STATE-DEPENDENT SEISMIC DAMAGE INCREMENT. To model degradation in structures with possible energy dissipation during seismic shaking (e.g., hysteretic behavior) an index measuring accumulating damage, and its effect on structural performance, is needed. This has been, and currently is, a relevant topic in the earthquake engineering research. According to the review in [13], damage indices may be grossly categorized in two main classes labeled as displacement-related and energy-related. In the former case, the principle is that the structure reaches the limit state of interest because it exceeds a maximum displacement threshold, that is, maximum strain. The latter case refers to structures in which damage is related to the amount of energy dissipated by hysteretic loops. Indeed, the most representative strain-based damage index is the maximum displacement demand imposed by the seismic shock, while hysteretic energy (i.e., the summation of the areas of plastic cycles during seismic shaking) is the most direct energy-based index. Hybrid damage indices, accounting for both damage phenomena in a single metric, also exist; the best known is that by Park and Ang [14]. The main issue in modeling the stochastic evolution of accumulating seismic damage is that the probability of observing a certain increment of deterioration (i.e., the vulnerability) in a generic earthquake shock may be dependent on the history prior to its occurrence. If the damage increment in one earthquake depends on the seismic history only via the state of the structure at the time the seismic event occurs, then the stochastic process describing/predicting the evolution of structural conditions in time may be categorized as a Markovian one. In structures, there are two concurrent seismic vulnerability issues which call for Markov-type reliability models: (1) the hysteretic behavior of the structure does not remain the same in subsequent earthquakes; and (2) the way damage is measured introduces a dependency on history, even if the structural behavior remains the same. In the following subsections these two issues are further discussed for two different structural systems, subsequently, it will be shown how state-dependent vulnerability, jointly with the classical probabilistic seismic hazard analysis, leads to a global Markovian failure risk process.
