Efficient Mode Superposition Method for Seismic Analysis of Nonlinear Systems

Abstract

A step-by-step integration method is proposed to compute within the framework of the conventional mode superposition technique the response of bilinear hysteretic structures subjected to earthquake ground motions. The method is computationally efficient because only a few modes need to be considered to obtain an accurate estimate of such a response, and because it does not require the use of excessively small time steps to avoid problems of accuracy or stability. It is developed on the basis that the nonlinear terms in the equations of motion for nonlinear systems may be considered as additional external forces, and on the fact that by doing so such equations of motion can be interpreted as the equations of motion of an equivalent linear system, excited by a modified ground motion. These linear equations are then subjected to a conventional modal decomposition and transformed, as with linear systems, into a set of independent differential equations, each representing the system’s response in one of its modes of vibration. To increase the efficiency of the method and properly account for the participation of higher modes, these independent equations are solved using Nigam-Jennings technique in conjunction with the so-called mode-acceleration method. In addition, an iterative scheme is introduced to avoid an inefficient recalculation of the system’s eigenvectors and eigenvalues every time there is a change in the stiffness of one of its elements. The accuracy and efficiency of the method is verified by means of a comparative study with solutions obtained with a conventional direct integration method. In this comparative study, with only a few modes considered, the proposed method accurately predicts the seismic response of three two-dimensional frame structures, but requiring only, on the average, about 43 per cent of the computer time spent when using the direct integration method.