Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Torsional flexural steady state response of monosymmetric thin-walled beams under harmonic loads

  • Hjaji, Mohammed A. (Department of Mechanical and Industrial Engineering, University of Tripoli) ;
  • Mohareb, Magdi (Civil Engineering Department, University of Ottawa)
  • Received : 2013.03.26
  • Accepted : 2014.07.13
  • Published : 2014.11.25
⟨ Previous Next ⟩

Abstract

Starting with Hamilton's variational principle, the governing field equations for the steady state response of thin-walled beams under harmonic forces are derived. The formulation captures shear deformation effects due to bending and warping, translational and rotary inertia effects and as well as torsional flexural coupling effects due to the cross section mono-symmetry. The equations of motion consist of four coupled differential equations in the unknown displacement field variables. A general closed form solution is then developed for the coupled system of equations. The solution is subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing field equations. A super-convergent finite element is then formulated based on the exact shape functions. Key features of the element developed include its ability to (a) isolate the steady state response component of the response to make the solution amenable to fatigue design, (b) capture coupling effects arising as a result of section mono-symmetry, (c) eliminate spatial discretization arising in commonly used finite elements, (d) avoiding shear locking phenomena, and (e) eliminate the need for time discretization. The results based on the present solution are found to be in excellent agreement with those based on finite element solutions at a small fraction of the computational and modelling cost involved.

Keywords

References

  1. Ambrosini, D. (2004), "On free vibration of nonsymmetrical thin-walled beams", Thin Wall. Struct., 47(6-7), 629-636.
  2. Back, S.Y. and Will, K.M. (1998), "A shear-flexible element with warping for thin-walled open beams", Int. J. Numer. Meth. Eng., 43(7), 1173-1191. https://doi.org/10.1002/(SICI)1097-0207(19981215)43:7<1173::AID-NME340>3.0.CO;2-4
  3. Banerjee, J.R., Guo, S. and Howson, W.P. (1996), "Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping", Comput. Struct., 59(4), 613-621. https://doi.org/10.1016/0045-7949(95)00307-X
  4. Bishop, R.E., Cannon, S. and Miao, S. (1989), "On coupled bending and torsional vibration of uniform beams", J. Sound Vib., 131(3), 457-464. https://doi.org/10.1016/0022-460X(89)91005-5
  5. Bishop, R.E. and Price, W.G. (1985), "A note on the dynamical behavior of uniform beams having open channel section", J. Sound Vib., 99(2), 155-167. https://doi.org/10.1016/0022-460X(85)90354-2
  6. Chen, X. and Tamma, K. (1994), "Dynamic response of elastic thin-walled structures influenced by coupling effects", Comput. Struct., 51(1), 91-105. https://doi.org/10.1016/0045-7949(94)90039-6
  7. Chen, W.F. and Lui, E.M. (1997), Handbook of Structural Engineering, CRC Press, New York, USA.
  8. Cortinez, V.H. and Piovan, M.T. (2001), "Vibration and buckling of composite thin-walled beams with shear deformability", J. Sound Vib., 258(4), 701-723.
  9. De Bordon, F. and Ambrosini, D. (2010), "On free vibration analysis of thin-walled beams axially loaded", Thin Wall. Struct., 48(12), 915-920. https://doi.org/10.1016/j.tws.2010.06.002
  10. Dokumaci, E. (1987), "An exact solution for coupled flexural and torsional vibrations of uniform beams having single cross-sectional symmetry", J. Sound Vib., 119(3), 443-449. https://doi.org/10.1016/0022-460X(87)90408-1
  11. Friberg, P.O. (1985), "Beam element matrices derived from Vlasov's theory of open thin-walled elastic beams", Int. J. Numer. Meth. Eng., 21(7), 1205-1228. https://doi.org/10.1002/nme.1620210704
  12. Giunta, G., Belouettar, S., Biscani, F. and Carrera, E. (2014), "Hierarchical theories for a linearised stability analysis of thin-walled beams with open and closed cross-section", Adv. Aircraft Spacecraft Sci., 1(3), 253-271 https://doi.org/10.12989/aas.2014.1.3.253
  13. Hashemi, S.M. and Richard, M.J. (2000a), "A dynamic finite element method for free vibrations of bending-torsion coupled beams", Aerosp. Sci. Tech., 4(1), 41-55. https://doi.org/10.1016/S1270-9638(00)00114-0
  14. Hashemi, S.M. and Richard, M.J. (2000b), "Free vibrational analysis of axially loaded bending-torsion coupled beams - a dynamic finite element", Comput. Struct., 77(6), 711-724. https://doi.org/10.1016/S0045-7949(00)00012-2
  15. Hjaji, M.A. and Mohareb, M. (2011a), "Steady state response of doubly symmetric thin-walled members under harmonic excitations - Closed-form solution", Second International Engineering Mechanics and Materials Specialty Conference, Ottawa, Canada, June.
  16. Hjaji, M.A. and Mohareb, M. (2011b), "Steady state response of doubly symmetric thin-walled members under harmonic excitations - Finite element formulation", Second International Engineering Mechanics and Materials Specialty Conference, Ottawa, Canada, June.
  17. Hjaji, M.A. and Mohareb, M. (2013a), "Harmonic response of doubly symmetric thin-walled members based on the Vlasov theory- I. Analytical solution", 3rd Specialty Conference on Material Engineering & Applied Mechanics, Montreal, Canada, May.
  18. Hjaji, M.A. and Mohareb, M. (2013b), "Harmonic response of doubly symmetric thin-walled members based on the Vlasov theory - II. Finite element formulation", 3rd Specialty Conference on Material Engineering & Applied Mechanics, Montreal, Canada, May.
  19. Hjaji, M.A. and Mohareb, M. (2014a), "Coupled flexural-torsional response of harmonically excited monosymmetric thin-walled Vlasov beams - II. Finite element solution", 4th International Structural Specialty Conference, Halifax, Nova Scotia, Canada, May
  20. Hjaji, M.A. and Mohareb, M. (2014b), "Coupled flexural-torsional response of harmonically excited monosymmetric thin-walled Vlasov beams - I. Closed form solution", 4th International Structural Specialty Conference, Halifax, Nova Scotia, Canada, May.
  21. Hu, Y., Jin, X. and Chen, B. (1996), "A finite element model for static and dynamic analysis of thin-walled beams with asymmetric cross-sections", Comput. Struct., 61(5), 897-908. https://doi.org/10.1016/0045-7949(96)00058-2
  22. Kim, M.Y., Kim, N. and Yun, H.T. (2003), "Exact dynamic and static stiffness matrices of shear deformable thin-walled beam-columns", J. Sound Vib., 267(1), 29-55. https://doi.org/10.1016/S0022-460X(02)01410-4
  23. Kim, N.I. and Kim, M.N. (2005), "Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects", Thin Wall. Struct., 43(5), 701-734. https://doi.org/10.1016/j.tws.2005.01.004
  24. Kim, N., Chung, C.F. and Kim, M.Y. (2007), "Stiffness matrices for flexural-torsional/lateral buckling and vibration analysis of thin-walled beam", J. Sound Vib., 299(4-5), 739-756. https://doi.org/10.1016/j.jsv.2006.06.062
  25. Kollar, J.P. (2001), "Flexural-torsional vibration of open section composite columns with shear deformation", Int. J. Solid. Struct., 38(42-43), 7543-7558. https://doi.org/10.1016/S0020-7683(01)00025-7
  26. Laudiero, F. and Savoia, M. (1991), The shear strain influence on the dynamics of thin-walled beams", Thin Wall. Struct., 11(5), 375-407. https://doi.org/10.1016/0263-8231(91)90035-H
  27. Leung, A.Y.T. (1991), "Natural shape functions of a compressed Vlasov element", Thin Wall. Struct., 11(5), 431-438. https://doi.org/10.1016/0263-8231(91)90037-J
  28. Lee, J. and Kim, S.E. (2002a), "Free vibration of thin-walled composite beams with I-shaped crosssections", Comput. Struct., 55(2), 205-215. https://doi.org/10.1016/S0263-8223(01)00150-7
  29. Lee, J. and Kim, S.E. (2002b), "Flexural-torsional coupled vibration of thin-walled composite beams with channel sections", Comput. Struct., 80(2), 133-144. https://doi.org/10.1016/S0045-7949(01)00171-7
  30. Librescu, L. and Song, O. (2006), Thin-Walled Composite Beams - Theory and Application, Springer, Netherland.
  31. Li, J., Hua, H., Shen, R. and Jin, X. (2004a), "Dynamic response of axially loaded monosymmetrical thinwalled Bernoulli-Euler beams", Thin Wall. Struct., 42(12), 1689-1707. https://doi.org/10.1016/j.tws.2004.05.005
  32. Li, J., Shen, R., Hua, H. and Jin, X. (2004b), "Response of monosymmetric thin-walled Timoshenko beams to random excitations", Int. J. Solid. Struct., 41(22-23), 6023-6040. https://doi.org/10.1016/j.ijsolstr.2004.05.030
  33. Li, J, Shen, R., Hua, N. and Jin, X. (2004c), "Coupled bending and torsional vibration of axially loaded thinwalled Timoshenko beams", Mech. Int. J. Mech. Sci., 46(2), 299-320. https://doi.org/10.1016/j.ijmecsci.2004.02.009
  34. Machado, S.P. and Cortinez, V.H. (2007), "Free vibration of thin-walled composite beams with static initial stresses and deformations", Eng. Struct., 29(3), 372-382. https://doi.org/10.1016/j.engstruct.2006.05.004
  35. Machado, S.P. (2007), "Geometrically non-linear approximations on stability and free vibration of composite beams", Eng. Struct., 29(12), 3567-3578. https://doi.org/10.1016/j.engstruct.2007.08.009
  36. Mei, C. (1970), "Coupled vibrations of thin-walled beams of open section using the finite element method", Int. J. Mech. Sci., 12(10), 883-891. https://doi.org/10.1016/0020-7403(70)90025-1
  37. Prokic, A. (2006), "On fivefold coupled vibrations of Timoshenko thin-walled beams", Eng. Struct., 28(1), 54-62. https://doi.org/10.1016/j.engstruct.2005.07.002
  38. Prokic, A., Lukic D. and Ladjinovic, D. (2014), "Automatic analysis of thin-walled laminated composite sections", Steel Compos. Struct., 16(3), 233-252. https://doi.org/10.12989/scs.2014.16.3.233
  39. Tanaka, M. and Bercin, A.N. (1997), "Finite element modeling of the coupled bending and torsional free vibration of uniform beams with an arbitrary cross-section", Appl. Math. Model., 21(6), 339-344. https://doi.org/10.1016/S0307-904X(97)00030-9
  40. Tanaka, M. and Bercin, A.N. (1998), "Free vibration solution for uniform beams of nonsymmetrical crosssection using mathematica", Comput. Struct., 71(1), 1-8.
  41. Vlasov, V. (1961), Thin-Walled Elastic Beams, Israel Program for Scientific Translation, Jerusalem.
  42. Voros, G.M. (2008), "On coupled vibrations of beams with lateral loads", J. Comput. Appl. Mech., 9, 1-14.
  43. Voros, G.M. (2009), "On coupled bending-torsional vibrations of beams with initial loads", Mech. Res. Commun., 36(5), 603-611. https://doi.org/10.1016/j.mechrescom.2009.01.006
  44. Vo, T.P. and Lee, J. (2009a), "Free vibration of axially loaded thin-walled composite box beams", Compos. Struct., 90(2) 233-241. https://doi.org/10.1016/j.compstruct.2009.03.010
  45. Vo, T.P. and Lee, J. (2009b), "On six-fold coupled buckling of thin-walled composite beams", Compos. Struct., 90(3), 295-303. https://doi.org/10.1016/j.compstruct.2009.03.008
  46. Vo, T.P. and Lee, J. (2009c), "Flexural-torsional coupled vibration and buckling of thin-walled open section composite beams using shear-deformable beam theory", Int. J. Mech. Sci., 51(9-10), 631-641. https://doi.org/10.1016/j.ijmecsci.2009.05.001
  47. Vo, T.P., Lee, J. and Ahn, N. (2009), "On sixfold coupled vibrations of thin-walled composite box beams", Compos. Struct., 89(4), 524-535. https://doi.org/10.1016/j.compstruct.2008.11.004
  48. Vo, T.P. and Lee, J. (2010), "Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments", Appl. Math. Model., 34, 3142-3157. https://doi.org/10.1016/j.apm.2010.02.003
  49. Vo, T.P., Lee, J. and Lee, K. (2010), "On triply coupled vibrations of axially loaded thin-walled composite beams", Compos. Struct., 88(3-4), 144-153. https://doi.org/10.1016/j.compstruc.2009.08.015
  50. Vo, T.P., Lee, J., Lee, K. and Ahn, N. (2011), "Vibration analysis of thin-walled composite beams with Ishaped cross-sections", Compos. Struct., 93(2), 812-820. https://doi.org/10.1016/j.compstruct.2010.08.001
  51. Wu, L. and Mohareb, M. (2011), "Buckling of shear deformable thin-walled members - I. variational principle and analytical solutions", Thin Wall. Struct., 49(1), 197-207. https://doi.org/10.1016/j.tws.2010.09.025