Summary
The stochastic response of hysteretic MDOF systems is most conveniently determined by statistical equivalent linearization (EQL). Traditionally, EQL assumes the nonlinear hysteretic stochastic response to be normally distributed, although it is well known that this assumption is not justified and may lead to considerable errors in the prediction of the second moment properties. In the present paper, the consideration of non-Gaussian response properties is shown. The suggested method utilizes nonlinear transformations between the nonlinear and linearized state vector and Nataf’s representation of the n-dimensional joint distribution by its marginal distributions. A method to evaluate the required nonlinear transformation is proposed. The applicability of the introduced concept is shown in a numerical example.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Lin, Y.K.: “Probabilistic Theory of Structural Dynamics”, McGraw-Hill, New York, 1967; Reprint Robert E. Krieger, Huntington, 1976.
Cai, G.Q., Lin, Y.K.: “On exact stationary solutions of equivalent non-linear stochastic systems”, Int. J. Non-Linear Mech., Vol. 23, No.4, 1988, pp.315–325.
Wu, W.F., Lin, Y.K.: “Cumulant-neglect Closure for Nonlinear Oscillators under Random Parametric and External Excitation”, International Journal of Non-Linear Mechanics, 19 (1984) pp.349–362.
Benaroya, H., Rehak, M: “Finite Element Methods in Probabilistic Structural Analysis: A selective review”, Applied Mechanics Review, 41(5) (1988) pp.201–213.
Schuëller, G.I., Pradlwarter, H.J., Bucher C.G.: “Efficient Computational Procedures For Reliability Estimates of MDOF Systems”, to appear: Int. J. Non-Linear Mechanics, 1991
Pradlwarter, H.J., Schuëller, G.I.: “Accuracy and Limitations of the Method of Equivalent Linearization for Hysteretic Multi-Storey Structures”, F. Ziegler, G.I. Schuëller (Eds.). Nonlinear Stochastic Dynamic Engineering Systems, Proc. IUTAM Symp., Innsbruck, Springer Verlag Berlin Heidelberg, 1988, pp.3–21.
Wen, Y.K.: “Method for Random Vibration of Hysteretic Systems”, Journal of the Engineering Mechanics Division, ASCE, April 1976, Vol.102, pp.249–263.
Spanos, P.-T.D.: “Stochastic Linearization in Structural Dynamics”, Applied Mechanics Review, Vol.34 (1981)pp.1–8.
Booton, R.C.: “The Analysis of Nonlinear Control System with Random Inputs”, Proc. MRI Symposium on Nonlinear Circuits, Polytechnic Inst. of Brooklyn, 1953, pp. 341–344.
Kazakov, I.E.: “An Approximate Technique for the Statistical Investigations of Nonlinear Systems”, Tr. VVIA im Prof. n.E. Zhukovskogo, No.394, 1952.
Caughey T.K.: “Equivalent linearization techniques”, Journal of the Acoustical Society of America, Vol.35, 1965, pp.1706–1711.
Kozin, F.: “The Method of Statistical Linearization for Nonlinear Stochastic Vibrations”, F. Ziegler, G.I. Schuëller (Eds): Nonlinear Stochastic Dynamic Engineering Systems, Proc, IUTAM Symposium Innsbruck/Igls, Austria, 1987, Springer Verlag Berlin Heidelberg, 1988, pp.44–56.
Pradlwarter, H.J.: “On the Existence of “True” Equivalent Linear Systems for the Evaluation of the Nonlinear Stochastic Response due to Nonstationary Gaussian Excitation”, Report 28–90, Institute of Engineering Mechanics, University of Innsbruck (1990).
Atalik, T.S., Utku, S.: “Stochastic Linearization of Multidegree of Freedom Nonlinear Systems”, Journal of Earthquake Engineering and Structual Dynamics, 1976, Vol.4, pp.411–420.
Chen, X.-W.: “The Response of Hysteretic MDOF-Systems Driven by an Evolutionary Process Evaluated by the Equivalent Linearization Technique”, Dissertation, Institute of Engineering Mechanics, University of Innsbruck (1988).
Hampl, N.C., Schuëller, G.I.: “Probability Densities of the Response of Non-Linear Structures under Stochastic Dynamic Excitation”, Probabilistic Engineering Mechanics, Vol.4, No.1, March 1989, pp.2–9.
Nataf, A.: “Determination des Distribution dont les Marges sont Donnees”, Comptes Rendus de l’Academie des Sciences, Paris, 1962, 225, pp.42–43.
Liu, P-L., Der Kiureghian, A.:“Multivariate Distribution Models with Prescribed Marginals and Covariances”, Probabilistic Engineering Mechanics, 1986, Vol.1, No.2, pp. 105–112.
Pradlwarter, H.J., Schuëller, G.I. 1989. Consideration of Non-Gaussian Hysteretic Properties of MDOF-Systems by Use of Equivalent Linearization. Proc. of ICOSSAR’89, the 5th Intern. Conf.on Structural Safety and Reliability, San Francisco, Aug. 7–11.1989. A.H-S. Ang, M. Shinozuka, G.I. Schuëller (eds.), Vol.II:1333–1340.New York:ASCE.
Pradlwarter, H.J.: “Non-Gaussian Linearization — an Efficient Tool to Analyse Nonlinear MDOF-Systems”, to appear: Nuclear Engineering and Design, 128, pp. 175–192, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pradlwarter, H.J., Schuëller, G.I. (1992). A Practical Approach to Predict the Stochastic Response on Many-DOF-Systems Modeled by Finite Elements. In: Bellomo, N., Casciati, F. (eds) Nonlinear Stochastic Mechanics. IUTAM Symposia. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-84789-9_37
Download citation
DOI: https://doi.org/10.1007/978-3-642-84789-9_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-84791-2
Online ISBN: 978-3-642-84789-9
eBook Packages: Springer Book Archive