One and two-dimensional maximum softening damage indicators for reinforced concrete structures under seismic excitation

Abstract

The maximum softening concept is based on the variation of the vibrational periods of a structure during a seismic event. Maximum softening damage indicators, which measure the maximum relative stiffness reduction caused by stiffness and strength deterioration of the actual structure, are calculated for an equivalent linear structure with slowly varying stiffness characteristics. In the paper, a one-dimensional scalar-valued and a two-dimensional vector-valued maximum softening damage indicator are defined. The one-dimensional damage indicator, defined considering the variation in the first period of the structure, analogous to a SDOF system, is a genuine global damage index representing the average damage throughout the whole structure. The two-dimensional damage indicator is defined considering the variations in the first and second periods, and thus is analogous to an equivalent linear two-degrees-of-freedom system. The components of the damage vector can be interpreted as damage indicators for the lower half and the upper half of the structure, hence representing a simple local description of the damage state of the structure. Since statements on the post earthquake reliability should be obtained solely from knowledge of the latest recorded values of the damage indicators above defined, these are required to possess a Markov property. This problem has been investigated based on numerical Monte-Carlo simulations, and it is observed that the Markov assumption of the one-dimensional damage indicator is justified for the mean value of the transition probability density function (TPDF), whereas some deviations are observed for the variance and higher order statistical moments. For the two-dimensional damage indicator the Markov assumption seems justified for both the mean values and the covariances of the TPDF. From these, it is concluded that the Markov properties of the mean and covariances of the two-dimensional indicators are superior to the Markov property of the first and second order moments of the one-dimensional damage indicator.