The Modified Energy-based Method for Seismic Evaluation of Structural Systems with Different Hardening Ratios and Deterioration Hysteresis Models

Prediction of target displacement in structural systems plays a significant role in performance-based design and rehabilitation of structures. In this study, the γ factor for different hardening ratios, including 1, 2, 3, 5, 7.5, 10, and 15 percentages, stiffness-strength-deterioration models, and soil type classes is determined to modify the energy balance equation in performance-based plastic design (PBPD). Statistical results indicate that the effect of the hardening ratio, deterioration, and soil type class on the capacity curve is considerable. Therefore, a simple equation based on the period of the vibration and ductility is suggested to estimate the γ factor in different structural systems. Moreover, four 1-, 3-, 7-, and 12-story moment steel structures with various hardening ratios in the material are designed to validate the proposed method. The suggested values for the γ factor show exact results compared to collected displacements from time history analysis, while the error in the previous work was considerable. Statistical results showed that the mean error in the previous method in estimating target displacement for 1-, 3-. 7-, and 12-story structures is about 15%, 20%, 20%, and 32%, respectively. Conversely, the mean error in this study for estimating target displacement of 1-, 3-. 7-, and 12-story structures is about 10%, 7%, 6%, and 15%, respectively. Finally, the proposed method is examined on the empirical reinforced concrete (RC) bridge pier simulated numerically with fiber-based modeling. Similarly, the suggested equation estimates the target displacement appropriately for the concrete model compared to achieved displacements from nonlinear dynamic analysis.