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Architectural Institute of Korea (대한건축학회)
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Static Analysis of Timoshenko Beams using Isogeometric Approach

  • Lee, Sang Jin (ADOPT Research Group, Department of Architectural Engineering, Gyeongsang National University) ;
  • Park, Kyoung Sub (ADOPT Research Group, Department of Architectural Engineering, Gyeongsang National University)
  • Received : 2013.05.20
  • Accepted : 2014.02.05
  • Published : 2014.06.30
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Abstract

A study on the static analysis of Timoshenko beams is presented. A beam element is developed by using isogeometric approach based on Timoshenko beam theory which allows the transverse shear deformation. The identification of transverse shear locking is conducted by three refinement schemes such as h-, p- and k-refinement and compared to other reference solutions. From numerical examples, the present beam element does not produce any shear locking in very thin beam situations even with full Gauss integration rule. Finally, the benchmark tests described in this study is provided as future reference solutions for Timoshenko beam problems based on isogeometric approach.

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References

  1. Cottrell, J.A., Bazilevs, Y. and Hughes, T.J.R. (2009). Isogeometric Analysis: Towards Integration of CAD and FEA. Wiley.
  2. Dawe, D.J. (1978) A finite element for the vibration analysis of Timoshenko beams. Journal of Sound and Vibration, vol. 60, pp.11-20. https://doi.org/10.1016/0022-460X(78)90397-8
  3. De Boor, C. (2001). A Practical Guide to Splines (revised edn.). Springer.
  4. Heyliger, P.R. and Reddy, J.N. (1988) A higher order beam finite element for bending and vibration problems. Journal of Sound and Vibration, 126, pp.309-326. https://doi.org/10.1016/0022-460X(88)90244-1
  5. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y. (2005). Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp.4135-4195. https://doi.org/10.1016/j.cma.2004.10.008
  6. Hughes, T.J.R. and Evans J.A. (2010). Isogeometric analysis. ICES Report 10-18, The Institute of Computational Engineering and Science, University of Texas Austin.
  7. Hughes, T.J.R., Reali, A. and Sangalli, G. (2008). Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp.4104-4124. https://doi.org/10.1016/j.cma.2008.04.006
  8. Kapur, K. K. (1966) Vibrations of a Timoshenko beam using finite element approach. Journal of the Acoustical Society of America, vol. 40, pp.1058-1063. https://doi.org/10.1121/1.1910188
  9. Kosmatka, J. B. (1995) An improved two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams. Computers and Structures, vol. 57, pp.141-149. https://doi.org/10.1016/0045-7949(94)00595-T
  10. Lee, S.J. (1998). Analysis and optimization of shells. University of Wales, Swansea. Ph.D. Thesis.
  11. Lee, S.J. and Ma, H. (1993). Analytical study of shear locking. Proceedings of Annual Symposium of Computational Structural Engineering Institute of Korea.
  12. Lee, S.J. and Park, K.S. (2013). Vibrations of Timoshenko beams with isogeometric approach. Applied Mathematical Modelling, 37, pp.9174-9190. https://doi.org/10.1016/j.apm.2013.04.034
  13. Levinson, M. (1981) A new rectangular beam theory. Journal of Sound and Vibration, 74, pp.81-87. https://doi.org/10.1016/0022-460X(81)90493-4
  14. Nickel, R.E. and Secor, G.A. (1972) Convergence of consistently derived Timoshenko beam finite elements. International Journal for Numerical Methods in Engineering, vol.5, pp.243-252. https://doi.org/10.1002/nme.1620050210
  15. Prathap, G. and Bhashyam, G. (1982) "Reduced integration and the shear-flexible beam element." International Journal for Numerical Methods in Engineering, 18, 195-210 https://doi.org/10.1002/nme.1620180205
  16. Reddy, J.N. (1997) On locking shear deformable beam finite elements. Computer methods in applied mechanics and engineering, 149, pp.113-132. https://doi.org/10.1016/S0045-7825(97)00075-3
  17. Rogers, D.F. (2000). An introduction to NURBS: With Historical Perspective. Morgan Kaufmann.
  18. Timoshenko, S.P. and Goodier .JN. (1970). Theory of Elasticity (3rd edn). McGraw Hill.
  19. Timoshenko, S.P. (1921). On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. 41, pp.744-746. https://doi.org/10.1080/14786442108636264
  20. Zienkiewicz, O.C. and Taylor, R.L. (1989). The finite element method (4th edn). McGraw-Hill.