THE CLOSED-FORM SOLUTION OF STEADY STATE RESPONSE OF HYSTERETIC SYSTEM UNDER STOCHASTIC EXCITATION

The hysteretic system is one of the typical strongly nonlinear systems. Hysteretic force depends not only on the instantaneous deformation but also on the past history of deformation. In the last few decades, random vibration of hysteretic system has been studied extensively, but no closed-form solution of random response of hysteretic systems is available so far. In this paper, the newly developed nonlinear random vibration scheme called iterative method of weighted residuals is explored to obtain the closed-form solution of steady-state probability density function (PDF) of the Bouc-Wen hysteretic system under Gaussian white noise excitation. First, a Gaussian PDF is obtained with equivalent linearization technique, which is used as a weighting function. Then, the method of weighted residuals is utilized to determine the non-Gaussian PDF of exponential polynomial type. Finally, an iterative procedure is introduced to improve the accuracy of the solutions obtained from the method of weighted residuals. As an illustrative example, the steadystate stochastic response of the steel fiber reinforced ceramsite concrete column under random excitation is studied, in which the hysteretic parameters associated with Bouc-Wen hysteretic model is identified from the pseudo-static test by using the method of least square. Compared to the Monte Carlo results, the accuracy of results obtained from equivalent linearization method is poor. The results obtained from weight residue method can show the nonlinearity of Bouc-Wen systems, but its accuracy is still unsatisfactory. The iterative method of weight residuals can lead to results with higher accuracy. In the case of stronger random excitation, the progressive iterative method of weighted residuals has high efficiency. The obtained solutions agree well with the Monte Carlo simulation data. The proposed closed-form solution of PDF of Bouc-Wen hysteretic system not only is significant to the civil engineering, but also can be a benchmark to examine the accuracy of solutions obtained by other methods.