Shear deformation effect in flexural-torsional buckling analysis of beams of arbitrary cross section by BEM

  • Sapountzakis, E.J. (School of Civil Engineering, National Technical University, Zografou Campus) ;
  • Dourakopoulos, J.A. (School of Civil Engineering, National Technical University, Zografou Campus)
  • Received : 2008.05.20
  • Accepted : 2010.01.04
  • Published : 2010.05.30


In this paper a boundary element method is developed for the general flexural-torsional buckling analysis of Timoshenko beams of arbitrarily shaped cross section. The beam is subjected to a compressive centrally applied concentrated axial load together with arbitrarily axial, transverse and torsional distributed loading, while its edges are restrained by the most general linear boundary conditions. The resulting boundary value problem, described by three coupled ordinary differential equations, is solved employing a boundary integral equation approach. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six coupled boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-walled theory and the significant influence of the boundary conditions and the shear deformation effect on the buckling load are investigated through examples with great practical interest.


  1. Attard, M.M. (1986), "Nonlinear theory of non-uniform torsion of thin-walled open beams", Thin Wall. Struct., 4, 101-134.
  2. Barsoum, R.S. and Gallagher, R.H. (1970), "Finite element analysis of torsional and torsional-flexural stability problems", Int. J. Numer. Meth. Eng., 2, 335-352.
  3. Bazant, Z.P. and Cedolin, L. (1991), Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories, Oxford University Press.
  4. Catal, S. and Catal, H.H. (2006), "Buckling analysis of partially embedded pile in elastic soil using differential transform method", Struct. Eng. Mech., 24(2), 247-268.
  5. Cowper, G.R. (1966), "The shear coefficient in timoshenko's beam theory", J. Appl. Mech., 33(2), 335-340.
  6. Euler, L. (1759), Sur la force des colonnes, Memoires Academic Royale des Sciences et Belle Lettres.
  7. Gadalla, M.A. and Abdalla, J.A. (2006), "Modeling and prediction of buckling behavior of compression members with variability in material and/or section properties", Struct. Eng. Mech., 22(5), 631-645.
  8. Hutchinson, J.R. (2001), "Shear coefficients for timoshenko beam theory", J. Appl. Mech., 68, 87-92.
  9. Ioannidis, G.I. and Kounadis, A.N. (1999), "Flexural-torsional postbuckling analysis of centrally compressed bars with open thin-walled cross-section", Eng. Struct., 21, 55-61.
  10. Katsikadelis, J.T. (2002), "The analog equation method, a boundary-only integral equation method for nonlinear static and dynamic problems in general bodies", Theor. Appl. Mech., 27, 13-38.
  11. Knothe, K. and Wessels, H. (1992), Finite Elemente, Springer Verlag, 2. Auflage, Berlin-New York.
  12. Kounadis, A.N. (1998), "Postbuckling analysis of bars with thin-walled cross sections under simultaneous bending and torsion due to central thrust", J. Construct. Steel Res., 45, 17-37.
  13. Li, Q.S. (2003), "Effect of shear deformation on the critical buckling of multi-step bars", Struct. Eng. Mech., 15(1), 71-81.
  14. Mohri, F., Azrar, L. and Potier-Ferry, M. (2001), "Flexural-torsional post-buckling analysis of thin-walled elements with open sections", Thin Wall.Struct., 39, 907-938.
  15. MSC/NASTRAN for Windows (1999), Finite Element Modeling and Postprocessing System, Help System Index,Version 4.0, USA.
  16. Rajasekaran, S. (2008), "Buckling of fully embedded non-prismatic columns using ifferential quadrature and differential transformation methods", Struct. Eng. Mech., 28(2), 221-238.
  17. Sapountzakis, E.J. and Katsikadelis, J.T. (2000), "Elastic deformation of ribbed plate systems under static, transverse and inplane loading", Comput. Struct., 74, 571-581.
  18. Sapountzakis, E.J. and Mokos, V.G. (2001), "Nonuniform torsion of composite bars by boundary element method", J. Eng. Mech-ASCE, 127(9), 945-953.
  19. Sapountzakis, E.J. and Mokos, V.G. (2003), "Warping shear stresses in nonuniform torsion by BEM", Comput. Mech., 30, 131-142.
  20. Sapountzakis, E.J. and Mokos, V.G. (2004), "Nonuniform torsion of bars of variable cross section", Comput. Struct., 82, 703-715.
  21. Sapountzakis, E.J. and Mokos, V.G. (2005), "A BEM solution to transverse shear loading of beams", Computat. Mech., 36, 384-397.
  22. Schramm, U., Kitis, L., Kang, W. and Pilkey, W.D. (1994), "On the shear deformation coefficient in beam theory", Finite Elem. Anal. Des., 16, 141-162.
  23. Schramm, U., Rubenchik, V. and Pilkey, W.D. (1997), "Beam stiffness matrix based on the elasticity equations", Int. J. Numer. Meth. Eng., 40, 211-232.<211::AID-NME58>3.0.CO;2-P
  24. Simitses, G.J. and Hodges, D.H. (2006), Fundamentals of Structural Stability, Elsevier, Boston.
  25. Stephen, N.G. (1980), "Timoshenko's shear coefficient from a beam subjected to gravity loading", J. Appl. Mech., 47, 121-127.
  26. Szymczak, C. (1980), "Buckling and initial post-buckling behavior of thin-walled I columns", Comput. Struct., 11(6), 481-487.
  27. Timoshenko, S.P. (1921). "On the correction for shear of the differential equation for transverse vibrations of prismatic bars", Philos. Mag., 41, 744-746.
  28. Timoshenko, S.P. and Gere, J.M. (1961), Theory of Elastic Stability, McGraw-Hill, Tokyo.
  29. Timoshenko, S.P. and Goodier, J.N. (1984), Theory of Elasticity, 3rd edition, McGraw-Hill, New York.
  30. Trahair, N.S. (1993), Flexural-torsional Buckling of Structures, Chapman and Hall, London.
  31. Vlasov, V.Z. (1961), Thin-walled Elastic Beams, Israel Program for Scientific Translations, Jerusalem.
  32. Yu, W., Hodges, D.H., Volovoi, V.V. and Fuchs, E.D. (2005), "A generalized vlasov theory for composite beams", Thin Wall. Struct., 43(9), 1493-1511.

Cited by

  1. Warping stresses of a rectangular single leaf flexure under torsion vol.59, pp.3, 2016,
  2. Torsional analysis of a single-bent leaf flexure vol.54, pp.1, 2015,
  3. A new analytical approach for determination of flexural, axial and torsional natural frequencies of beams vol.55, pp.3, 2015,