An analytical procedure for the derivation of state-dependent fragility curves for masonry buildings

The seismic capacity of a structure is a function of the characteristics of the system and of its state, which is mainly affected by previous damage and deterioration. The effect of repeated shocks (for example during a seismic sequence or due to multiple events affecting an unrepaired building stock) on the seismic vulnerability of masonry buildings is considered in this work by developing state-dependent fragility curves. These curves only take into account cumulated seismic damage, neglecting possible ageing effects. An analytical procedure for the derivation of fragility curves is proposed, which is based on nonlinear dynamic analyses with a large set of unscaled natural records. To speed up computational time, analyses are carried out using an equivalent single-degree-of-freedom system, properly calibrated to reproduce the cyclic dynamic response of the selected building prototype. The seismic safety of a system can be measured by its capacity to withstand a certain intensity of the seismic action. Hence, the seismic capacity of the structure can be expressed in terms of intensity measures, the latter being expressed either by scalar or vector parameters. Since seismic safety has to be evaluated for different limit states, the seismic capacity can be expressed by a set of values of the intensity measures corresponding to the levels of seismic action inducing the attainment of the considered limit states, i.e. the attainment of given thresholds for selected engineering demand parameters. The seismic capacity is a function of the characteristics of the system and of its state, which is mainly affected by damage and deterioration. Damage state is considered as the cumulated effect of previous seismic events. It mainly depends on the seismic history of the site since the system was established (construction time in case of structures) and on the seismic vulnerability. For sake of simplicity the effect of possible damages other than those caused by seismic event are not considered in this work as well as ageing effects are not considered. If repair, retrofitting or maintenance interventions are not considered, the seismic safety is assumed to monotonically decrease with time. The damage accumulation due to seismic events is by nature a discontinuous process. In general, the effect of the event intensity on the damage evolution and hence on the reduction of the residual capacity is highly nonlinear and only events whose intensity at the site exceed a certain threshold level affect damage accumulation and hence the state of the structures. Such an intensity threshold can be also regarded as a function of the previous damage state. Based on the aforementioned considerations, the problem of defining time-dependent vulnerability of systems can be split into two separate sub-problems: 1. Defining the system state as a function of time, e.g. considering the combined effect of ageing and cumulated seismic damage; 2. Defining state-dependent fragility curves representing the seismic vulnerability of the system given the system state (at a given time).