DOI QR코드

DOI QR Code

On eigenvalue problem of bar structures with stochastic spatial stiffness variations

  • Rozycki, B. (Voivodeship Roads Administration in Opole) ;
  • Zembaty, Z. (Faculty of Civil Engineering, The Opole University of Technology)
  • Received : 2010.09.03
  • Accepted : 2011.05.25
  • Published : 2011.08.25

Abstract

This paper presents an analysis of stochastic eigenvalue problem of plane bar structures. Particular attention is paid to the effect of spatial variations of the flexural properties of the structure on the first four eigenvalues. The problem of spatial variations of the structure properties and their effect on the first four eigenvalues is analyzed in detail. The stochastic eigenvalue problem was solved independently by stochastic finite element method (stochastic FEM) and Monte Carlo techniques. It was revealed that the spatial variations of the structural parameters along the structure may substantially affect the eigenvalues with quite wide gap between the two extreme cases of zero- and full-correlation. This is particularly evident for the multi-segment structures for which technology may dictate natural bounds of zero- and full-correlation cases.

Keywords

References

  1. Augusti, G., Baratta, A. and Casciati, F. (1984), Probabilistic Methods in Structural Engineering, Chapman and Hall, London.
  2. Baecher, G.B. and Ingra, T.S. (1981), "Stochastic FEM in settlement predictions", J. Geotech. Eng. Div., 107(4), 449-463.
  3. Benaroya, H. and Rehak, M. (1988), "Finite element methods in probabilistic structural analysis: A selective review", Appl. Mech. Rev., 41(5), 201-213. https://doi.org/10.1115/1.3151892
  4. Bathe, K.J. (1982), Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, New Jersey.
  5. Collins, J.D. and Thomson, W.T. (1969), "The eigenvalue problem for structural systems with statistical properties", Am. Inst. Aero. Astronaut. J., 7(4), 642-648. https://doi.org/10.2514/3.5180
  6. Doebling, S.W., Farrar, C.R., Prime, M.B. and Shevitz, D.W. (1996), "Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review", Report LA-13070-MS, Los Alamos National Laboratory.
  7. Fox, R.L. and Kapoor, M.P. (1968), "Rates of change of eigenvalues and eigenvectors", Am. Inst. Aero. Astronaut. J., 6(12), 2426-2429. https://doi.org/10.2514/3.5008
  8. Ghanem, R.G. and Spanos, P.D. (1991a), "Spectral stochastic finite-element formulation for reliability analysis", J. Eng. Mech., 117(10), 2351-2372. https://doi.org/10.1061/(ASCE)0733-9399(1991)117:10(2351)
  9. Ghanem, R.G. and Spanos, P.D. (1991b), Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York.
  10. Hasselman, T.K. and Hart, G.C. (1972), "Modal analysis of random structural systems", J. Eng. Mech. Div., 98(3), 561-579.
  11. Hisada, T. and Nakagiri, S. (1985), "Role of stochastic finite element methods in structural safety and reliability", Proceedings of 4th ICOSSAR, 385-394.
  12. Hoshiya, M. and Shah, H.C. (1971), "Free vibration of stochastic beam-column", J. Eng. Mech. Div., 97(4), 1239-1255.
  13. Ibrachim, R.A. (1987), "Structural dynamics with parameter uncertainties", Appl. Mech. Rev., 40(3), 309-328. https://doi.org/10.1115/1.3149532
  14. Kaleta, B. and Zembaty, Z. (2007), "Eigenvalue problem of a beam on stochastic Vlasov foundation", Arch. Civil Eng., 53(3), 447-477.
  15. Kleiber, M. and Hien, T.D. (1992), The Stochastic Finite Element Method. Basic Perturbation Technique and Comput. Implementation, John Wiley & Sons, Chichester.
  16. Lin, Y.K. and Cai, G.Q. (1995), Probabilistic Structural Dynamics. Advance Theory and Applications, Mc Graw- Hill, Singapore.
  17. Liu, W.K., Belytschko, T. and Mani, A. (1986), "Probabilistic finite elements for nonlinear structural dynamics", Comput. Meth. Appl. Mech. Eng., 56, 61-81. https://doi.org/10.1016/0045-7825(86)90136-2
  18. Marek, P., Brozzetti, J., Gustar, M. and Tikalsky, P. (editors) (2003), Probabilistic Assessment of Structures Using Monte Carlo Simulation. Background, Exercises and Software, Academy of Science of the Czech Republic, Praha.
  19. Mehlhose, S., vom Scheidt, J. and Wunderlich, R. (1999), "Random eigenvalue problems for bending vibrations of beams", J. Appl. Math. Mech., 79(10), 693-702.
  20. Mironowicz, W. and Sniady, P. (1987), "Dynamics of machine foundations with random parameters", J. Sound Vib., 112(1), 23-30. https://doi.org/10.1016/S0022-460X(87)80090-1
  21. Pradlwarter, H.J., Schueller, G.I. and Szekely, G.S. (2002), "Random eigenvalue problems for large systems", Comput. Struct., 80(5), 2415-2424. https://doi.org/10.1016/S0045-7949(02)00237-7
  22. Qiu, Z., Chen, S. and Elishakoff, I. (1996), "Non-probabilistic eigenvalue problem for structures with uncertain parameters via interval analysis", Chaos Solit. Fract., 7(3), 303-308. https://doi.org/10.1016/0960-0779(95)00087-9
  23. Ramu, S.A. and Ganesan, R. (1991), "Free vibration of stochastic beam-column using stochastic FEM", Comput. Struct., 41(5), 987-994. https://doi.org/10.1016/0045-7949(91)90292-T
  24. Ramu, S.A. and Ganesan, R. (1993), "A Galerkin finite element technique for stochastic field problems", Comput. Meth. Appl. Mech. Eng., 105, 315-331. https://doi.org/10.1016/0045-7825(93)90061-2
  25. Sohn, H., Farrar, C.R., Hemez, F.M., Shunk, D.D., Stinemates, D.W. and Nadler, B.R. (2003), "A review of structural health monitoring literature: 1996-2001", Report LA-13976-MS, Los Alamos National Laboratory.
  26. Shinozuka, M. (1972), "Probabilistic modeling of concrete structures", J. Eng. Mech. Div., 98(6), 1433-1451.
  27. Shinozuka, M. and Astill, C.J. (1972), "Random eigenvalue problems in structural analysis", Amer. Inst. Aero. Astronaut. J., 10(4), 456-462. https://doi.org/10.2514/3.50119
  28. Sobczyk, K. (1972), "Free vibrations of elastic plate with random properties-the eigenvalue problem", J. Sound Vib., 22(1), 33-39. https://doi.org/10.1016/0022-460X(72)90842-5
  29. Song, D., Chen, S. and Qiu, Z. (1995), "Stochastic sensitivity analysis of eigenvalues and eigenvectors", Comput. Struct., 54(5), 891-896. https://doi.org/10.1016/0045-7949(94)00386-H
  30. Soong, T.T. and Bogdanoff, J.L. (1963), "On the natural frequencies of a disordered linear chain of n degrees of freedom", Int. J. Mech. Sci., 5, 237-265. https://doi.org/10.1016/0020-7403(63)90052-3
  31. Spanos, P.D. and Ghanem, R. (1989), "Stochastic finite element expansion for random media", J. Eng. Mech., 115(5), 1035-1053. https://doi.org/10.1061/(ASCE)0733-9399(1989)115:5(1035)
  32. Szekely, G.S. and Schueller, G.I. (2001), "Computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties", Comput. Meth. Appl. Mech. Eng., 191, 799-816. https://doi.org/10.1016/S0045-7825(01)00290-0
  33. Vanmarcke, E. (1984), Random Fields: Analysis and Synthesis, The MIT Press, Cambridge.
  34. Vanmarcke, E. and Shinozuka, M., Nakagiri, S., Schueller, G.I. and Grigoriu, M. (1986), "Random fields and stochastic finite elements", Struct. Safe., 3, 143-166. https://doi.org/10.1016/0167-4730(86)90002-0
  35. vom Scheidt, J. and Purkert, W. (1983), Random Eigenvalue Problems, Akademie-Verlag, Berlin.
  36. Xia, Y., Weng, S., Xu, Y.L. and Zhu, H.P. (2010), "Calculation of eigenvalue and eigenvector derivatives with the improved Kron's substructuring method", Struct. Eng. Mech., 36(1), 37-54. https://doi.org/10.12989/sem.2010.36.1.037
  37. Yamazaki, F., Shinozuka, M. and Dasgupta, G. (1988), "Neumann expansion for stochastic finite element analysis", J. Eng. Mech., 114(8), 1335-1354. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:8(1335)
  38. Zhu, W.Q. and Wu, W.Q. (1991), "A stochastic finite element method for real eigenvalue problems", Probab. Eng. Mech., 6(3/4), 228-232. https://doi.org/10.1016/0266-8920(91)90014-U
  39. Zhu, Z.Q. and Chen, J.J. (2009), "Dynamic eigenvalue analysis of structures with interval parameters based on affine arithmetic", Struct. Eng. Mech., 33(4), 539-542. https://doi.org/10.12989/sem.2009.33.4.539
  40. Zielinski, R. (1970), Monte Carlo Methods, WNT, Warszawa. (in Polish)

Cited by

  1. Spatial variability and stochastic strength prediction of unreinforced masonry walls in vertical bending vol.59, 2014, https://doi.org/10.1016/j.engstruct.2013.11.031
  2. Moment Lyapunov exponents of the Parametrical Hill's equation under the excitation of two correlated wideband noises vol.52, pp.3, 2014, https://doi.org/10.12989/sem.2014.52.3.525