Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
권호 표지

Thin- Walled Curved Beam Theory Based on Centroid-Shear Center Formulation

  • Kim Nam-Il (Department of Civil and Environmental Engineering, Sungkyunkwan University) ;
  • Kim Moon-Young (Department of Civil and Environmental Engineering, Sungkyunkwan University)
  • Published : 2005.02.01
Download PDF
⟨ Previous Next ⟩

Abstract

To overcome the drawback of currently available curved beam theories having non-symmetric thin-walled cross sections, a curved beam theory based on centroid-shear center formulation is presented for the spatially coupled free vibration and elastic analysis. For this, the displacement field is expressed by introducing displacement parameters defined at the centroid and shear center axes, respectively. Next the elastic strain and kinetic energies considering the thickness-curvature effect and the rotary inertia of curved beam are rigorously derived by degenerating the energies of the elastic continuum to those of curved beam. And then the equilibrium equations and the boundary conditions are consistently derived for curved beams having non-symmetric thin-walled cross section. It is emphasized that for curved beams with L- or T-shaped sections, this thin-walled curved beam theory can be easily reduced to the solid beam theory by simply putting the sectional properties associated with warping to zero. In order to illustrate the validity and the accuracy of this study, FE solutions using the Hermitian curved beam elements are presented and compared with the results by previous research and ABAQUS's shell elements.

Keywords

References

  1. Chucheepsakul, S. and Saetiew, W., 2002, 'Free Vibrations of Inclined Arches Using Finite Elements,' Structural Engineering and Mechanics, Vol. 13, pp.713-730 https://doi.org/10.12989/sem.2002.13.6.713
  2. Cortinez, V. H. and Piovan, M. T., 1999, 'Out of Plane Vibrations of Thin-Walled Curved Beams Considering Shear Flexibility,' Structural Engineering and Mechanics, Vol. 8, pp. 257-272 https://doi.org/10.12989/sem.1999.8.3.257
  3. Dabrowski, R., 1968, Curved Thin-walled Girders. (Translated from the German), Cement and Concrete Association, London
  4. Fam, A. R. M. and Turkstra, C., 1976, 'Model Study of Horizontally Curved Box Girder,' Journal of the Structural Division (ASCE), Vol. 102, pp. 1097-1108
  5. Gendy, A. S. and Saleeb, A. F., 1994, 'Vibration Analysis of Coupled Extensional/Flexural/Torsional Modes of Curved Beams with Arbitrary Thin-Walled Sections,' Journal of Sound and Vibration, Vol. 174, pp. 261-274 https://doi.org/10.1006/jsvi.1994.1275
  6. Gendy, A. S. and Saleeb, A. F., 1992, 'On the Finite Element Analysis of the Spatial Response of Curved Beams with Arbitrary Thin-Walled Sections,' Computers & Structures, Vol. 44, pp. 639-652 https://doi.org/10.1016/0045-7949(92)90396-H
  7. Gjelsvik, A., 1981, The Theory of Thin Walled Bars. John Wiley & Sons, Inc.
  8. Gruttmann, F., Sauer, R. and Wagner, W., 2000, 'Theory and Numerics of Three-Dimensional Beams with Elastoplastic Material Behaviour,' International Journal for Numerical Methods in Engineering, Vol. 48, pp. 1675-1702 https://doi.org/10.1002/1097-0207(20000830)48:12<1675::AID-NME957>3.0.CO;2-6
  9. Gruttmann, F., Sauer, R. and Wagner, W., 1998, 'A Geometrical Nonlinear Eccentric 3D-Beam Element with Arbitrary Cross-Sections,' Computer Methods in Applied Mechanics and Engineering, Vol. 160, pp.383-400 https://doi.org/10.1016/S0045-7825(97)00305-8
  10. Gupta, A. K. and Howson, W. P., 1994, 'Exact Natural Frequencies of Plane Structures Composed of Slender Elastic Curved Members,' Journal of Sound and Vibration, Vol. 175, pp. 145-157 https://doi.org/10.1006/jsvi.1994.1319
  11. Heins, C. P., 1975, Bending and Torsional Design in Structural Members. D. C. Health and Company
  12. Howson, W. P. and Jemah, A. K., 1999, 'Exact Out-of-Plane Natural Frequencies of Curved Timoshenko Beams,' Journal of Engineering Mechanics, Vol. 125, pp. 19-25 https://doi.org/10.1061/(ASCE)0733-9399(1999)125:1(19)
  13. Hu, N., Hu, B., Fukunaga, H. and Sekine, H., 1999, 'Two Kinds of Co-Type Eelements for Buckling Analysis of Thin-Walled Curved Beams,' Computer Methods in Applied Mechanics and Engineering, Vol. 171, pp. 87-108 https://doi.org/10.1016/S0045-7825(98)00246-1
  14. Kang, K. J. and Han, J. W., 1998, 'Analysis of a Curved Beam Using Classical and Shear Deformable Beam Theories,' KSME International Journal, Vol. 12, pp.244-256 https://doi.org/10.1007/BF02947169
  15. Kang, Y. J. and Yoo, C. H., 1994, 'Thin-Walled Curved Beams. I: Formulation of Nonlinear Equations,' Journal of Engineering Mechanics (ASCE), Vol. 120, pp.2072-2101 https://doi.org/10.1061/(ASCE)0733-9399(1994)120:10(2072)
  16. Kawakami, M., Sakiyama, T., Matsuda, H. and Morita, C., 1995, 'In-Plane and Out-of-Plane Free Vibrations of Curved Beams with Variable Sections,' Journal of Sound and Vibration, Vol. 187, pp. 381-401 https://doi.org/10.1006/jsvi.1995.0531
  17. Lee, J. H., 2003, 'In-Plane Free Vibration Analysis of Curved Timoshenko Beams by the Pseudospectral Method,' KSME International Journal, Vol. 17, pp.1156-1163
  18. Kim, M. Y., Kim, N. I. and Min, B. C., 2002, 'Analytical and Numerical Study on Spatial Free Vibration of Non-Symmetric Thin-Walled Curved Beams,' Journal of Sound and Vibration, Vol. 258, pp. 595-618 https://doi.org/10.1006/jsvi.2002.5090
  19. Kim, M. Y., Min, B. C. and Suh, M. W., 2000a, 'Spatial Stability of Non-Symmetric Thin-Walled Curved Beams I: Analytical Approach,' Journal of Engineering Mechanics (ASCE), Vol. 126, pp. 497-505 https://doi.org/10.1061/(ASCE)0733-9399(2000)126:5(497)
  20. Kim, M. Y., Min, B. C. and Suh, M. W., 2000b, 'Spatial Stability of Non-Symmetric Thin-Walled Curved Beams II: Numerical Approach,' Journal of Engineering Mechanics (ASCE), Vol. 126, 506-514 https://doi.org/10.1061/(ASCE)0733-9399(2000)126:5(506)
  21. Saleeb, A. F., Chang, T. Y., Graf, W. and Yingyeunyoung, S., 1990, 'A Hybrid/mixed Model for Nonlinear shell Analysis and its Applications to Large-Rotation Problems,' International Journal for Numerical Methods in Engineering, Vol. 29, pp. 407-446 https://doi.org/10.1002/nme.1620290213
  22. Piovan, M. T. Cortinez, V. H. and Rossi, R. E., 2000, 'Out-of-Plane Vibrations of Shear Deformable Continuous Horizontally Curved Thin-Walled Beams,' Journal of Sound and Vibration, Vol. 237, pp. 101-118 https://doi.org/10.1006/jsvi.2000.3055
  23. Raveendranath, P. Singh, G. and Pradhan, B., 2000, 'Free Vibration of Arches Using a Curved Beam Element Based on a Coupled Polynomial Displacement Field,' Computers & Structures, Vol. 78, pp. 583-590 https://doi.org/10.1016/S0045-7949(00)00038-9
  24. Saleeb, A. F. and Gendy, A. S., 1991, 'Shear-Flexible Models for Spatial Bucking of Thin-Walled Curved Beams,' International Journal for Numerical Methods in Engineering, Vol. 31, pp.729-757 https://doi.org/10.1002/nme.1620310407
  25. Timoshenko, S. P. and Gere, J. M., 1961, Theory of Elastic Stability. 2nd Ed., McGraw-Hill, N.Y.
  26. Tong, G. and Xu, Q., 2002, 'An Exact Theory for Curved Beams with Any Thin-Walled Open Sections,' Advances in Structural Engineering, Vol. 5, pp. 195-209 https://doi.org/10.1260/136943302320974572
  27. Vlasov, V. Z., 1961, Thin-Walled Elastic Beams, 2nd Ed., National Science Foundation, Washington, D.C.
  28. Wilson, J. F. and Lee, B. K., 1995, 'In-Plane Free Vibrations of Catenary Arches with Un-symmetric Axes,' Structural Engineering and Mechanics, Vol. 3, pp. 511-525 https://doi.org/10.12989/sem.1995.3.5.511
  29. Yang, Y. B. and Kuo, S. R., 1987, 'Effect of Curvature on Stability of Curved Beams,' Journal of Structural Engineering (ASCE), Vol. 113, pp. 1185-1202 https://doi.org/10.1061/(ASCE)0733-9445(1987)113:6(1185)
  30. Yang, Y. B. and Kuo, S. R., 1986, 'Static Stability of Curved Thin-Walled Beams,' Journal of Engineering Mechanics (ASCE), Vol. 112, pp.821-841 https://doi.org/10.1061/(ASCE)0733-9399(1986)112:8(821)