A Comprehensive Stationary Non-Gaussian Analysis of BWB Hysteresis

  • Mohammad Noori
  • Hamid Davoodi
  • Thomas T. Baber
Conference paper
Part of the IUTAM Symposia book series (IUTAM)


A thorough non-Gaussian stationary analysis of a nondimensionalized form of a Bouc-Wen-Baber smooth hysteresis model is presented. Stationary moments of up to 6th order are generated via 3000 sample Monte Carlo simulation in order to assess the convergence of the non-Gaussian response statistics. The degree of deviation from Gaussian response is assessed by comparison of the simulated probability density functions of various response variables with a corresponding unit normal density function. The effect of variation of hysteresis shape control parameters and other important system and excitation parameters, such as the postyield to preyield stiffness ratio, and power spectral density level on response coordinate moments of various orders is studied. The effect of these parameters on correlation coefficients, probability density of the displacement, and other statistical indices are thoroughly studied. This study provides considerable insight about the range of important system and excitation parameters influencing system response. These information on the characteristic behaviors of BWB hysteresis model is useful for non-Gaussian analysis of MDOF hysteresis models and comparative studies with approximation techniques.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baber, T.T. and Wen, Y.K.: Stochastic Response of Multistory Yielding Frames. J. of Earthq. Eng. and Struc. Dyn., 1982, (10), 403–416.CrossRefGoogle Scholar
  2. 2.
    Baber, T.T.: Nonzero Mean Random Vibration of Hysteretic Systems. ASCE, J. of Eng. Mech. 1984, (110), 1036–1049.Google Scholar
  3. 3.
    Baber, T.T.: Nonzero Mean Random Vibration of Hysteretic Frames. Computers and Structures, 1986, (23), 265–277.CrossRefGoogle Scholar
  4. 4.
    Baber, T.T. and Noori, M.: Random Vibration of Degrading, Pinching Systems. ASCE, J. of Eng. Mech., 1985, (111), 1010–1027.CrossRefGoogle Scholar
  5. 5.
    Baber, T.T. and Noori, M.: Modeling General Hysteresis Behavior and Random Vibration Application. Trans. of ASME, J. of Vib., Acous., Str., and Des., 1986, (108), 411–420.Google Scholar
  6. 6.
    Casciati, F. and Faravelli, L.: Stochastic equivalent linearization in 3-D hysteretic frames. Trans. of 9th Int. Conf. on Struc. Mech. in Reactor Technology, 1987, (M), pp 453–458.Google Scholar
  7. 7.
    Casciati, F. and Faravelli, L.: Non-linear seismic analysis of three-dimensional frames. Proc. of Int. Conf. on Des., Const. and Repair of Building Structures in Earthq. Zone, 1987, 104–108.Google Scholar
  8. 8.
    Davoodi, H. and Noori, M.N.: Extension of An Ito-Based Approximation Technique for Random Vibration of A BBW General Hysteresis Model, Part II: Non-Gaussian Analysis. J. of Sound and Vib., 1990, (139), No. 3.Google Scholar
  9. 9.
    Minami, T. and Osawa, Y.: Elastic-Plastic Response Spectra for Different Hysteretic Rules. Earthq. Eng. and Struc. Dyn., 1988 (16), 555–568.CrossRefGoogle Scholar
  10. 10.
    Noori, M., Davoodi, H. and Choi, J.D.: Zero and Nonzero Mean Random Vibration Analysis of a New General Hysteresis Model. J. of Prob. Eng. Mech., 1986, (4), 192–201.CrossRefGoogle Scholar
  11. 11.
    Noori, M. and Padula, M.: Application of A New Approximation Method To Random Vibration of A General Hysteresis. Journal of Nonlinear Dynamics (to appear).Google Scholar
  12. 12.
    Noori, M.N., Davoodi, H. and Saffar, A.: An Ito-Based General Approximation Method for Random Vibration of Hysteretic Systems, Part I: Gaussian Analysis. J. of Sound and Vib., 1988, (127), No. 2, 331–342.CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Noori, M.N., Davoodi, H.: Comparison Between Equivalent Linearization and Gaussian Closure for Random Vibration of Several Nonlinear Systems. Int. J. of Eng. Science, 1990, (28), No. 9, 897–905.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Roberts, J.B. and Spanos, P.D.: Random Vibration and Statistical Linearization, 1990, John Wiley and Sons, New York.MATHGoogle Scholar
  15. 15.
    Thyagarajan, R.S.: Modeling and Analysis of Hysteretic Structural Behavior. Report No. EERL 89-03, Caltech Pasadena, California, 1989.Google Scholar
  16. 16.
    Wen, Y. K.: Method for Random Vibration of Hysteretic Systems. J. of the Eng. Mech. Div., ASCE, 1976, (102), EM2, 249–263.Google Scholar
  17. 17.
    Wen, Y.K.: Methods of Random Vibration for Inelastic Structures. Appl. Mech. Rev., 1989, (42), No. 2, 39–52CrossRefADSGoogle Scholar
  18. 18.
    Bouc, R.: Modele Mathematique d’Hysteresis. Acustica, 1961, (24), 16–25, Marseille, France.Google Scholar
  19. 19.
    Mohammadi, J. and Amin, M.: Nonlinear Stochastic Finite Element Analysis of Pipes on Hysteretic Supports Under Seismic Excitation. Comp. Prob. Meth., Proc. of The Joint ASME/SES Conf., 1988, Berkeley, California, AMD 93, 123–133.Google Scholar
  20. 20.
    Simulescu, I., Mochio, T. and Shinozuka, M.: Equivalent Linearization Method in Nonlinear FEM. ASCE, J. of Eng. Mech., 1989, pp 475–492.Google Scholar
  21. 21.
    Constantinou, M.C. and Tadjbakhsh, I.G.: Hysteretic Dampers in Base Isolation: Random Approach. ASCE, J. of Struc. Eng., 1985, (4), 705–721.Google Scholar
  22. 22.
    Igusa, T.: Response Characteristics of an Inelastic Two-Degree-of-Freedom Primary-Secondary System. Report No. 87-7/TI-01, Department of Civil Eng., Northwestern University, 1987.Google Scholar
  23. 23.
    Su, L., Ahmadi, G. and Tadjabaksh I.G.: A Comparative Study of Different Base Isolators. Proc. of ASCE, Struc. Congress 87, 1987, Orlando, Florida. 15–26.Google Scholar
  24. 24.
    Mohammad Yar, A. and Hammond, T.: Modelling and Response of Bilinear Hysteretic Systems. ASCE, J. of Eng. Mech., 1987, (113), 1000–1013.CrossRefGoogle Scholar
  25. 25.
    Orabi, I., Ahmadi, G. and Su, L.: Hysteretic Column under Earthquake Excitations. ASCE, J. of Eng. Mech., 1989, (1), 33–51.CrossRefGoogle Scholar
  26. 26.
    Wu, W.F. and Lin, Y.K.: Cumulant-Neglect Closure for Nonlinear Oscillators Under Random Parametric and 8External Excitations. Int. J. of Nonl. Mech., 1984, (19), 349–362.CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Minai, R. and Suzuki, Y.: Stochastic Estimate and Control of Hysteretic Structural Systems. Presented at the Second International Conference on Stochastic Structural Dynamics, Boca Raton, FL, May 9–11, 1990.Google Scholar
  28. 28.
    Yang, C.Y., Cheng, A.H. and Roy, V.: Chaotic and Stochastic Dynamics for a Nonlinear Structure with Hysteresis and Degradation. J. of Prob. Eng. Mech. (To Appear).Google Scholar
  29. 29.
    Park, Y.J., Wen Y.K and Ang, A.H-S.: Random Vibration of Hysteretic Systems Under Bi-Directional Ground Motions. Earth. Eng. and Struc. Dyn., 1986, (14), 543–557.CrossRefGoogle Scholar
  30. 30.
    Spanos, P-T.: Stochastic Analysis of An Osci llator with Non-Linear Damping, Int. J. of Nonl. Mech., 1978, (13), 249–259.CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Spanos, P-T.: Formulation of Stochastic Linearization for Symmetric or Assymetric MDOF Nonlinear Systems. Trans. of ASME, J. of Appl. Mech., 1980, (1), 209–211.CrossRefADSGoogle Scholar
  32. 32.
    Spanos, P-T.: Stochastic Linearization in Structural Dynamics. Applied Mech. Rev., 1981, (1), 1–8.MathSciNetGoogle Scholar
  33. 33.
    Sues, R.H., Wen Y.K. and Ang, A.H-S.: Safety Evaluation of Structures to Earthquakes. ASCE, Proc. of the Symp. on Prob. Meth. in Struc. Eng., 1981, St. Louis, Missouri., 358–377.Google Scholar
  34. 34.
    Wen, Y.K.: Stochastic Response and Damage Analysis of Inelastic Structures. J. of Prob. Eng. Mech., 1986, (1), No. 1, pp 49–57.CrossRefGoogle Scholar
  35. 35.
    Hampl, N.C., and Schuller, G.I.: Probability Densities of the Response of Nonlinear Structures Under Stochastic Dynamic Excitation. J. of Probabilistic Engineering Mechanics, 1989, (4), No. 1, 2–9.CrossRefGoogle Scholar
  36. 36.
    Lin, Y.K.: A New Solution Technique For Randomly Excited Hysteretic Structures. Technical Report NCEER-88-0012, Center for Applied Stochastic Research, Boca Raton, Florida, 1988.Google Scholar
  37. 37.
    Liu, Q.: Response Analysis of a Hysteretic System Under Random Excitation. Department of Mechanical Engineering, University of New Brunswick, Canada, 1990.Google Scholar
  38. 38.
    Ibrahim, R.A.: Parametric Random Vibration, Letchworth, Hertfordshire, England: Research Studies Press Ltd, 1985.Google Scholar
  39. 39.
    Ibrahim, R.A., Soundararajan, A. and Heo, H.: Stochastic Response of Nonlinear Dynamic Systems Based on Non-Gaussian Closure. Trans. of the ASME, J. of Appl. Mech., 1985, (52), 965–970.CrossRefMATHADSMathSciNetGoogle Scholar
  40. 40.
    Crandall, S.H.: Non-Gaussian Closure for Random Vibration of a Non-Linear Oscillator. Int. J. of Nonlinear Mechanics, 1980, (20), 303–313.CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Fan, F.G., and Ahmadi, G.: Loss of Accuracy and Nonuniqueness of Sound and Vibration of Solutions Generated by Equivalent Linearization and Cumulant-Neglect Methods. (To Appear).Google Scholar
  42. 42.
    Sun, J.Q., and Hsu, C.S.: Cumulant-Neglect Closure Method for Nonlinear Systems Under Random Excitations. ASME Journal of Applied Mechanics, 1987, (54), 649–655.CrossRefMATHADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Mohammad Noori
    • 1
  • Hamid Davoodi
    • 2
  • Thomas T. Baber
    • 3
  1. 1.Mechanical Engineering DepartmentWorcester Polytechnic InstituteWorcesterUSA
  2. 2.Department of Mechanical EngineeringUniversity of Puerto RicoMayaguezPuerto Rico
  3. 3.Civil Engineering DepartmentUniversity of VirginiaCharlottesvilleUSA

Personalised recommendations