Displacement-based damage measures. It has been clarified why the type of hysteretic behavior can induce state-dependent seismic response and then state-dependent vulnerability. However, even if the hysteretic loop of the structure may be considered non-evolutionary, that is, its shape does not change through shocks, as in the EPP-SDOF, the damage-criterion may still let the damage increment to be history-dependent. Indeed, considering a maximum- displacement-based (i.e., maximum strain) criterion, damage accumulation in an earthquake occurs only once the maximum displacement reached in it is larger than the maximum in those previous [13], which makes the probability of observing a certain damage increment dependent on the distance between the residual displacement from which shaking starts (depending on seismic history) and the maximum transient until that shock. This may be argued by Figure 3, where the effect of a generic seismic shock is depicted referring to the envelope response of an EPP-SDOF (the same reasoning applies to cyclic response). In the figure it is illustrated that, while the area of hysteretic loops do not depend on the previous behavior of the structure, the maximum absolute displacement (and then the damage increment) does. Indeed, the amount of damage in a shock, which is the strain increment in this case, depends on the maximum recorded strain in previous shocks and also on the residual displacement. This is because the amount of plastic displacement required to increase damage is the difference between the maximum transient ever recorded and the residual displacement at the time of the shock. Therefore in this case the vulnerability is state-dependent also if the response is non evolutionary. In a series of previous studies, stochastic modeling of damage accumulation in structures was addressed: [4], [11-12]. In these works, an energy-based criterion was considered with reference to EPP-SDOF systems, therefore a series of reliability models based on independent increments was developed. Herein, conversely, a Markovian model is developed to accommodate state-dependent vulnerability because either of the (1) hysteretic behavior and/or (2) the considered damage measure. In particular, to isolate the effect of (2), the developed model is calibrated for an EPP-SDOF system when damage is based on the maximum displacement demand (or an equivalent of it; e.g., the drift ratio), while, to consider also the effect of (1), a PIN-SDOF systems is investigated, with the same dam...
