Simple equivalent linearization of yielding MDF structures subjected to random excitation

A simple equivalent linear multi-degree-of-freedom (MDF) model is proposed to approximate the stationary rms response levels of yielding MDF structures subjected to Guassian random base excitation. The linear model is characterized by an equivalent stiffness for each yielding element, and a damping matrix which is proportional to that in the unyielded system. The yielding systems under study herein have nearly elasto-plastic bilinear hysteretic elements. No exact analytical solutions are known for such systems. A semi-empirical approach is used to determine the equivalent parameters of the simple model used here. The equivalent stiffness values used here are not based on response velocity values, but rather on the distribution of rms distortion over the height of the structure. The new equivalent damping is taken as r times the original damping, and the factor r is evaluated using the assumption that the damping matrix affects the magnitudes of rms story distortions,- but not their distribution over the structure. With the equivalent stiffness and damping, the Lyapunov covariance matrix equation of the simple linear model is then solved to compute the stationary rms story distortions. Overall, the analytical results thus obtained by this model appear to agree quite well with simulation values. An apparently satisfactory analytical relationship is found between story distortion and equivalent stiffness. Further study will be required to find a corresponding relationship for the damping factor r. It is shown, however, that the equivalent damping is quite small for a variety of MDF systems