianluca Ranzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francesco D’Annibale
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianluca Ranzi.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Taig, G., Ranzi, G. & D’Annibale, F. An unconstrained dynamic approach for the Generalised Beam Theory. Continuum Mech. Thermodyn. 27, 879–904 (2015). https://doi.org/10.1007/s00161-014-0358-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-014-0358-5

Keywords

Search

Navigation