Maximum response occurrence time and spectra

Typically, in seismic response analysis, the focus is on absolute maximum response, without specific interest in its attainment time. The definition of response spectrum, for example, does not provide information about such a time and the issue is somewhat scarcely investigated in the literature, while it is the subject of the presented very simple study, where linear and non-linear single-degree-of-freedom systems are investigated when excited by a large set of ground motion records. For the linear systems, the occurrence time of maximum displacements is measured with respect to: (i) the initial instant defining the significant duration of the record; (ii) the initial instant defining the bracketed duration of the record; (iii) the time of occurrence of peak ground acceleration (PGA). For the non-linear systems, the maximum displacements are measured with respect to the time of occurrence of peak ground acceleration, for two different hysteretic behaviors, stable and evolutionary, considering several vibration periods and ductility levels. The occurrence time of inelastic response is compared between the two non-linear systems and to that of the corresponding elastic system. A characteristic behavior is found for the time of occurrence of the maximum displacement in linear systems that, as a trend, systematically occurs after the PGA, independently of the vibration period. The occurrence time tends to get closer to the PGA time as the damping increases. Finally, the comparison of the time of maximum response for the two non-linear systems and those linear, shows some relationships between the two.