Methodology for obtaining equivalent linearly-elastic to inelastic shear-type models and vice-versa

This paper is concerned with the development of a reliable methodology that yields an equivalent linearly-elastic model for a given inelastic shear-type model, and returns it to the inelastic state after applying a retrofit procedure of linear structures of some sort. The direct-displacement-based-design (DDBD) approach is adopted in calculating the effective stiffness of the equivalent model and the consequent classical inherent damping, while the structural mass matrix is unchanged. Two developed procedures are described. The first procedure calculates the effective story stiffness coefficients in the equivalent linear-elastic shear-type model to obtain inter-story drifts equal to those of the inelastic model. For a given ensemble of ground motion records, the effective story stiffness coefficients are obtained for each ground motion record separately and the average is taken between records as the final effective story stiffness coefficient. The second procedure calculates the pre-yield story stiffness coefficients of the inelastic shear-type model by reversing the equivalence approach of the first procedure. A numerical example of a 10-story inelastic shear-type model is studied. The equivalent linear-elastic model is assigned with stiffness and mass changes and then transformed back into “retrofitted” inelastic shear-type models. Results show that the peak dynamic responses of inter-story drifts and the absolute accelerations are approximately the same.