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GBT pre-buckling and buckling analyses of thin-walled members under axial and transverse loads


This paper presents an analytical approach for pre-buckling and buckling analyses of thin-walled members implemented within the framework of the Generalised Beam Theory (GBT). With the proposed GBT cross-sectional analysis, the set of deformation modes used in the analysis is represented by the dynamic modes obtained for an unrestrained frame representing the cross-section. In this manner, it is possible to account for the deformability of the cross-section in both pre-buckling and buckling analyses. Different loading conditions, including both axial and transverse arrangements, are considered in the applications to highlight under which circumstances the use of the GBT deformation modes is required for an adequate representation of the pre-buckling and buckling response. The numerical results have been validated against those determined using a shell element model developed in the finite element software ABAQUS.

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  1. 1

    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)

  2. 2

    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)

  3. 3

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

  4. 4

    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)

  5. 5

    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)

  6. 6

    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)

  7. 7

    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)

  8. 8

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

  9. 9

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

  10. 10

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

  11. 11

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

  12. 12

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

  13. 13

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

  14. 14

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

  15. 15

    Schardt R.: Generalised beam theory—an adequate method for coupled stability problems. Thin Walled Struct. 19, 161–180 (1994)

  16. 16

    Luongo A., Zulli D.: Mathematical Models of Beams and Cables. Wiley, Hoboken (2013)

  17. 17

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

  18. 18

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

  19. 19

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

  20. 20

    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)

  21. 21

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

  22. 22

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

  23. 23

    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)

  24. 24

    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)

  25. 25

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

  26. 26

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

  27. 27

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

  28. 28

    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)

  29. 29

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

  30. 30

    Contro R., Poggi C., Cazzani A.: Numerical analysis of fire effects on beam structures. Eng. Comput. 5, 53–58 (1988)

  31. 31

    Cazzani A., Contro R., Corradi L.: On the evaluation of the shakedown boundary for temperature-dependent elastic properties. Eur. J. Mech. A Solids 11, 539–550 (1992)

  32. 32

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

  33. 33

    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)

  34. 34

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

  35. 35

    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)

  36. 36

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

  37. 37

    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)

  38. 38

    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)

  39. 39

    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)

  40. 40

    Davies J.M., Leach P.: First-order generalised beam theory. J. Constr. Steel Res. 31, 187–220 (1994)

  41. 41

    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)

  42. 42

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

  43. 43

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

  44. 44

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

  45. 45

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

  46. 46

    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)

  47. 47

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

  48. 48

    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)

  49. 49

    Casafront 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)

  50. 50

    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)

  51. 51

    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)

  52. 52

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

  53. 53

    Nedelcu M., Cucu H.L.: Buckling modes identification from FEA of thin-walled members using only GBT cross-sectional deformation modes. Thin Walled Struct. 81, 150–158 (2014)

  54. 54

    Silvestre N., Camotim D.: A shear deformable generalized beam theory for the analysis of thin-walled composite members. J. Eng. Mech. ASCE 139(8), 1010–1024 (2013)

  55. 55

    Abambres M., Camotim D., Silvestre N., Rasmussen K.J.: GBT-based structural analysis of elastic–plastic thin-walled members. Comput. Struct. 136, 1–23 (2014)

  56. 56

    de Miranda S., Gutierrez A., Miletta R.: Equilibrium-based reconstruction of three-dimensional stresses in GBT. Thin Walled Struct. 74, 146–154 (2014)

  57. 57

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

  58. 58

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

  59. 59

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

  60. 60

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

  61. 61

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

  62. 62

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

  63. 63

    Taig G., Ranzi G.: Generalised beam theory (GBT) for stiffened sections. Int. J. Steel Struct. 14(2), 381–397 (2014)

  64. 64

    Ranzi G., Luongo A.: An analytical approach for the cross-sectional analysis of GBT. Proc. Inst. Civ. Eng. Struct. Build. 167(7), 414–425 (2013)

  65. 65

    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). doi:10.1016/j.tws.2013.09.008

  66. 66

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

  67. 67

    Taig, G., Ranzi, G., D’Annibale, F.: An unconstrained dynamic approach for the generalised beam theory. Continuum Mech. Thermodyn. 1–26 (2014) . doi:10.1007/s00161-014-0358-5

  68. 68

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

  69. 69

    Pignataro M., Rizzi N., Luongo A.: Stability, Bifurcation, and Postcritical Behaviour of Elastic Structures. Elsevier, Amsterdam (1991)

  70. 70

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

  71. 71

    Cook R.D., Malkus D.S., Plesha M.E.: Concepts and Applications of Finite Element Analysis, 4th edn. Wiley, New York (2002)

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Correspondence to Gianluca Ranzi.

Additional information

Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.

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Taig, G., Ranzi, G. & Luongo, A. GBT pre-buckling and buckling analyses of thin-walled members under axial and transverse loads. Continuum Mech. Thermodyn. 28, 41–66 (2016). https://doi.org/10.1007/s00161-014-0399-9

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  • Buckling analysis
  • Dynamic approach
  • Generalised Beam Theory
  • Pre-buckling analysis
  • Thin-walled members