Inelastic Spectra Normalized By Two Characteristic Periods

Dynamic analysis of single-degree-of-freedom (SDOF) systems under impact and harmonic loads considering P-?effect are studied. Energy and displacement response of SDOF system under elastic, shakedown and plastic loads are analyzed. The hysteretic models are elastic-perfectly plastic (EPP), elastic-linearly-hardening (ELH), shear-slipped (SSP) and bilinear-elastic (BIL) model. The influence of damping ratio, energy dissipation capacity and P-?effect on structural responses are studied. It is found that the P-?effect reduces the elastic limit and shakedown loads, and enlarges input energy and maximum displacement. Post yield stiffness ratio can make up for the adverse impact of the P-?effect. Damping is very important for elastic system in dissipating energy and almost half of total energy is due to the damping. While for elasto-plastic systems, hysteretic energy plays the most important role, especially when considering P-?effect.Due to the complexity of nonlinear dynamic analysis, the seismic force modification factor R is commonly used to reduce elastic seismic force to get the strength demand. Based on elastic-plastic time-history earthquake analysis of EPP hysteretic SDOF system and MDOF system, total 370 earthquake records belonging to 4 site are used to calculate the structural response. Results indicate that peak feature of R spectra around characteristic periods are well retained with lateral axis being normalized by Tga and TgR. Seismic force modification factors becomes smaller when considering P-?effect. Using R of SDOF system for MDOF structure is dangerous. RM factor is introduced to consider MDOF effect (R?,MDOF=R?,SDOF×RM).RM decreases with increase of the number of stories. RM is presented for different periods.By using inelastic response spectra, seismic force of elasto-plastic system can be obtained more easily. The direct method has more accuracy than the indirect method. Inelastic response spectra between systems with and without considering P-?effect are different. The effects of damping ratio on elastic and inelastic spectra are independent on sites. Elastic spectra decreases with the increase of damping ratio. Damping effects on elastic spectra and inelastic spectra have little relation to P-?effect and ductility, respectively. Elastic spectra are more sensitive to damping ratio than inelastic spectra. The influence of damping ratio on inelastic spectra is larger for system considering P-?effect.Inelastic displacement ratio Cd is studied to estimate the maximum inelastic displacement according to the maximum elastic displacement. It is found that the equal displacement rule is applicable for long period region. Maximum inelastic displacement of structure considering the P-?effect is larger than those without P-?effect. The influence of P-?effect is significant for structure with period being equal to characteristic periods, especially when period being equal to TgR.The relation between sectional ductility and member ductility is studied. Strength reduction factors based on curvature ductility and displacement ductility are analyzed. Curved segment in the moment-curvature relationship between elastic stage and hardening stage is proved to be favorable. The responses can be conservatively predicted using the simple bilinear-elastic model. The results based on displacement ductility are more stable. Larger element number of structures is required to guarantee calculation precision when the analysis is based on curvature ductility than on displacement ductility.