Statistical Linearization Techniques of Hysteretic Structures to Earthquake Excitations

Abstract

To evaluate the reliability of structures subjected to severe earthquake excitations in relation to the ultimate aseismic design method which guarantees the safety of structures in the ultimate state, the basic statistical characteristics of the response of elasto-plastic structures should be understood. At the present time there is no useful analytical method which can be used to treat problems of the response of stochastically excited hysteretic structures with strong nonlinearity. The Fokker-Planck equation approach is inapplicable to problems which involve the hysteretic nonlinearity or the non-white excitation. As an approximate analytical method, the statistical linearization technique has been widely used, since this technique overcomes the above restrictions. However, this technique requires small nonlinearity, and so may be not applicable to find statistical characteristics of the response of hysteretic structures with strong nonlinearity which is considered important in relation to the ultimate aseismic safety. The purpose of this paper is to introduce a new statistical linearization technique that will allow as to obtain solutions to the problem of the response of hysteretic structures with strong nonlinearity to random excitation. From experimental results of the response of hysteretic structures with strong nonlinearity by means of the simulation technique, it is pointed out that the response possesses broad-band characteristics and the center of hysteresis eccentrically fluctuates. With these points as background, a new statistical linearization technique is introduced by considering the scatter of frequency and the fluctuation of the center of hysteretic oscillation. The equivalent linearization coefficients are determined by the least mean-square error method. In the present study, the average used in minimization procedure is a combination of ensemble and time averages. Though the technique developed by T.K. Caughey involves only amplitude of hysteretic oscillation as random variable, the shift of center, amplitude and frequency of hysteretic oscillation are treated as random variable. The joint probability density function of these random variables is approximately evaluated under the assumption that the response is a stationary Gaussian process. The numerical analysis is carried out in the case of a single-degree-of-freedom structure with bilinear hysteresis to a band-limited white Gaussian excitation. From the statistics of the stationary response computed by the statistical linearization techniques and the simulation technique, it is shown that the scatter of frequency and the fluctuation of the center of hysteretic oscillation affect strongly on the displacement response of bilinear structures with severe nonlinearity, and so the statistical linearization technique is considerably improved by taking them into account. Another fact of importance is the existence of the optimum rigidity ratio which minimizes the variance of the displacement response. It is suggested that the above optimum ratio has a great significance in doing the aseismic design of the elasto-plastic structures.