DOI QR코드

DOI QR Code

Convergence studies on static and dynamic analysis of beams by using the U-transformation method and finite difference method

  • Yang, Y. (School of Civil Engineering and Transportation, South China University of Technology) ;
  • Cai, M. (Department of Mechanics, Zhongshan University) ;
  • Liu, J.K. (Department of Mechanics, Zhongshan University)
  • 투고 : 2006.03.03
  • 심사 : 2009.01.31
  • 발행 : 2009.03.10

초록

The static and dynamic analyses of simply supported beams are studied by using the U-transformation method and the finite difference method. When the beam is divided into the mesh of equal elements, the mesh may be treated as a periodic structure. After an equivalent cyclic periodic system is established, the difference governing equation for such an equivalent system can be uncoupled by applying the U-transformation. Therefore, a set of single-degree-of-freedom equations is formed. These equations can be used to obtain exact analytical solutions of the deflections, bending moments, buckling loads, natural frequencies and dynamic responses of the beam subjected to particular loads or excitations. When the number of elements approaches to infinity, the exact error expression and the exact convergence rates of the difference solutions are obtained. These exact results cannot be easily derived if other methods are used instead.

키워드

참고문헌

  1. Abarbanel, S., Ditkowski, A. and Gustafsson, B. (2000), "On error bounds of finite difference approximations to partial differential equations - temporal behavior and rate of convergence", J. Scientific Comput., 15(1), 79-116 https://doi.org/10.1023/A:1007688522777
  2. Akio, Y. (2005), "Convergence property of response matrix method for various finite-difference formulations used in the nonlinear acceleration method", Nuclear Sci. Eng., 149(3), 259-269
  3. Borzi, A., Kunisch, K. and Kwak, D.Y. (2003), "Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system", SIAM J. Control Optim., 41(5), 1477-1497
  4. Brezzi, F., Lipnikov, K. and Shashkov, M. (2005), "Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes", SIAM J. Numer. Anal., 43(5), 1872-1896 https://doi.org/10.1137/040613950
  5. Cai, C.W., Liu, J.K. and Chan, H.C. (2002), Exact Analysis of Bi-Periodic Structures. World Scientific, Singapore
  6. Caldwell, C.S. (1976), "Convergence estimates for a locally one dimensional finite difference scheme for parabolic initial-boundary value problems", SIAM J. Numer. Anal., 13(4), 514-519 https://doi.org/10.1137/0713044
  7. Chan, H.C., Cai, C.W. and Cheung, Y.K. (1998), Exact Analysis of Structures with Periodicity using UTransformation. World Scientific, Singapore
  8. Liu, J.K., Yang, Y. and Cai, C.W. (2006), "Convergence studies on static and dynamic analyses of plates by using the U-transformation and the finite difference method", J. Sound Vib., 289, 66-76 https://doi.org/10.1016/j.jsv.2005.01.036
  9. Liu, J.K., Yang, Y. and Cai, M. (2003), "Convergence study of the finite difference method: a new approach", Acta Mech. Sinica, 35(6), 757-760
  10. Runchal, A.K. (1972), "Convergence and accuracy of three finite difference schemes for a two-dimensional conduction and convection problem", Int. J. Numer. Method. Eng., 4(4), 541-550 https://doi.org/10.1002/nme.1620040408
  11. Tetsuro, Y. (2002), "Convergence of consistent and inconsistent finite difference schemes and an acceleration technique", J. Comput. Appl. Math., 140(1-2), 849-866 https://doi.org/10.1016/S0377-0427(01)00522-2

피인용 문헌

  1. Convergence and exact solutions of spline finite strip method using unitary transformation approach vol.32, pp.11, 2011, https://doi.org/10.1007/s10483-011-1511-9
  2. Analytical studies on stress concentration due to a rectangular small hole in thin plate under bending loads vol.36, pp.6, 2010, https://doi.org/10.12989/sem.2010.36.6.669
  3. Stress concentration around a rectangular cuboid hole in a three-dimensional elastic body under tension loading vol.88, pp.8, 2018, https://doi.org/10.1007/s00419-018-1369-7
  4. Analysis of elastic foundation plates with internal and perimetric stiffening beams on elastic foundations by using Finite Differences Method vol.45, pp.2, 2009, https://doi.org/10.12989/sem.2013.45.2.169