To approximate nonlinear response of structures subjected to an earthquake excitation, the displacement method in U.S. or the energy method in Japan has been practically used. However, unless these methods include the nature of dynamics of nonlinear structure well, the nonlinear response is not adequately calculated. Applying the equivalent linearization technique to a Single Degree Of Freedom (SDOF) system with bilinear hysteresis subjected to white noise base acceleration, this paper mathematically quantifies deterioration in a spring constant and increase in a damping coefficient with the progress of nonlinearity in the restoring force system as a function of the ensemble ductility ratio. As the nonlinearity progresses, the spring constant rapidly deteriorates and the damping substantially increases. Increments of damping of the lightly damped linear SDOF system are more than that of the moderately damped one. A comparison of the response of the equivalently linearized system to that of the corresponding linear system reveals the capability for growth in the nonlinear response. The nonlinear response predicable by either method is identified by values of the ensemble ductility ratio and damping ratio of the linear SDOF system. In addition, in a range of the ensemble ductility ratio where the bulk of the engineered structures are included, neither method can properly evaluate the nonlinear response. Although the results presented herein give the mean nature of the nonlinear response and phase and amplitude characteristics of accelerograms make the nonlinear response vary around the mean, uniform application of either displacement method or energy method to approximating the nonlinear response may be reconsidered.

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