Structural Engineering and Mechanics

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Non-stochastic interval arithmetic-based finite element analysis for structural uncertainty response estimate

  • Lee, Dongkyu (Architectural Engineering Research Department, Steel Structure Research Laboratory, Research Institute of Industrial Science & Technology) ;
  • Park, Sungsoo (Department of Architectural Engineering, Pusan National University) ;
  • Shin, Soomi (Department of Architectural Engineering, Pusan National University)
  • Received : 2006.05.10
  • Accepted : 2008.05.19
  • Published : 2008.07.30
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Abstract

Finite element methods have often been used for structural analyses of various mechanical problems. When finite element analyses are utilized to resolve mechanical systems, numerical uncertainties in the initial data such as structural parameters and loading conditions may result in uncertainties in the structural responses. Therefore the initial data have to be as accurate as possible in order to obtain reliable structural analysis results. The typical finite element method may not properly represent discrete systems when using uncertain data, since all input data of material properties and applied loads are defined by nominal values. An interval finite element analysis, which uses the interval arithmetic as introduced by Moore (1966) is proposed as a non-stochastic method in this study and serves a new numerical tool for evaluating the uncertainties of the initial data in structural analyses. According to this method, the element stiffness matrix includes interval terms of the lower and upper bounds of the structural parameters, and interval change functions are devised. Numerical uncertainties in the initial data are described as a tolerance error and tree graphs of uncertain data are constructed by numerical uncertainty combinations of each parameter. The structural responses calculated by all uncertainty cases can be easily estimated so that structural safety can be included in the design. Numerical applications of truss and frame structures demonstrate the efficiency of the present method with respect to numerical analyses of structural uncertainties.

Keywords

References

  1. Alefeld, G. (1983), Introductions to Interval Computations, New York, Academic Press
  2. Alefeld, G. and Claudio, D. (1998), 'The basic properties of interval arithmetic, its software realizations and some applications', Comput. Struct., 67, 3-8 https://doi.org/10.1016/S0045-7949(97)00150-8
  3. Alefeld, G. and Herzberger, J. (1983), Introduction to Interval Computations, Academic Press, New York
  4. Alefeld, G. and Mayer, G. (2000), 'Interval analysis: Theory and applications', J. Comput. Appl. Math., 121, 421-464 https://doi.org/10.1016/S0377-0427(00)00342-3
  5. Chen, S.H. and Qiu, Z.P. (1994), 'A new method for computing the upper and lower bounds on frequencies of structures with interval parameters', Mech. Res. Commun., 2, 583-592
  6. Chen, S.H. and Qiu, Z.P. (1994), 'Perturbation method for computing eigenvalue bounds in vibration system with interval parameters', Commun. Numer. Math. Eng., 10, 121-134 https://doi.org/10.1002/cnm.1640100204
  7. Chen, S.H. and Yang, X.W. (2000), 'Interval finite element method for beam structures', Finite Elem. Anal. Des., 34, 75-88 https://doi.org/10.1016/S0168-874X(99)00029-3
  8. Dwyer, P.S. (1951), Linear Computation, Wiley
  9. Geschwindner, L.F. and Disque, R.O. (1994), Load and Resistance Factor Design of Steel Structures, Prentice-hall
  10. Jansson, C. (2001), 'Quasic-convex relation based on interval arithmetic', Linear Algebra and Its Applications, 342, 27-53
  11. Karni, R. and Belikoff, S. (1996), 'Concurrent engineering design using interval methods', Int. Trans. Opl. Res., 3, 77-87 https://doi.org/10.1111/j.1475-3995.1996.tb00037.x
  12. Kim, U.H. and Yoo, Y.H. (1999), Matrix Structural Analysis, Korea, Kong-Sung Press
  13. Klir, G.J. and Folger, T.A. (1998), Fuzzy set. Uncertainty and Information, Prentice-Hall International
  14. Kulisch, U. (1969), Grundzuge der Intervallrechnung, Jahrbuch Ueberblicke Mathematik, Vol. 2, Bibliographisches Institut, Mannheim
  15. Kulpa, Z., Pownuk, A. and Skalna, I. (1998), 'Analysis linear mechanical structures with uncertainties by means of interval methods', Comput. Assisted Mech. Eng. Sci., 5
  16. Koyluoglu, U. and Elishakoff, I. (1998), 'A comparison of stochastic and interval finite elements applied to shear frames with uncertain stiffness properties', Comput. Struct., 67(1-3), 91-98 https://doi.org/10.1016/S0045-7949(97)00160-0
  17. Koyluoglu, U., Cakmak, A.S. and Nielsen, S.R.K. (1995), 'Interval algebra to deal with pattern loading and structural uncertainty', J. Eng. Mech., 1149-1157
  18. McWilliam, S. (2001), 'Anti-optimization of uncertain structures using interval analysis', Comput. Struct., 79, 421-430 https://doi.org/10.1016/S0045-7949(00)00143-7
  19. Moore, R.E. (1962), Interval Arithmetic and Automatic Error Analysis in Digital Computing, PhD Thesis, Stanford University
  20. Moore, R.E. (1966), Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ
  21. Moore, R.E. (1979), 'Methods and applications of interval analysis', SIAM Studies in Applied Mathematics, SIAM, Philadelphia, PA
  22. Noh, H.I. (1998), Reinforce Concrete Structure, Sanup-Dose, 17-18
  23. Qiu, Z. and Elishakoff, I. (1998), 'Anti-optimization of structures with large uncertain-but-non-random parameters via interval analysis', Comput. Meth. Appl. Mech., 152, 361-372 https://doi.org/10.1016/S0045-7825(96)01211-X
  24. Qu, Y. (1999), 'On using alpha-cuts to evaluate fuzzy equations', Fuzzy Set Syst., 38, 309-312 https://doi.org/10.1016/0165-0114(90)90204-J
  25. Rao, S.S. and Berke, L. (1997), 'Analysis of uncertain structural systems using interval analysis', AIAA J., 35(4), 727-735 https://doi.org/10.2514/2.164
  26. Sen, M. and Powers, J. (2001), LECTURE NOTES ON 24. MATHEMATICAL METHODS, 86-109
  27. Skrzypczyk, J. (1997), 'Fuzzy finite element methods-A new methodology', Comput. Meth. Mech., 4, 1187-1194
  28. Sunaga, T. (1958), 'Theory of an interval algebra and its application to numerical analysis', RAAG Memoirs, 2, 29-46
  29. Wiener, N. (1914), 'A contribution to the theory of relative position', Proc. Cambridge Philos. Soc., 17, 441-449
  30. Wiener, N. (1921), 'A new theory of measurement: a study in the logic of mathematics', Proceedings of the London Mathematical Society
  31. Yang, T.Y. (1986), Finite Element Structural Analysis, Prentice-Hall, Inc. Englewood Cliffs, N.J
  32. Yang, X., Chen, S. and Lian, H. (2001), 'Bounds of complex eigenvalues of structures with interval parameters', Eng. Struct., 23, 557-563 https://doi.org/10.1016/S0141-0296(00)00049-3

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