Structural Engineering and Mechanics

  • 제28권5호
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  • Pages.549-570
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  • 2008
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Nonlinear dynamic analysis by Dynamic Relaxation method

  • Rezaiee-Pajand, M. (Department of Civil Engineering, Ferdowsi University) ;
  • Alamatian, J. (Department of Civil Engineering, Ferdowsi University)
  • 투고 : 2007.01.17
  • 심사 : 2008.01.11
  • 발행 : 2008.03.30
⟨ 이전 논문 다음 논문 ⟩

초록

Numerical integration is an efficient approach for nonlinear dynamic analysis. In this paper, general category of the implicit integration errors will be discussed. In order to decrease the errors, Dynamic Relaxation method with modified time step (MFT) will be used. This procedure leads to an alternative algorithm which is very general and can be utilized with any implicit integration scheme. For numerical verification of the proposed technique, some single and multi degrees of freedom nonlinear dynamic systems will be analyzed. Moreover, results are compared with both exact and other available solutions. Suitable accuracy, high efficiency, simplicity, vector operations and automatic procedures are the main merits of the new algorithm in solving nonlinear dynamic problems.

키워드

참고문헌

  1. Anvoner, S. (1970), Solution of Problems in Mechanics of Machines 1. Pitman Paperbacks: London
  2. Bathe, K.J. and Baig, M.M.I. (2005), "On a composite implicit time integration procedure for nonlinear dynamics", Comput. Struct., 83, 2513-2524 https://doi.org/10.1016/j.compstruc.2005.08.001
  3. Chen, C.N. (2000), "Efficient and reliable solutions of static and dynamic nonlinear structural mechanics problems by an integration numerical approach using DQFEM and direct time integration with accelerated equilibrium iteration schemes", Appl. Math. Model., 24, 637-655 https://doi.org/10.1016/S0307-904X(00)00007-X
  4. Chung, J. and Hulbert, G. (1993), "A time integration method for structural dynamics with improved numerical dissipation: The generalized-method", J. Appl. Mech., 30, 371-384
  5. Clough, R.W. and Penzien, J. (1993) Dynamics of Structures, McGraw Hill: New York
  6. Felippa, C.A. (1999), Nonlinear Finite Element Methods. ASEN 5017, Course Material, www.colorado.edu/courses.d/nfemd/; Spring
  7. Frankel, S.P. (1950), "Convergence rates of iterative treatments of partial differential equations", Math. Tables Aids Comp., 4, 65-75 https://doi.org/10.2307/2002770
  8. Fung, T.C. (1998), "Complex-time step newmark methods with controllable numerical dissipation", Int. J. Numer. Meth. Eng., 41, 65-93
  9. Hoff, C. and Taylor, R.L. (1990), "Higher derivative explicit one step methods for non-linear dynamic problems. Part I: Design and theory", Int. J. Numer. Meth. Eng., 29, 275-290 https://doi.org/10.1002/(SICI)1097-0207(19980115)41:1<65::AID-NME270>3.0.CO;2-F
  10. Hoff, C. and Taylor, R.L. (1990), "Higher derivative explicit one step methods for non-linear dynamic problems. Part II: Practical calculations and comparison with other higher order methods", Int. J. Numer. Meth. Eng., 29, 291-301 https://doi.org/10.1002/nme.1620290205
  11. Hulbert, G. and Chung, J. (1996), "Explicit time integration algorithm for structural dynamics with optimal numerical dissipation", Comput. Meth. Appl. Mech. Eng., 137, 175-188 https://doi.org/10.1002/nme.1620290206
  12. Hulbert, G.M. (1994), "A unified set of single-step asymptotic annihilation algorithms for structural dynamics", Comput. Meth. Appl. Mech. Eng., 113, 1-9 https://doi.org/10.1016/S0045-7825(96)01036-5
  13. Kadkhodayan, M., Alamatian, J. and Turvey, G.J. (2007), "A new fictitious time for the Dynamic Relaxation (DXDR) method", Int. J. Numer. Meth. Eng. https://doi.org/10.1016/0045-7825(94)90208-9
  14. Katona, M. and Zienkiewicz, O.C. (1985), "A unified set of single step algorithms Part 3: The beta-m method, A generalization of the Newmark scheme", Int. J. Numer. Meth. Eng., 21, 1345-1359 https://doi.org/10.1002/nme.1620210713
  15. Kim, S.E., Huu, N.C. and Lee, D.H. (2005), "Second order inelastic dynamic analysis of 3-D steel frames", Int. J. Solids Struct. https://doi.org/10.1002/nme.1620210713
  16. Kim, S.J., Cho, J.Y. and Kim, W.D. (1997), "From the trapezoidal rule to higher order accurate and unconditionally stable time-integration method for structural dynamics", Comput. Meth. Appl. Mech. Eng., 149, 73-88 https://doi.org/10.1016/S0045-7825(97)00061-3
  17. Lee, I.J., Oh, I.K., Lee, I. and Rhiu, J.J. (2003), "Nonlinear static and dynamic instability of complete spherical shells using mixed finite element method", Int. J. Nonlinear Mech., 38, 923-934 https://doi.org/10.1016/S0045-7825(97)00061-3
  18. Lei, Z. and Qui, C. (2000), "A stochastic variational formulation for nonlinear dynamic analysis of structure", Comput. Meth. Appl. Mech. Eng., 190, 597-608 https://doi.org/10.1016/S0020-7462(02)00038-0
  19. Mickens, R.E. (2005), "A numerical integration technique for conservative oscillators combining non-standard finite differences methods with a Hamilton's principle", J. Sound Vib., 285, 477-482 https://doi.org/10.1016/S0045-7825(99)00431-4
  20. Modak, S. and Sotelino, E. (2002), "The generalized method for structural dynamic applications", Adv. Eng. Software, 33, 565-575 https://doi.org/10.1016/j.jsv.2004.09.027
  21. Ozkul, T.A. (2004), "A finite element formulation for dynamic analysis of shells of general shape by using the Wilson-${\theta}$ method", Thin Wall. Struct., 42, 497-503 https://doi.org/10.1016/S0965-9978(02)00079-0
  22. Papadrakakis, M. (1881), "A method for the automatic evaluation of the dynamic relaxation parameters", Comput. Meth. Appl. Mech. Eng., 25, 35-48 https://doi.org/10.1016/j.tws.2003.12.008
  23. Paz, M. (1978), Structural Dynamics: Theory and Computation, John Willy & Sons https://doi.org/10.1016/0045-7825(81)90066-9
  24. Penry, S.N. and Wood, W.L. (1985), "Comparison of some single-step methods for the numerical solution of the structural dynamic equation", Int. J. Numer. Meth. Eng., 21, 1941-1955 https://doi.org/10.1002/nme.1620211102
  25. Qu, W.L., Chen, Z.H. and Xu, Y.L. (2001), "Dynamic analysis of wind-excited truss tower with friction dampers", Comput. Struct., 79, 2817-2831 https://doi.org/10.1002/nme.1620211102
  26. Qu, W.L., Chen, Z.H. and Xu, Y.L. (2001), "Dynamic analysis of wind-excited truss tower with friction dampers", Comput. Struct., 79, 2817-2831 https://doi.org/10.1016/S0045-7949(01)00151-1
  27. Rao, A.R.M. (2002), "A parallel mixed time integration for nonlinear dynamic analysis", Adv. Eng. Software, 33, 261-271
  28. Rao, A.R.M. (2005), "MPI-based parallel finite element approach for implicit nonlinear dynamic analysis employing sparse PCG solvers", Adv. Eng. Software, 36, 181-198 https://doi.org/10.1016/S0965-9978(02)00021-2
  29. Rao, A.R.M., Rao, T.V.S.R.A. and Dattaguru, B. (2003), "A new parallel overlapped domain decomposition method for nonlinear dynamic finite element analysis", Comput. Struct., 81, 2441-2454 https://doi.org/10.1016/j.advengsoft.2004.10.004
  30. Soares, D. and Mansur, W.J. (2005), "A frequency-domain FEM approach based on implicit Green's functions for non-linear dynamic analysis", Int. J. Solids Struct., 42(23), 6003-6014 https://doi.org/10.1016/S0045-7949(03)00312-2
  31. Soroushian, A., Wriggeres, P. and Farjoodi, J. (2005), "On practical integration of semi-discritized nonlinear equation of motions, Part I: Reasons for probable instability and improper convergence", J. Sound Vib., 284, 705-731 https://doi.org/10.1016/j.ijsolstr.2005.05.047
  32. Tamma, K.K., Zhou, X. and Sha, D. (2001), "A theory of development and design of generalized integration operators for computational structural dynamics", Int. J. Numer. Meth. Eng., 60, 1619-1664 https://doi.org/10.1016/j.jsv.2004.07.008
  33. Undewood, P. (1983), Computational Method for Transient Analysis Chapter 5: Dynamic Relaxation, McGraw Hill, Amsterdam
  34. Wood, W.L. (1984), "A unified set of single step algorithms Part 2: Theory", Int. J. Numer. Meth. Eng., 20, 2303-2309 https://doi.org/10.1002/nme.1620201210
  35. Zhai, W.M. (1996), "Two simple fast integration methods for large-scale dynamic problems in engineering", Int. J. Numer. Meth. Eng., 39, 4199-4214 https://doi.org/10.1002/nme.1620201210
  36. Zhang, L.C. and Yu, T.X. (1989), "Modified adaptive dynamic relaxation method and its application to elasticplastic bending and wrinkling of circular plates", Comput. Struct., 33, 609-614 https://doi.org/10.1002/(SICI)1097-0207(19961230)39:24<4199::AID-NME39>3.0.CO;2-Y
  37. Zhang, L.C., Kadkhodayan, M. and Mai, Y.W. (1994), "Development of the maDR method", Comput. Struct., 52, 1-8 https://doi.org/10.1016/0045-7949(89)90035-7
  38. Zienkiewicz, O.C., Wood, W.L. and Taylor, R.L. (1984), "A unified set of single step algorithms Part 1: General formulation and applications", Int. J. Numer. Meth. Eng., 20, 1529-1552 https://doi.org/10.1016/0045-7949(94)90249-6

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