Accuracy and Limitations of the Method of Equivalent Linearization for Hysteretic Multi-Storey Structures
A method of solution for determing the response statistics of a nonlinear hysteretic shear building subjected to nonstationary stochastic excitation is suggested by utilizing a stochastic equivalent linearization technique. This paper concentrates on the accuracy of the predicted response for an excitation ranging from low to high intensity and on the computational efficiency of the time step procedure. The nonlinear hysteretic properties of the shear building are modelled by Bouc’s  model in terms of auxiliary variables following nonlinear differential equations. The auxiliary variables are linearized leading to a set of linear differential equations. They are solved numerically using the state vector formulation and a complex modal analysis time-step procedure in order to consider the time varying linearization coefficients. The procedure is applied to a six storey shear building with no residual linear stiffness to demonstrate the applicability to this special case. For the numerical efficiency of the time step procedure, a generalized Jacobi-iteration is suggested to solve in each time step efficiently the characteristic value problem. The predicted variances are then compared with the variances obtained by applying simulations procedures. The agreement is excellent for the velocity response, but less satisfactory for the displacement response. Finally, probability densities of response quantities are shown (based on 3000 simulated samples).
Unable to display preview. Download preview PDF.
- BOUC, R.: “Forced Vibration of Mechanical Systems with Hysteresis”, Proceedings of the 4th Conference on Nonlinear Oscillation, Prague, Czechoslovakia, 1967Google Scholar
- LIN, Y.K.: “Probabilistic Theory of Structural Dynamics”, R.E. Krieger Publ. Comp., N.Y., (1976)Google Scholar
- BRUCKNER, A., LIN, Y.K.: “Generalization of the Equivalent Linearization Method for Nonlinear Random Vibration Problems”, Rep. CAS 86–2, Florida Atlantic University, 1986Google Scholar
- WEN, Y.K.: “Method for Random Vibration of Hysteretic Systems”, Journal of the Engineering Mechanics Division, ASCE, April 1976, 102, 249–263Google Scholar
- BABER, T.T. and WEN, Y.K.: “Stochastic Equivalent Linearization for Hysteretic, Degrading, Multistory Structures”, Civil Engineering, University of Illinois, Urbana, Illinois, April 1980Google Scholar
- CHANG, T-P., MOCHIO, T., SAMARAS, E.: “Seismic Response Analysis of Nonlinear Structures”, Probabilistic Engineering Mechanics, 1986, Vol. I., No. 3, 157–166Google Scholar
- PRADLWARTER, H.J., CHEN, X.W.: “Evaluation of the Covariance Matrix of the Response of MDOF-Systems Loaded by an Evolutionary Spectrum”, Report -87, Institute of Engineering Mechanics, University of Innsbruck, Austria, (to appear 1987 )Google Scholar
- BATHE, K.-J.: “Finite Element Procedures in Engineering Analysis”, Prentice Hall, New Jersey, (1982)Google Scholar
- HAMPL, N.C.: “Zur Wahrscheinlichkeitsdichte der Reaktion nichtlinearer Systeme auf Zufallslasten”, (in German) Report 10–87, Institute of Engineering Mechanics, University of Innsbruck, Austria, 1987Google Scholar