Inelastic Response Analysis of MDOFs to Earthquake Motions Based on Superposition of Elastoplastic Response of Each Mode

This paper develops the time history analysis of the elastoplastic response of a Multi-Degree-Of-Freedom (MDOF) system, consisting of the arbitrary number of masses and springs that exhibit bilinear hysteresis, being subjected to a base excitation, by superposing the time history of the elastoplastic response of each mode. Its essence is that the mode frequencies and shapes are readily specified by the eigenvalue problem with the mass and the corresponding spring stiffness that the MDOF system shows. The uncoupled equation of motion of each mode regardless of the combination of the state of the spring of the MDOF system is described in detail. The time history of each mode is numerically and simultaneously analyzed and superposed them each time to check the yielding and the turnaround of the spring of interest, which can only be verifiable in the global coordinate system. If the yielding or turn-around of the spring is found, the uncoupled equation of motion of each mode is updated to that derived from the corresponding combination of the state of the spring of the MDOF system. Employing a 4DOF system, the accuracy of the proposed method is thoroughly discussed. Moreover, the force-displacement diagram of the spring of interest induced by each mode is presented. The contribution of the mode response to the force-displacement diagram is examined.